Package 'AICcmodavg'

Title: Model Selection and Multimodel Inference Based on (Q)AIC(c)
Description: Functions to implement model selection and multimodel inference based on Akaike's information criterion (AIC) and the second-order AIC (AICc), as well as their quasi-likelihood counterparts (QAIC, QAICc) from various model object classes. The package implements classic model averaging for a given parameter of interest or predicted values, as well as a shrinkage version of model averaging parameter estimates or effect sizes. The package includes diagnostics and goodness-of-fit statistics for certain model types including those of 'unmarkedFit' classes estimating demographic parameters after accounting for imperfect detection probabilities. Some functions also allow the creation of model selection tables for Bayesian models of the 'bugs', 'rjags', and 'jagsUI' classes. Functions also implement model selection using BIC. Objects following model selection and multimodel inference can be formatted to LaTeX using 'xtable' methods included in the package.
Authors: Marc J. Mazerolle <[email protected]>
Maintainer: Marc J. Mazerolle <[email protected]>
License: GPL (>= 2)
Version: 2.3-3
Built: 2024-10-01 07:01:07 UTC
Source: CRAN

Help Index

Model Selection and Multimodel Inference Based on (Q)AIC(c)

Description

Description: This package includes functions to create model selection tables based on Akaike's information criterion (AIC) and the second-order AIC (AICc), as well as their quasi-likelihood counterparts (QAIC, QAICc). The package also features functions to conduct classic model averaging (multimodel inference) for a given parameter of interest or predicted values, as well as a shrinkage version of model averaging parameter estimates. Other handy functions enable the computation of relative variable importance, evidence ratios, and confidence sets for the best model. The present version supports Cox proportional hazards models and conditional logistic regression (coxph and coxme classes), linear models (lm class), generalized linear models (glm, glm.nb, vglm, hurdle, and zeroinfl classes), linear models fit by generalized least squares (gls class), linear mixed models (lme class), generalized linear mixed models (mer, merMod, and glmmTMB classes), multinomial and ordinal logistic regressions (multinom, polr, clm, and clmm classes), robust regression models (rlm class), beta regression models (betareg class), parametric survival models (survreg class), nonlinear models (nls and gnls classes), nonlinear mixed models (nlme and nlmerMod classes), univariate models (fitdist and fitdistr classes), and certain types of latent variable models (lavaan class). The package also supports various models of unmarkedFit and maxLikeFit classes estimating demographic parameters after accounting for imperfect detection probabilities. Some functions also allow the creation of model selection tables for Bayesian models of the bugs and rjags classes. Objects following model selection and multimodel inference can be formatted to LaTeX using xtable methods included in the package.

Details

Package: AICcmodavg
Type: Package
Version: 2.3-3
Date: 2023-11-16
License: GPL (>=2 )
LazyLoad: yes

Many functions of the package require a list of models as the input to conduct model selection and multimodel inference. Thus, you should start by organizing the output of the models in a list (See 'Examples' below).

This package contains several useful functions for model selection and multimodel inference for several model classes:

AICc

Computes AIC, AICc, and their quasi-likelihood counterparts (QAIC, QAICc).

aictab

Constructs model selection tables with number of parameters, AIC, delta AIC, Akaike weights or variants based on AICc, QAIC, and QAICc for a set of candidate models.

bictab

Constructs model selection tables with number of parameters, BIC, delta BIC, BIC weights for a set of candidate models.

boot.wt

Computes summary statistics from detection histories.

confset

Determines the confidence set for the best model based on one of three criteria.

DIC

Extracts DIC.

dictab

Constructs model selection tables with number of parameters, DIC, delta DIC, DIC weights for a set of candidate models.

evidence

Computes the evidence ratio between the highest-ranked model based on the information criteria selected and a lower-ranked model.

importance

Computes importance values (w+) for the support of a given parameter among set of candidate models.

modavg

Computes model-averaged estimate, unconditional standard error, and unconditional confidence interval of a parameter of interest among a set of candidate models.

modavgEffect

Computes model-averaged effect sizes between groups based on the entire candidate model set.

modavgShrink

Computes shrinkage version of model-averaged estimate, unconditional standard error, and unconditional confidence interval of a parameter of interest among entire set of candidate models.

modavgPred

Computes model-average predictions, unconditional SE's, and confidence intervals among entire set of candidate models.

multComp

Performs multiple comparisons across levels of a factor in a model selection framework.

useBIC

Computes BIC or a quasi-likelihood counterparts (QBIC).

For models not yet supported by the functions above, the following can be useful for model selection and multimodel inference conducted from input values supplied by the user:

AICcCustom

Computes AIC, AICc, QAIC, and QAICc from user-supplied input values of log-likelihood and number of parameters.

aictabCustom

Creates model selection tables based on (Q)AIC(c) from user-supplied input values of log-likelihood and number of parameters.

bictabCustom

Creates model selection tables based on (Q)BIC from user-supplied input values of log-likelihood and number of parameters.

ictab

Creates model selection tables from user-supplied values of an information criterion.

modavgCustom

Computes model-averaged parameter estimate based on (Q)AIC(c) from user-supplied input values of log-likelihood, number of parameters, parameter estimates, and standard errors.

modavgIC

Computes model-averaged parameter estimate from user-supplied values of information criterion, parameter estimates, and standard errors.

useBICCustom

Computes BIC and QBIC from user-supplied input values of log-likelihoods and number of parameters.

A number of functions for model diagnostics are available:

c_hat

Estimates variance inflation factor for binomial or Poisson GLM's based on various estimators.

checkConv

Checks the convergence information of the algorithm for the model.

checkParms

Checks the occurrence of parameter estimates with high standard errors in a model.

countDist

Computes summary statistics from distance sampling data.

countHist

Computes summary statistics from count history data.

covDiag

Computes covariance diagnostics for lambda in N-mixture models.

detHist

Computes summary statistics from detection histories.

detTime

Computes summary statistics from time-to-detection data.

extractCN

Extracts condition number from models of certain classes.

mb.gof.test

Computes the MacKenzie and Bailey goodness-of-fit test for single season and dynamic occupancy models using the Pearson chi-square statistic.

Nmix.gof.test

Computes goodness-of-fit test for N-mixture models based on the Pearson chi-square statistic.

Other utility functions include:

anovaOD

Computes likelihood-ratio test statistic corrected for overdispersion between two models.

extractLL

Extracts log-likelihood from models of certain classes.

extractSE

Extracts standard errors from models of certain classes and adds the labels.

extractX

Extracts the predictors and associated information on variables from a list of candidate models.

fam.link.mer

Extracts the distribution family and link function from a generalized linear mixed model of classes mer and merMod.

predictSE

Computes predictions and associated standard errors models of certain classes.

summaryOD

Displays summary of model output adjusted for overdispersion.

xtable

Formats various objects resulting from model selection and multimodel inference to LaTeX or HTML tables.

Author(s)

Marc J. Mazerolle <[email protected]>.

References

Anderson, D. R. (2008) Model-based inference in the life sciences: a primer on evidence. Springer: New York.

Burnham, K. P., and Anderson, D. R. (2002) Model selection and multimodel inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Examples

##Example 1:  Poisson GLM with offset
##anuran larvae example from Mazerolle (2006) 
data(min.trap)
##assign "UPLAND" as the reference level as in Mazerolle (2006)          
min.trap$Type <- relevel(min.trap$Type, ref = "UPLAND") 

##set up candidate models in a list
Cand.mod <- list()
##global model          
Cand.mod[[1]] <- glm(Num_anura ~ Type + log.Perimeter + Num_ranatra,
                     family = poisson, offset = log(Effort),
                     data = min.trap) 
Cand.mod[[2]] <- glm(Num_anura ~ Type + log.Perimeter, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[3]] <- glm(Num_anura ~ Type + Num_ranatra, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[4]] <- glm(Num_anura ~ Type, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[5]] <- glm(Num_anura ~ log.Perimeter + Num_ranatra,
                     family = poisson, offset = log(Effort),
                     data = min.trap) 
Cand.mod[[6]] <- glm(Num_anura ~ log.Perimeter, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[7]] <- glm(Num_anura ~ Num_ranatra, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[8]] <- glm(Num_anura ~ 1, family = poisson,
                     offset = log(Effort), data = min.trap) 
          
##check c-hat for global model
c_hat(Cand.mod[[1]], method = "pearson") #uses Pearson's chi-square/df
##note the very low overdispersion: in this case, the analysis could be
##conducted without correcting for c-hat as its value is reasonably close
##to 1  

##output of model corrected for overdispersion
summaryOD(Cand.mod[[1]], c.hat = 1.04)

##assign names to each model
Modnames <- c("type + logperim + invertpred", "type + logperim",
              "type + invertpred", "type", "logperim + invertpred",
              "logperim", "invertpred", "intercept only") 

##model selection table based on AICc
aictab(cand.set = Cand.mod, modnames = Modnames)

##compute evidence ratio
evidence(aictab(cand.set = Cand.mod, modnames = Modnames))

##compute confidence set based on 'raw' method
confset(cand.set = Cand.mod, modnames = Modnames, second.ord = TRUE,
        method = "raw")  

##compute importance value for "TypeBOG" - same number of models
##with vs without variable
importance(cand.set = Cand.mod, modnames = Modnames, parm = "TypeBOG") 

##compute model-averaged estimate of "TypeBOG" using the natural average
modavg(cand.set = Cand.mod, modnames = Modnames, parm = "TypeBOG")

##compute model-averaged estimate of "TypeBOG" using shrinkage estimator
##same number of models with vs without variable
modavgShrink(cand.set = Cand.mod, modnames = Modnames,
             parm = "TypeBOG")

##compute model-averaged predictions for two types of ponds
##create a data set for predictions
dat.pred <- data.frame(Type = factor(c("BOG", "UPLAND")),
                       log.Perimeter = mean(min.trap$log.Perimeter),
                       Num_ranatra = mean(min.trap$Num_ranatra),
                       Effort = mean(min.trap$Effort))

##model-averaged predictions across entire model set
modavgPred(cand.set = Cand.mod, modnames = Modnames,
           newdata = dat.pred, type = "response")

##compute model-averaged effect size between two groups
##'newdata' data frame must be limited to two rows
modavgEffect(cand.set = Cand.mod, modnames = Modnames,
             newdata = dat.pred, type = "link")


## Not run: 
##Example 2:  single-season occupancy model example modified from ?occu
require(unmarked)
##single season
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
## add some fake covariates for illustration
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)),
                                sitevar2 = rnorm(numSites(pferUMF))) 
     
## observation covariates are in site-major, observation-minor order
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) *
                                 obsNum(pferUMF))) 

##check detection history data from data object
detHist(pferUMF)

##set up candidate model set
fm1 <- occu(~ obsvar1 ~ sitevar1, pferUMF)
##check detection history data from model object
detHist(fm1)

fm2 <- occu(~ 1 ~ sitevar1, pferUMF)
fm3 <- occu(~ obsvar1 ~ sitevar2, pferUMF)
fm4 <- occu(~ 1 ~ sitevar2, pferUMF)
Cand.models <- list(fm1, fm2, fm3, fm4)

##assign names to elements in list
##alternative to using 'modnames' argument
names(Cand.models) <- c("fm1", "fm2", "fm3", "fm4")

##check GOF of global model and estimate c-hat
mb.gof.test(fm4, nsim = 100) #nsim should be > 1000

##check for high SE's in models
lapply(Cand.models, checkParms, simplify = FALSE)

##compute table
print(aictab(cand.set = Cand.models,
             second.ord = TRUE), digits = 4)

##export as LaTeX table
if(require(xtable)) {
xtable(aictab(cand.set = Cand.models,
              second.ord = TRUE))
}

##compute evidence ratio
evidence(aictab(cand.set = Cand.models))
##evidence ratio between top model vs lowest-ranked model
evidence(aictab(cand.set = Cand.models), model.high = "fm2", model.low = "fm3")

##compute confidence set based on 'raw' method
confset(cand.set = Cand.models, second.ord = TRUE,
        method = "raw")  

##compute importance value for "sitevar1" on occupancy
##same number of models with vs without variable
importance(cand.set = Cand.models, parm = "sitevar1",
           parm.type = "psi") 

##compute model-averaged estimate of "sitevar1" on occupancy
##(natural average)
modavg(cand.set = Cand.models, parm = "sitevar1",
       parm.type = "psi")

##compute model-averaged estimate of "sitevar1"
##(shrinkage estimator)
##same number of models with vs without variable
modavgShrink(cand.set = Cand.models,
             parm = "sitevar1", parm.type = "psi")

##compute model-average predictions
##check explanatory variables appearing in models
extractX(Cand.models, parm.type = "psi")

##create a data set for predictions
dat.pred <- data.frame(sitevar1 = seq(from = min(siteCovs(pferUMF)$sitevar1),
                         to = max(siteCovs(pferUMF)$sitevar1), by = 0.5),
                       sitevar2 = mean(siteCovs(pferUMF)$sitevar2))

##model-averaged predictions of psi across range of values
##of sitevar1 and entire model set
modavgPred(cand.set = Cand.models, newdata = dat.pred,
           parm.type = "psi")
detach(package:unmarked)

## End(Not run)

## Not run: 
##Example 3:  example with user-supplied values of log-likelihoods and
##number of parameters

##vector with model LL's
LL <- c(-38.8876, -35.1783, -64.8970)
     
##vector with number of parameters
Ks <- c(7, 9, 4)
     
##create a vector of names to trace back models in set
Modnames <- c("Cm1", "Cm2", "Cm3")

##generate AICc table
aictabCustom(logL = LL, K = Ks, modnames = Modnames, nobs = 121,
             sort = TRUE) 
##generate AIC table
aictabCustom(logL = LL, K = Ks, modnames = Modnames,
             second.ord = FALSE, nobs = 121, sort = TRUE)

##model averaging parameter estimate
##vector of beta estimates for a parameter of interest
model.ests <- c(0.0478, 0.0480, 0.0478)
     
##vector of SE's of beta estimates for a parameter of interest
model.se.ests <- c(0.0028, 0.0028, 0.0034)
     
##compute model-averaged estimate and unconditional SE based on AICc
modavgCustom(logL = LL, K = Ks, modnames = Modnames, 
             estimate = model.ests, se = model.se.ests, nobs = 121)
##compute model-averaged estimate and unconditional SE based on BIC
modavgCustom(logL = LL, K = Ks, modnames = Modnames, 
             estimate = model.ests, se = model.se.ests, nobs = 121,
             useBIC = TRUE)

## End(Not run)

## Not run: 
##Example 4:  example with user-supplied values of information criterion
##model selection based on WAIC

##WAIC values
waic <- c(105.74, 107.36, 108.24, 100.57)
##number of effective parameters
effK <- c(7.45, 5.61, 6.14, 6.05)

##create a vector of names to trace back models in set
Modnames <- c("global model", "interactive model",
              "additive model", "invertpred model")

##generate WAIC model selection table
ictab(ic = waic, K = effK, modnames = Modnames,
      sort = TRUE, ic.name = "WAIC")

##compute model-averaged estimate
##vector of predictions
Preds <- c(0.106, 0.137, 0.067, 0.050)
##vector of SE's for prediction
Ses <- c(0.128, 0.159, 0.054, 0.039)

##compute model-averaged estimate and unconditional SE based on WAIC
modavgIC(ic = waic, K = effK, modnames = Modnames, 
         estimate = Preds, se = Ses,
         ic.name = "WAIC")

##export as LaTeX table
if(require(xtable)) {
##model-averaged estimate and confidence interval
xtable(modavgIC(ic = waic, K = effK, modnames = Modnames, 
       estimate = Preds, se = Ses,
       ic.name = "WAIC"))
##model selection table with estimate and SE's from each model
xtable(modavgIC(ic = waic, K = effK, modnames = Modnames, 
       estimate = Preds, se = Ses,
       ic.name = "WAIC"), print.table = TRUE)
}

## End(Not run)

Computing AIC, AICc, QAIC, and QAICc

Description

Functions to compute Akaike's information criterion (AIC), the second-order AIC (AICc), as well as their quasi-likelihood counterparts (QAIC, QAICc).

Usage

AICc(mod, return.K = FALSE, second.ord = TRUE, nobs = NULL, ...) 

## S3 method for class 'aov'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'betareg'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'clm'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...) 

## S3 method for class 'clmm'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)  

## S3 method for class 'coxme'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...) 

## S3 method for class 'coxph'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...) 

## S3 method for class 'fitdist'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'fitdistr'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'glm'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, c.hat = 1, ...)

## S3 method for class 'glmmTMB'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, c.hat = 1, ...)

## S3 method for class 'gls'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'gnls'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'hurdle'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'lavaan'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'lm'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'lme'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'lmekin'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...) 

## S3 method for class 'maxlikeFit'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, c.hat = 1, ...)

## S3 method for class 'mer'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'merMod'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...) 

## S3 method for class 'lmerModLmerTest'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...) 

## S3 method for class 'multinom'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, c.hat = 1, ...)

## S3 method for class 'negbin'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'nlme'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'nls'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'polr'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'rlm'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'survreg'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

## S3 method for class 'unmarkedFit'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, c.hat = 1, ...)

## S3 method for class 'vglm'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, c.hat = 1, ...)

## S3 method for class 'zeroinfl'
AICc(mod, return.K = FALSE, second.ord = TRUE,
     nobs = NULL, ...)

Arguments

mod

an object of class aov, betareg, clm, clmm, clogit, coxme, coxph, fitdist, fitdistr, glm, glmmTMB, gls, gnls, hurdle, lavaan, lm, lme, lmekin, maxlikeFit, mer, merMod, lmerModLmerTest, multinom, negbin, nlme, nls, polr, rlm, survreg, vglm, zeroinfl, and various unmarkedFit classes containing the output of a model.

return.K

logical. If FALSE, the function returns the information criterion specified. If TRUE, the function returns K (number of estimated parameters) for a given model.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, AICc will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

...

additional arguments passed to the function.

Details

AICc computes one of the following four information criteria:

Akaike's information criterion (AIC, Akaike 1973),

−2∗log−likelihood+2∗K,-2 * log-likelihood + 2 * K,

where the log-likelihood is the maximum log-likelihood of the model and K corresponds to the number of estimated parameters.

Second-order or small sample AIC (AICc, Sugiura 1978, Hurvich and Tsai 1989, 1991),

−2∗log−likelihood+2∗K∗(n/(n−K−1)),-2 * log-likelihood + 2 * K * (n/(n - K - 1)),

where n is the sample size of the data set.

Quasi-likelihood AIC (QAIC, Burnham and Anderson 2002),

QAIC=−2∗log−likelihoodc−hat+2∗K,QAIC = \frac{-2 * log-likelihood}{c-hat} + 2 * K,

where c-hat is the overdispersion parameter specified by the user with the argument c.hat.

Quasi-likelihood AICc (QAICc, Burnham and Anderson 2002),

QAIC=−2∗log−likelihoodc−hat+2∗K∗(n/(n−K−1))QAIC = \frac{-2 * log-likelihood}{c-hat} + 2 * K * (n/(n - K - 1))

.

Note that AIC and AICc values are meaningful to select among gls or lme models fit by maximum likelihood. AIC and AICc based on REML are valid to select among different models that only differ in their random effects (Pinheiro and Bates 2000).

Value

AICc returns the AIC, AICc, QAIC, or QAICc, or the number of estimated parameters, depending on the values of the arguments.

Note

The actual (Q)AIC(c) values are not really interesting in themselves, as they depend directly on the data, parameters estimated, and likelihood function. Furthermore, a single value does not tell much about model fit. Information criteria become relevant when compared to one another for a given data set and set of candidate models.

Author(s)

Marc J. Mazerolle

References

Akaike, H. (1973) Information theory as an extension of the maximum likelihood principle. In: Second International Symposium on Information Theory, pp. 267–281. Petrov, B.N., Csaki, F., Eds, Akademiai Kiado, Budapest.

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Hurvich, C. M., Tsai, C.-L. (1989) Regression and time series model selection in small samples. Biometrika 76, 297–307.

Hurvich, C. M., Tsai, C.-L. (1991) Bias of the corrected AIC criterion for underfitted regression and time series models. Biometrika 78, 499–509.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Pinheiro, J. C., Bates, D. M. (2000) Mixed-effect models in S and S-PLUS. Springer Verlag: New York.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Sugiura, N. (1978) Further analysis of the data by Akaike's information criterion and the finite corrections. Communications in Statistics: Theory and Methods A7, 13–26.

See Also

AICcCustom, aictab, confset, importance, evidence, c_hat, modavg, modavgShrink, modavgPred, useBIC,

Examples

##cement data from Burnham and Anderson (2002, p. 101)
data(cement)
##run multiple regression - the global model in Table 3.2
glob.mod <- lm(y ~ x1 + x2 + x3 + x4, data = cement)

##compute AICc with full likelihood
AICc(glob.mod, return.K = FALSE)

##compute AIC with full likelihood 
AICc(glob.mod, return.K = FALSE, second.ord = FALSE)
##note that Burnham and Anderson (2002) did not use full likelihood
##in Table 3.2 and that the MLE estimate of the variance was
##rounded to 2 digits after decimal point  


##compute AICc for mixed model on Orthodont data set in Pinheiro and
##Bates (2000)
## Not run: 
require(nlme)
m1 <- lme(distance ~ age, random = ~1 | Subject, data = Orthodont,
          method= "ML")
AICc(m1, return.K = FALSE)

## End(Not run)

Compute AIC, AICc, QAIC, and QAICc from User-supplied Input

Description

This function computes Akaike's information criterion (AIC), the second-order AIC (AICc), as well as their quasi-likelihood counterparts (QAIC, QAICc) from user-supplied input instead of extracting the values automatically from a model object. This function is particularly useful for output imported from other software or for model classes that are not currently supported by AICc.

Usage

AICcCustom(logL, K, return.K = FALSE, second.ord = TRUE, nobs = NULL,
           c.hat = 1)

Arguments

logL

the value of the model log-likelihood.

K

the number of estimated parameters in the model.

return.K

logical. If FALSE, the function returns the information criterion specified. If TRUE, the function returns K (number of estimated parameters) for a given model.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

the sample size required to compute the AICc or QAICc.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season or dynamic occupancy models (MacKenzie et al. 2002, 2003), N-mixture models (Royle 2004, Dail and Madsen 2011), or capture-mark-recapture models (e.g., Lebreton et al. 1992). If c.hat > 1, AICcCustom will return the quasi-likelihood analogue of the information criterion requested.

Details

AICcCustom computes one of the following four information criteria:

Akaike's information criterion (AIC, Akaike 1973), the second-order or small sample AIC (AICc, Sugiura 1978, Hurvich and Tsai 1989, 1991), the quasi-likelihood AIC (QAIC, Burnham and Anderson 2002), and the quasi-likelihood AICc (QAICc, Burnham and Anderson 2002).

Value

AICcCustom returns the AIC, AICc, QAIC, or QAICc, or the number of estimated parameters, depending on the values of the arguments.

Note

The actual (Q)AIC(c) values are not really interesting in themselves, as they depend directly on the data, parameters estimated, and likelihood function. Furthermore, a single value does not tell much about model fit. Information criteria become relevant when compared to one another for a given data set and set of candidate models.

Author(s)

Marc J. Mazerolle

References

Akaike, H. (1973) Information theory as an extension of the maximum likelihood principle. In: Second International Symposium on Information Theory, pp. 267–281. Petrov, B.N., Csaki, F., Eds, Akademiai Kiado, Budapest.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Hurvich, C. M., Tsai, C.-L. (1989) Regression and time series model selection in small samples. Biometrika 76, 297–307.

Hurvich, C. M., Tsai, C.-L. (1991) Bias of the corrected AIC criterion for underfitted regression and time series models. Biometrika 78, 499–509.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67–118.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Sugiura, N. (1978) Further analysis of the data by Akaike's information criterion and the finite corrections. Communications in Statistics: Theory and Methods A7, 13–26.

See Also

AICc, aictabCustom, confset, evidence, c_hat, modavgCustom

Examples

##cement data from Burnham and Anderson (2002, p. 101)
data(cement)
##run multiple regression - the global model in Table 3.2
glob.mod <- lm(y ~ x1 + x2 + x3 + x4, data = cement)

##extract log-likelihood
LL <- logLik(glob.mod)[1]

##extract number of parameters
K.mod <- coef(glob.mod) + 1

##compute AICc with full likelihood
AICcCustom(LL, K.mod, nobs = nrow(cement))

Defunct Functions in AICcmodavg Package

Description

The functions listed below have been removed from the AICcmodavg package.

Usage

AICc.mult(...)
AICc.unmarked(...)
extract.LL(...)
extract.LL.coxph(...)
extract.LL.unmarked(...)
aictab.clm(...)
aictab.clmm(...)
aictab.coxph(...)
aictab.glm(...)
aictab.gls(...)
aictab.lm(...)
aictab.lme(...)
aictab.mer(...)
aictab.merMod(...)
aictab.mult(...)
aictab.nlme(...)
aictab.nls(...)
aictab.polr(...)
aictab.rlm(...)
aictab.unmarked(...)
dictab.bugs(...)
dictab.rjags(...)
modavg.clm(...)
modavg.clmm(...)
modavg.coxph(...)
modavg.glm(...)
modavg.gls(...)
modavg.lme(...)
modavg.mer(...)
modavg.merMod(...)
modavg.mult(...)
modavg.polr(...)
modavg.rlm(...)
modavg.unmarked(...)
modavg.effect(...)
modavg.effect.glm(...)
modavg.effect.gls(...)
modavg.effect.lme(...)
modavg.effect.mer(...)
modavg.effect.merMod(...)
modavg.effect.rlm(...)
modavg.effect.unmarked(...)
modavg.shrink(...)
modavg.shrink.clm(...)
modavg.shrink.clmm(...)
modavg.shrink.coxph(...)
modavg.shrink.glm(...)
modavg.shrink.gls(...)
modavg.shrink.lme(...)
modavg.shrink.mer(...)
modavg.shrink.merMod(...)
modavg.shrink.mult(...)
modavg.shrink.polr(...)
modavg.shrink.rlm(...)
modavg.shrink.unmarked(...)
modavgpred(...)
modavgpred.glm(...)
modavgpred.gls(...)
modavgpred.lme(...)
modavgpred.mer(...)
modavgpred.merMod(...)
modavgpred.rlm(...)
modavgpred.unmarked(...)
mult.comp(...)
predictSE.zip(...)

Arguments

...

arguments passed to the function.

Details

AICc.mult has been replaced by AICc.multinom.

AICc.unmarked has been replaced by AICc.unmarkedFit.

extract.LL has been replaced by extractLL.

extract.LL.coxph has been replaced by extractLL.coxph.

extract.LL.unmarked has been replaced by extractLL.unmarkedFit.

aictab.clm has been replaced by aictab.AICsclm.clm.

aictab.clmm has been replaced by aictab.AICclmm.

aictab.coxph has been replaced by aictab.AICcoxph.

aictab.glm has been replaced by aictab.AICglm.lm.

aictab.gls has been replaced by aictab.AICgls.

aictab.lm has been replaced by aictab.AIClm.

aictab.lme has been replaced by aictab.AIClme.

aictab.mer has been replaced by aictab.AICmer.

aictab.merMod has been replaced by aictab.AIClmerMod, aictab.AICglmerMod, or aictab.AICnlmerMod, depending on the class of the objects.

aictab.mult has been replaced by aictab.AICmultinom.nnet.

aictab.nlme has been replaced by aictab.AICnlme.

aictab.nls has been replaced by aictab.AICnls.

aictab.polr has been replaced by aictab.AICpolr.

aictab.rlm has been replaced by aictab.AICrlm.lm.

aictab.unmarked has been replaced by aictab.AICunmarkedFitOccu, aictab.AICunmarkedFitColExt, aictab.AICunmarkedFitOccuRN, aictab.AICunmarkedFitPCount, aictab.AICunmarkedFitPCO, aictab.AICunmarkedFitDS, aictab.AICunmarkedFitGDS, aictab.AICunmarkedFitOccuFP, aictab.AICunmarkedFitMPois, aictab.AICunmarkedFitGMM, or aictab.AICunmarkedFitGPC, depending on the class of the objects.

dictab.bugs has been replaced by dictab.AICbugs.

dictab.jags has been replaced by dictab.AICjags.

modavg.clm has been replaced by modavg.AICsclm.clm.

modavg.clmm has been replaced by modavg.AICsclm.clm.

modavg.coxph has been replaced by modavg.AICcoxph.

modavg.glm has been replaced by modavg.AIClm or modavg.AICglm.lm, depending on the class of the objects.

modavg.gls has been replaced by modavg.AICgls.

modavg.lme has been replaced by modavg.AIClme.

modavg.mer has been replaced by modavg.AICmer.

modavg.merMod has been replaced by modavg.AIClmerMod or modavg.AICglmerMod, depending on the class of the objects.

modavg.mult has been replaced by modavg.AICmultinom.nnet.

modavg.polr has been replaced by modavg.AICpolr.

modavg.rlm has been replaced by modavg.AICrlm.lm.

modavg.unmarked has been replaced by modavg.AICunmarkedFitOccu, modavg.AICunmarkedFitColExt, modavg.AICunmarkedFitOccuRN, modavg.AICunmarkedFitPCount, modavg.AICunmarkedFitPCO, modavg.AICunmarkedFitDS, modavg.AICunmarkedFitGDS, modavg.AICunmarkedFitOccuFP, modavg.AICunmarkedFitMPois, modavg.AICunmarkedFitGMM, or modavg.AICunmarkedFitGPC, depending on the class of the objects.

modavg.effect has been replaced by modavgEffect.

modavg.effect.glm has been replaced by modavgEffect.AICglm.lm or modavgEffect.AIClm, depending on the class of the objects.

modavg.effect.gls has been replaced by modavgEffect.AICgls.

modavg.effect.lme has been replaced by modavgEffect.AIClme.

modavg.effect.mer has been replaced by modavgEffect.AICmer.

modavg.effect.merMod has been replaced by modavgEffect.AICglmerMod or modavgEffect.AIClmerMod, depending on the class of the objects.

modavg.effect.rlm has been replaced by modavgEffect.AICrlm.lm.

modavg.effect.unmarked has been replaced by modavgEffect.AICunmarkedFitOccu, modavgEffect.AICunmarkedFitColExt, modavgEffect.AICunmarkedFitOccuRN, modavgEffect.AICunmarkedFitPCount, modavgEffect.AICunmarkedFitPCO, modavgEffect.AICunmarkedFitDS, modavgEffect.AICunmarkedFitGDS, modavgEffect.AICunmarkedFitOccuFP, modavgEffect.AICunmarkedFitMPois, modavgEffect.AICunmarkedFitGMM, or modavgEffect.AICunmarkedFitGPC, depending on the class of the objects.

modavg.shrink has been replaced by modavgShrink.

modavg.shrink.clm has been replaced by modavgShrink.AICsclm.clm.

modavg.shrink.clmm has been replaced by modavgShrink.AICclmm.

modavg.shrink.coxph has been replaced by modavgShrink.AICcoxph.

modavg.shrink.glm has been replaced by modavgShrink.AICglm.lm or modavgShrink.AICglm.lm, depending on the class of the objects.

modavg.shrink.gls has been replaced by modavgShrink.AICgls.

modavg.shrink.lme has been replaced by modavgShrink.AIClme.

modavg.shrink.mer has been replaced by modavgShrink.AICmer.

modavg.shrink.merMod has been replaced by modavgShrink.AICglmerMod or modavgShrink.AIClmerMod, depending on the class of the objects.

modavg.shrink.mult has been replaced by modavgShrink.AICmultinom.nnet.

modavg.shrink.polr has been replaced by modavgShrink.AICpolr.

modavg.shrink.rlm has been replaced by modavgShrink.AICrlm.lm

modavg.shrink.unmarked has been replaced by modavgShrink.AICunmarkedFitOccu, modavgShrink.AICunmarkedFitColExt, modavgShrink.AICunmarkedFitOccuRN, modavgShrink.AICunmarkedFitPCount, modavgShrink.AICunmarkedFitPCO, modavgShrink.AICunmarkedFitDS, modavgShrink.AICunmarkedFitGDS, modavgShrink.AICunmarkedFitOccuFP, modavgShrink.AICunmarkedFitMPois, modavgShrink.AICunmarkedFitGMM, or modavgShrink.AICunmarkedFitGPC, depending on the class of the objects.

modavgpred has been replaced by modavgPred.

modavgpred.glm has been replaced by modavgpred.AICglm.lm or modavgPred.AIClm, depending on the class of the objects.

modavgpred.gls has been replaced by modavgPred.AICgls.

modavgpred.lme has been replaced by modavgPred.AIClme.

modavgpred.mer has been replaced by modavgPred.AICmer.

modavgpred.merMod has been replaced by modavgpred.AICglmerMod or modavgPred.AIClmerMod, depending on the class of the objects.

modavgpred.rlm has been replaced by modavgPred.AICrlm.lm.

modavgpred.unmarked has been replaced by modavgPred.AICunmarkedFitOccu, modavgPred.AICunmarkedFitColExt, modavgPred.AICunmarkedFitOccuRN, modavgPred.AICunmarkedFitPCount, modavgPred.AICunmarkedFitPCO, modavgPred.AICunmarkedFitDS, modavgPred.AICunmarkedFitGDS, modavgPred.AICunmarkedFitOccuFP, modavgPred.AICunmarkedFitMPois, modavgPred.AICunmarkedFitGMM, or modavgPred.AICunmarkedFitGPC, depending on the class of the objects.

mult.comp has been replaced by multComp.

predictSE.zip has been replaced by predictSE.

Author(s)

Marc J. Mazerolle

See Also

aictab, confset, dictab, importance, evidence, extractLL, c_hat, modavg, modavgEffect, modavgShrink, modavgPred, multComp, predictSE

Create Model Selection Tables

Description

This function creates a model selection table based on one of the following information criteria: AIC, AICc, QAIC, QAICc. The table ranks the models based on the selected information criteria and also provides delta AIC and Akaike weights. aictab selects the appropriate function to create the model selection table based on the object class. The current version works with lists containing objects of aov, betareg, clm, clmm, clogit, coxme, coxph, fitdist, fitdistr, glm, glmmTMB, gls, gnls, hurdle, lavaan, lm, lme, lmekin, maxlikeFit, mer, merMod, lmerModLmerTest, multinom, negbin, nlme, nls, polr, rlm, survreg, vglm, and zeroinfl classes as well as various models of unmarkedFit classes but does not yet allow mixing of different classes.

Usage

aictab(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, ...)

## S3 method for class 'AICaov.lm'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICbetareg'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICsclm.clm'
aictab(cand.set, modnames = NULL,
       second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICclmm'
aictab(cand.set, modnames = NULL,
       second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICclm'
aictab(cand.set, modnames = NULL,
       second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICcoxme'
aictab(cand.set, modnames = NULL,
       second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICcoxph'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICfitdist'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICfitdistr'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICglm.lm'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)  

## S3 method for class 'AICglmmTMB'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)  

## S3 method for class 'AICgls'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICgnls.gls'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIChurdle'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClavaan'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClm'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClme'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClmekin'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICmaxlikeFit.list'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...) 

## S3 method for class 'AICmer'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClmerMod'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClmerModLmerTest'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICglmerMod'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...) 

## S3 method for class 'AICnlmerMod'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICmultinom.nnet'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICnegbin.glm.lm'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...) 

## S3 method for class 'AICnlme.lme'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...) 

## S3 method for class 'AICnls'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICpolr'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICrlm.lm'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICsurvreg'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICunmarkedFitOccu'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitColExt'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuRN'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitPCount'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitPCO'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitDS'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitGDS'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitMPois'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitGMM'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitGPC'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitMMO'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitDSO'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICvglm'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICzeroinfl'
aictab(cand.set, modnames = NULL,
        second.ord = TRUE, nobs = NULL, sort = TRUE, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models (i.e., a named list). If no names appear in the list and no character vector is provided, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

sort

logical. If TRUE, the model selection table is ranked according to the (Q)AIC(c) values.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, aictab will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

...

additional arguments passed to the function.

Details

aictab internally creates a new class for the cand.set list of candidate models, according to the contents of the list. The current function is implemented for clogit, coxme, coxph, fitdist, fitdistr, glm, glmmTMB, gls, gnls, hurdle, lavaan, lm, lme, lmekin, maxlikeFit, mer, merMod, lmerModLmerTest, multinom, negbin, nlme, nls, polr, rlm, survreg, vglm, and zeroinfl classes as well as various unmarkedFit classes.

The function constructs a model selection table based on one of the four information criteria: AIC, AICc, QAIC, and QAICc.

Ten guidelines for model selection:

1) Carefully construct your candidate model set. Each model should represent a specific (interesting) hypothesis to test.

2) Keep your candidate model set short. It is ill-advised to consider as many models as there are data.

3) Check model fit. Use your global model (most complex model) or subglobal models to determine if the assumptions are valid. If none of your models fit the data well, information criteria will only indicate the most parsimonious of the poor models.

4) Avoid data dredging (i.e., looking for patterns after an initial round of analysis).

5) Avoid overfitting models. You should not estimate too many parameters for the number of observations available in the sample.

6) Be careful of missing values. Remember that values that are missing only for certain variables change the data set and sample size, depending on which variable is included in any given model. I suggest to remove missing cases before starting model selection.

7) Use the same response variable for all models of the candidate model set. It is inappropriate to run some models with a transformed response variable and others with the untransformed variable. A workaround is to use a different link function for some models (e.g., identity vs log link).

8) When dealing with models with overdispersion, use the same value of c-hat for all models in the candidate model set. For binomial models with trials > 1 (i.e., success/trial or cbind(success, failure) syntax) or with Poisson GLM's, you should estimate the c-hat from the most complex model (global model). If c-hat > 1, you should use the same value for each model of the candidate model set (where appropriate) and include it in the count of parameters (K). Similarly, for negative binomial models, you should estimate the dispersion parameter from the global model and use the same value across all models.

9) Burnham and Anderson (2002) recommend to avoid mixing the information-theoretic approach and notions of significance (i.e., P values). It is best to provide estimates and a measure of their precision (standard error, confidence intervals).

10) Determining the ranking of the models is just the first step. Akaike weights sum to 1 for the entire model set and can be interpreted as the weight of evidence in favor of a given model being the best one given the candidate model set considered and the data at hand. Models with large Akaike weights have strong support. Evidence ratios, importance values, and confidence sets for the best model are all measures that assist in interpretation. In cases where the top ranking model has an Akaike weight > 0.9, one can base inference on this single most parsimonious model. When many models rank highly (i.e., delta (Q)AIC(c) < 4), one should model-average effect sizes for the parameters with most support across the entire set of models. Model averaging consists in making inference based on the whole set of candidate models, instead of basing conclusions on a single 'best' model. It is an elegant way of making inference based on the information contained in the entire model set.

Value

aictab creates an object of class aictab with the following components:

Modname

the name of each model of the candidate model set.
K

the number of estimated parameters for each model.
(Q)AIC(c)

the information criterion requested for each model (AIC, AICc, QAIC, QAICc).
Delta_(Q)AIC(c)

the appropriate delta AIC component depending on the information criteria selected.
ModelLik

the relative likelihood of the model given the data (exp(-0.5*delta[i])). This is not to be confused with the likelihood of the parameters given the data. The relative likelihood can then be normalized across all models to get the model probabilities.
(Q)AIC(c)Wt

the Akaike weights, also termed "model probabilities" sensu Burnham and Anderson (2002) and Anderson (2008). These measures indicate the level of support (i.e., weight of evidence) in favor of any given model being the most parsimonious among the candidate model set.
Cum.Wt

the cumulative Akaike weights. These are only meaningful if results in table are sorted in decreasing order of Akaike weights (i.e., sort = TRUE).
c.hat

if c.hat was specified as an argument, it is included in the table.
LL

if c.hat = 1 and parameters estimated by maximum likelihood, the log-likelihood of each model.
Quasi.LL

if c.hat > 1, the quasi log-likelihood of each model.
Res.LL

if parameters are estimated by restricted maximum-likelihood (REML), the restricted log-likelihood of each model.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICc, aictabCustom, bictab, confset, c_hat, evidence, importance, modavg, modavgEffect, modavgShrink, modavgPred

Examples

##Mazerolle (2006) frog water loss example
data(dry.frog)

##setup a subset of models of Table 1
Cand.models <- list( )
Cand.models[[1]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2,
                       data = dry.frog)
Cand.models[[2]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2 +
                       Shade:Substrate, data = dry.frog)
Cand.models[[3]] <- lm(log_Mass_lost ~ cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[4]] <- lm(log_Mass_lost ~ Shade + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[5]] <- lm(log_Mass_lost ~ Substrate + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)

##create a vector of names to trace back models in set
Modnames <- paste("mod", 1:length(Cand.models), sep = " ")

##generate AICc table
aictab(cand.set = Cand.models, modnames = Modnames, sort = TRUE)
##round to 4 digits after decimal point and give log-likelihood
print(aictab(cand.set = Cand.models, modnames = Modnames, sort = TRUE),
      digits = 4, LL = TRUE)



## Not run: 
##Burnham and Anderson (2002) flour beetle data
data(beetle)
##models as suggested by Burnham and Anderson p. 198          
Cand.set <- list( )
Cand.set[[1]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link = "logit"), weights = Number_tested,
                     data = beetle)
Cand.set[[2]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link = "probit"), weights = Number_tested,
                     data = beetle)
Cand.set[[3]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link ="cloglog"), weights = Number_tested,
                     data = beetle)

##check c-hat
c_hat(Cand.set[[1]])
c_hat(Cand.set[[2]])
c_hat(Cand.set[[3]])
##lowest value of c-hat < 1 for these non-nested models, thus use
##c.hat = 1 
       
##set up named list
names(Cand.set) <- c("logit", "probit", "cloglog")

##compare models
##model names will be taken from the list if modnames is not specified
res.table <- aictab(cand.set = Cand.set, second.ord = FALSE)
##note that delta AIC and Akaike weights are identical to Table 4.7
print(res.table, digits = 2, LL = TRUE) #print table with 2 digits and
##print log-likelihood in table
print(res.table, digits = 4, LL = FALSE) #print table with 4 digits and
##do not print log-likelihood

## End(Not run)


##two-way ANOVA with interaction
data(iron)
##full model
m1 <- lm(Iron ~ Pot + Food + Pot:Food, data = iron)
##additive model
m2 <- lm(Iron ~ Pot + Food, data = iron)
##null model
m3 <- lm(Iron ~ 1, data = iron)

##candidate models
Cand.aov <- list(m1, m2, m3)
Cand.names <- c("full", "additive", "null")
aictab(Cand.aov, Cand.names)



##single-season occupancy model example modified from ?occu
## Not run: 
require(unmarked)
##single season example modified from ?occu
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
##add fake covariates
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)),
                                sitevar2 = runif(numSites(pferUMF))) 
     
##observation covariates
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) *
                                 obsNum(pferUMF))) 

##set up candidate model set
fm1 <- occu(~ obsvar1 ~ sitevar1, pferUMF)
fm2 <- occu(~ 1 ~ sitevar1, pferUMF)
fm3 <- occu(~ obsvar1 ~ sitevar2, pferUMF)
fm4 <- occu(~ 1 ~ sitevar2, pferUMF)

##assemble models in named list (alternative to using 'modnames' argument)
Cand.mods <- list("fm1" = fm1, "fm2" = fm2, "fm3" = fm3, "fm4" = fm4)

##compute table
aictab(cand.set = Cand.mods, second.ord = TRUE)

detach(package:unmarked)

## End(Not run)

Create Model Selection Tables from User-supplied Input Based on (Q)AIC(c)

Description

This function creates a model selection table from model input (log-likelihood, number of estimated parameters) supplied by the user instead of extracting the values automatically from a list of candidate models. The models are ranked based on one of the following information criteria: AIC, AICc, QAIC, QAICc. The table ranks the models based on the selected information criteria and also provides delta AIC and Akaike weights.

Usage

aictabCustom(logL, K, modnames = NULL, second.ord = TRUE, nobs = NULL,
             sort = TRUE, c.hat = 1)

Arguments

logL

a vector of log-likelihood values for the models in the candidate model set.

K

a vector containing the number of estimated parameters for each model in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models (i.e., a named list). If no names appear in the list and no character vector is provided, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

the sample size required to compute the AICc or QAICc.

sort

logical. If TRUE, the model selection table is ranked according to the (Q)AIC(c) values.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season or dynamic occupancy models (MacKenzie et al. 2002, 2003), N-mixture models (Royle 2004, Dail and Madsen 2011), or capture-mark-recapture models (e.g., Lebreton et al. 1992). If c.hat > 1, aictabCustom will return the quasi-likelihood analogue of the information criterion requested.

Details

aictabCustom constructs a model selection table based on one of the four information criteria: AIC, AICc, QAIC, and QAICc. This function is most useful when model input is imported into R from other software (e.g., Program MARK, PRESENCE) or for model classes that are not yet supported by aictab.

Value

aictabCustom creates an object of class aictab with the following components:

Modname

the name of each model of the candidate model set.
K

the number of estimated parameters for each model.
(Q)AIC(c)

the information criteria requested for each model (AICc, AICc, QAIC, QAICc).
Delta_(Q)AIC(c)

the appropriate delta AIC component depending on the information criteria selected.
ModelLik

the relative likelihood of the model given the data (exp(-0.5*delta[i])). This is not to be confused with the likelihood of the parameters given the data. The relative likelihood can then be normalized across all models to get the model probabilities.
(Q)AIC(c)Wt

the Akaike weights, also termed "model probabilities" sensu Burnham and Anderson (2002) and Anderson (2008). These measures indicate the level of support (i.e., weight of evidence) in favor of any given model being the most parsimonious among the candidate model set.
Cum.Wt

the cumulative Akaike weights. These are only meaningful if results in table are sorted in decreasing order of Akaike weights (i.e., sort = TRUE).
c.hat

if c.hat was specified as an argument, it is included in the table.
LL

if c.hat = 1 and parameters estimated by maximum likelihood, the log-likelihood of each model.
Quasi.LL

if c.hat > 1, the quasi log-likelihood of each model.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67–118.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICcCustom, bictabCustom, confset, c_hat, evidence, ictab, modavgCustom

Examples

##vector with model LL's
LL <- c(-38.8876, -35.1783, -64.8970)

##vector with number of parameters
Ks <- c(7, 9, 4)

##create a vector of names to trace back models in set
Modnames <- c("Cm1", "Cm2", "Cm3")

##generate AICc table
aictabCustom(logL = LL, K = Ks, modnames = Modnames, nobs = 121,
             sort = TRUE)

Likelihood-Ratio Test Corrected for Overdispersion

Description

Compute likelihood-ratio test between a given model and a simpler model.

Usage

anovaOD(mod.simple, mod.complex, c.hat = 1, 
        nobs = NULL, ...)

## S3 method for class 'glm'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitOccu'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitColExt'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitOccuRN'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitPCount'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitPCO'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitDS'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitGDS'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitOccuFP'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitMPois'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitGMM'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitGPC'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitOccuMS'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitOccuTTD'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitMMO'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'unmarkedFitDSO'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'glmerMod'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'maxlikeFit'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'multinom'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

## S3 method for class 'vglm'
anovaOD(mod.simple, mod.complex, c.hat = 1,  
        nobs = NULL, ...)

Arguments

mod.simple

an object of class glm, glmmTMB, maxlikeFit, mer, merMod, multinom, vglm, and various unmarkedFit classes containing the output of a model. This model should be a simpler version of mod.complex resulting from a deletion of certain terms (i.e., nested model).

mod.complex

an object of the same class as mod.simple.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat, mb.gof.test, or Nmix.gof.test. Typically, this value should be computed for the most complex model and applied to simpler models.

nobs

the number of observations used in the analysis. If nobs = NULL, the total number of rows are used as the sample size to compute the residual degrees of freedom as nobs−Knobs - K, where K is the number of estimated parameters. This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

...

additional arguments passed to the function.

Details

This function applies a correction for overdispersion on the likelihood-ratio test between a model and its simpler counterpart. The simpler model must be nested within the more complex model, typically as the result of deleting terms. You should supply the c.hat value of the most complex of the two models you are comparing.

When 1<c.hat<41 < c.hat < 4, the likelihood-ratio test is computed as:

LR=−2∗(LL.simple−LL.complex)(K.complex−K.simple)∗c.hatLR = \frac{-2 * (LL.simple - LL.complex)}{(K.complex - K.simple) * c.hat}

where LL.simple and LL.complex are the log-likelihoods of the simple and complex models, respectively, and where K.complex and K.simple are the number of estimated parameters in each model. The test statistic is approximately distributed as FK.complex−K.simple,n−K.complexF_{K.complex - K.simple, n - K.complex}, where n is the number of observations (i.e., nobs) used in the analysis (Venables and Ripley 2002).

When nobs = NULL, the number of observations is based on the number of rows of the data frame used in the analysis. For mixed models or various models of unmarkedFit, sample size is less straightforward, and nobs could be based on the total number of observations or on the number of independent clusters (e.g., sites), among other choices.

When c.hat = 1, the likelihood-ratio test simplifies to:

LR=−2∗(LL.simple−LL.complex)LR = -2 * (LL.simple - LL.complex)

where in this case the test statistic is distributed as a χK.complex−K.simple2\chi^2_{K.complex - K.simple} (McCullagh and Nelder 1989).

The function supports different model types such as Poisson GLM's and GLMM's, single-season and dynamic occupancy models (MacKenzie et al. 2002, 2003), and various N-mixture models (Royle 2004, Dail and Madsen 2011).

Value

anovaOD returns an object of class anovaOD as a list with the following components:

form.simple

a character string of the parameters estimated in mod.simple.
form.complex

a character string of the parameters estimated in mod.complex.
c.hat

the c.hat estimate used to adjust the likelihood-ratio test.
devMat

a matrix storing as columns the number of parameters estimated (K), the log-likelihood of each model logLik, the difference in estimated parameters between the two models (Kdiff), minus twice the difference in log-likelihoods between the models (-2LL), the test statistic, and the associated P-value.

Author(s)

Marc J. Mazerolle

References

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

McCullagh, P., Nelder, J. A. (1989) Generalized Linear Models. Second edition. Chapman and Hall: New York.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Venables, W. N., Ripley, B. D. (2002) Modern Applied Statistics with S. Second edition. Springer-Verlag: New York.

See Also

c_hat, mb.gof.test, Nmix.gof.test, summaryOD

Examples

##anuran larvae example from Mazerolle (2006)
data(min.trap)
##assign "UPLAND" as the reference level as in Mazerolle (2006)          
min.trap$Type <- relevel(min.trap$Type, ref = "UPLAND") 

##run model
m1 <- glm(Num_anura ~ Type + log.Perimeter + Num_ranatra,
          family = poisson, offset = log(Effort),
          data = min.trap) 
##null model
m0 <- glm(Num_anura ~ 1,
          family = poisson, offset = log(Effort),
          data = min.trap) 

##check c-hat for global model
c_hat(m1) #uses Pearson's chi-square/df

##likelihood ratio test corrected for overdispersion
anovaOD(mod.simple = m0, mod.complex = m1, c.hat = c_hat(m1))
##compare without overdispersion correction
anovaOD(mod.simple = m0, mod.complex = m1)


##example with occupancy model
## Not run: 
##load unmarked package
if(require(unmarked)){
   
   data(bullfrog)
     
   ##detection data
   detections <- bullfrog[, 3:9]

   ##assemble in unmarkedFrameOccu
   bfrog <- unmarkedFrameOccu(y = detections)
     
   ##run model
   fm <- occu(~ 1 ~ Reed.presence, data = bfrog)
   ##null model
   fm0 <- occu(~ 1 ~ 1, data = bfrog)

   ##check GOF
   ##GOF <- mb.gof.test(fm, nsim = 1000)
   ##estimate of c-hat:  1.89

   ##display results after overdispersion adjustment
   anovaOD(fm0, fm, c.hat = 1.89)

   detach(package:unmarked)
}

## End(Not run)

Flour Beetle Data

Description

This data set illustrates the acute mortality of flour beetles (Tribolium confusum) following 5 hour exposure to carbon disulfide gas.

Usage

data(beetle)

Format

A data frame with 8 rows and 4 variables.

Dose

dose of carbon disulfide in mg/L.

Number_tested

number of beetles exposed to given dose of carbon disulfide.

Number_killed

number of beetles dead after 5 hour exposure to given dose of carbon disulfide.

Mortality_rate

proportion of total beetles found dead after 5 hour exposure.

Details

Burnham and Anderson (2002, p. 195) use this data set originally from Young and Young (1998) to show model selection for binomial models with different link functions (logit, probit, cloglog).

Source

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Young, L. J., Young, J. H. (1998) Statistical Ecology. Kluwer Academic Publishers: London.

Examples

data(beetle)
## maybe str(beetle) ; plot(beetle) ...

Create Model Selection Tables Based on BIC

Description

This function creates a model selection table based on the Bayesian information criterion (Schwarz 1978, Burnham and Anderson 2002). The table ranks the models based on the BIC and also provides delta BIC and BIC model weights. The function adjusts for overdispersion in model selection by using the QBIC when c.hat > 1. bictab selects the appropriate function to create the model selection table based on the object class. The current version works with lists containing objects of aov, betareg, clm, clmm, clogit, coxme, coxph, fitdist, fitdistr, glm, glmmTMB, gls, gnls, hurdle, lavaan, lm, lme, lmekin, maxlikeFit, mer, merMod, lmerModLmerTest, multinom, nlme, nls, polr, rlm, survreg, vglm, and zeroinfl classes as well as various models of unmarkedFit classes but does not yet allow mixing of different classes.

Usage

bictab(cand.set, modnames = NULL, nobs = NULL,
       sort = TRUE, ...)

## S3 method for class 'AICaov.lm'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICbetareg'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICsclm.clm'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICclmm'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICclm'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICcoxme'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICcoxph'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICfitdist'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICfitdistr'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICglm.lm'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, c.hat = 1, ...)  

## S3 method for class 'AICglmmTMB'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, c.hat = 1, ...)  

## S3 method for class 'AICgls'
bictab(cand.set, modnames = NULL,
        nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICgnls.gls'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIChurdle'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClavaan'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClm'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClme'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClmekin'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICmaxlikeFit.list'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...) 

## S3 method for class 'AICmer'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClmerMod'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AIClmerModLmerTest'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICglmerMod'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...) 

## S3 method for class 'AICnlmerMod'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICmultinom.nnet'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICnlme.lme'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...) 

## S3 method for class 'AICnls'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICpolr'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICrlm.lm'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICsurvreg'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

## S3 method for class 'AICunmarkedFitOccu'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitColExt'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuRN'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitPCount'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitPCO'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitDS'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitGDS'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitMPois'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitGMM'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitGPC'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitMMO'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICunmarkedFitDSO'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICvglm'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, c.hat = 1, ...)

## S3 method for class 'AICzeroinfl'
bictab(cand.set, modnames = NULL,
         nobs = NULL, sort = TRUE, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models (i.e., a named list). If no names appear in the list and no character vector is provided, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

nobs

this argument allows to specify a numeric value other than total sample size to compute the BIC (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

sort

logical. If TRUE, the model selection table is ranked according to the BIC values.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, bictab will return the quasi-likelihood analogue of the BIC (QBIC) and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

...

additional arguments passed to the function.

Details

BIC tends to favor simpler models than AIC whenever n > 8 (Schwarz 1978, Link and Barker 2006, Anderson 2008). BIC assigns uniform prior probabilities across all models (i.e., equal 1/R), whereas in AIC and AICc, prior probabilities increase with sample size (Burnham and Anderson 2004, Link and Barker 2010). Some authors argue that BIC requires the true model to be included in the model set, whereas AIC or AICc does not (Burnham and Anderson 2002). However, Link and Barker (2006, 2010) consider both as assuming that a model in the model set approximates truth.

bictab internally creates a new class for the cand.set list of candidate models, according to the contents of the list. The current function is implemented for clogit, coxme, coxph, fitdist, fitdistr, glm, glmmTMB, gls, gnls, hurdle, lavaan, lm, lme, lmekin, maxlikeFit, mer, merMod, lmerModLmerTest, multinom, nlme, nls, polr, rlm, survreg, vglm, and zeroinfl classes as well as various unmarkedFit classes. The function constructs a model selection table based on BIC.

Value

bictab creates an object of class bictab with the following components:

Modname

the name of each model of the candidate model set.
K

the number of estimated parameters for each model.
(Q)BIC

the Bayesian information criterion for each model.
Delta_(Q)BIC

the delta BIC component.
ModelLik

the relative likelihood of the model given the data (exp(-0.5*delta[i])). This is not to be confused with the likelihood of the parameters given the data. The relative likelihood can then be normalized across all models to get the model probabilities.
(Q)BICWt

the BIC model weights, also termed "model probabilities" (Burnham and Anderson 2002, Link and Barker 2006, Anderson 2008). These measures indicate the level of support (i.e., weight of evidence) in favor of any given model being the most parsimonious among the candidate model set.
Cum.Wt

the cumulative BIC weights. These are only meaningful if results in table are sorted in decreasing order of BIC weights (i.e., sort = TRUE).
c.hat

if c.hat was specified as an argument, it is included in the table.
LL

the log-likelihood of each model.
Res.LL

if parameters are estimated by restricted maximum-likelihood (REML), the restricted log-likelihood of each model.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Link, W. A., Barker, R. J. (2006) Model weights and the foundations of multimodel inference. Ecology 87, 2626–2635.

Link, W. A., Barker, R. J. (2010) Bayesian Inference with Ecological Applications. Academic Press: Boston.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461–464.

See Also

aictab, bictabCustom, confset, evidence, importance, useBIC,

Examples

##Mazerolle (2006) frog water loss example
data(dry.frog)

##setup a subset of models of Table 1
Cand.models <- list( )
Cand.models[[1]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2,
                       data = dry.frog)
Cand.models[[2]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2 +
                       Shade:Substrate, data = dry.frog)
Cand.models[[3]] <- lm(log_Mass_lost ~ cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[4]] <- lm(log_Mass_lost ~ Shade + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[5]] <- lm(log_Mass_lost ~ Substrate + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)

##create a vector of names to trace back models in set
Modnames <- paste("mod", 1:length(Cand.models), sep = " ")

##generate BIC table
bictab(cand.set = Cand.models, modnames = Modnames, sort = TRUE)
##round to 4 digits after decimal point and give log-likelihood
print(bictab(cand.set = Cand.models, modnames = Modnames, sort = TRUE),
      digits = 4, LL = TRUE)



## Not run: 
##Burnham and Anderson (2002) flour beetle data
data(beetle)
##models as suggested by Burnham and Anderson p. 198          
Cand.set <- list( )
Cand.set[[1]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link = "logit"), weights = Number_tested,
                     data = beetle)
Cand.set[[2]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link = "probit"), weights = Number_tested,
                     data = beetle)
Cand.set[[3]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link ="cloglog"), weights = Number_tested,
                     data = beetle)

##set up named list
names(Cand.set) <- c("logit", "probit", "cloglog")

##compare models
##model names will be taken from the list if modnames is not specified
bictab(cand.set = Cand.set)

## End(Not run)


##two-way ANOVA with interaction
data(iron)
##full model
m1 <- lm(Iron ~ Pot + Food + Pot:Food, data = iron)
##additive model
m2 <- lm(Iron ~ Pot + Food, data = iron)
##null model
m3 <- lm(Iron ~ 1, data = iron)

##candidate models
Cand.aov <- list(m1, m2, m3)
Cand.names <- c("full", "additive", "null")
bictab(Cand.aov, Cand.names)



##single-season occupancy model example modified from ?occu
## Not run: 
require(unmarked)
##single season example modified from ?occu
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
##add fake covariates
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)),
                                sitevar2 = runif(numSites(pferUMF))) 
     
##observation covariates
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) *
                                 obsNum(pferUMF))) 

##set up candidate model set
fm1 <- occu(~ obsvar1 ~ sitevar1, pferUMF)
fm2 <- occu(~ 1 ~ sitevar1, pferUMF)
fm3 <- occu(~ obsvar1 ~ sitevar2, pferUMF)
fm4 <- occu(~ 1 ~ sitevar2, pferUMF)

##assemble models in named list (alternative to using 'modnames' argument)
Cand.mods <- list("fm1" = fm1, "fm2" = fm2, "fm3" = fm3, "fm4" = fm4)

##compute table based on QBIC that accounts for c.hat
bictab(cand.set = Cand.mods, c.hat = 3.9)

detach(package:unmarked)

## End(Not run)

Create Model Selection Tables from User-supplied Input Based on (Q)BIC

Description

This function creates a model selection table from model input (log-likelihood, number of estimated parameters) supplied by the user instead of extracting the values automatically from a list of candidate models. The models are ranked based on the BIC (Schwarz 1978) or on a quasi-likelihood analogue (QBIC) corrected for overdispersion. The table ranks the models based on the selected information criteria and also provides delta BIC and BIC weights.

Usage

bictabCustom(logL, K, modnames = NULL, nobs = NULL, sort = TRUE,
             c.hat = 1)

Arguments

logL

a vector of log-likelihood values for the models in the candidate model set.

K

a vector containing the number of estimated parameters for each model in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models (i.e., a named list). If no names appear in the list and no character vector is provided, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

nobs

the sample size required to compute the AICc or QAICc.

sort

logical. If TRUE, the model selection table is ranked according to the (Q)BIC values.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season or dynamic occupancy models (MacKenzie et al. 2002, 2003), N-mixture models (Royle 2004, Dail and Madsen 2011), or capture-mark-recapture models (e.g., Lebreton et al. 1992). If c.hat > 1, bictabCustom will return the quasi-likelihood analogue of the information criterion requested.

Details

bictabCustom constructs a model selection table based on BIC or QBIC. This function is most useful when model input is imported into R from other software (e.g., Program MARK, PRESENCE) or for model classes that are not yet supported by bictab.

Value

bictabCustom creates an object of class bictab with the following components:

Modname

the name of each model of the candidate model set.
K

the number of estimated parameters for each model.
(Q)BIC

the information criteria requested for each model (BIC, QBIC).
Delta_(Q)BIC

the appropriate delta BIC component depending on the information criteria selected.
ModelLik

the relative likelihood of the model given the data (exp(-0.5*delta[i])). This is not to be confused with the likelihood of the parameters given the data. The relative likelihood can then be normalized across all models to get the model probabilities.
(Q)BICWt

the BIC weights, also termed "model probabilities" sensu Burnham and Anderson (2002) and Anderson (2008). These measures indicate the level of support (i.e., weight of evidence) in favor of any given model being the most parsimonious among the candidate model set.
Cum.Wt

the cumulative BIC weights. These are only meaningful if results in table are sorted in decreasing order of BIC weights (i.e., sort = TRUE).
c.hat

if c.hat was specified as an argument, it is included in the table.
LL

if c.hat = 1 and parameters estimated by maximum likelihood, the log-likelihood of each model.
Quasi.LL

if c.hat > 1, the quasi log-likelihood of each model.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67–118.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461–464.

See Also

AICcCustom, aictabCustom, confset, c_hat, evidence, ictab, modavgCustom

Examples

##vector with model LL's
LL <- c(-38.8876, -35.1783, -64.8970)

##vector with number of parameters
Ks <- c(7, 9, 4)

##create a vector of names to trace back models in set
Modnames <- c("Cm1", "Cm2", "Cm3")

##generate BIC table
bictabCustom(logL = LL, K = Ks, modnames = Modnames, nobs = 121,
             sort = TRUE)

Compute Model Selection Relative Frequencies

Description

This function computes the model selection relative frequencies based on the nonparametric bootstrap (Burnham and Anderson 2002). Models are ranked based on the AIC, AICc, QAIC, or QAICc. The function currently supports objects of aov, betareg, clm, glm, hurdle, lm, multinom, polr, rlm, survreg, vglm, and zeroinfl classes.

Usage

boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
        sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICaov.lm'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICsurvreg'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICsclm.clm'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICglm.lm'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, c.hat = 1, ...)

## S3 method for class 'AIChurdle'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)
       
## S3 method for class 'AIClm'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICmultinom.nnet'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, c.hat = 1, ...)

## S3 method for class 'AICpolr'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICrlm.lm'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICsurvreg'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

## S3 method for class 'AICvglm'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, c.hat = 1, ...)

## S3 method for class 'AICzeroinfl'
boot.wt(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
       sort = TRUE, nsim = 100, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for certain types of models such as mixed models where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

sort

logical. If TRUE, the model selection table is ranked according to the (Q)AIC(c) values.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax) or with Poisson GLM's. If c.hat > 1, boot.wt will return the quasi-likelihood analogue of the information criterion requested.

nsim

the number of bootstrap iterations. Burnham and Anderson (2002) recommend at least 1000 and up to 10 000 iterations for certain problems.

...

additional arguments passed to the function.

Details

boot.wt is implemented for aov, betareg, glm, hurdle, lm, multinom, polr, rlm, survreg, vglm, and zeroinfl classes. During each bootstrap iteration, the data are resampled with replacement, all the models specified in cand.set are updated with the new data set, and the top-ranked model is saved. When all iterations are completed, the relative frequency of selection is computed for each model appearing in the candidate model set.

Relative frequencies of the models are often similar to Akaike weights, and the latter are often preferred due to their link with a Bayesian perspective (Burnham and Anderson 2002). boot.wt is most useful for teaching purposes of sampling-theory based relative frequencies of model selection. The current implementation is only appropriate with completely randomized designs. For more complex data structures (e.g., blocks or random effects), the bootstrap should be modified accordingly.

Value

boot.wt creates an object of class boot.wt with the following components:

Modname

the names of each model of the candidate model set.
K

the number of estimated parameters for each model.
(Q)AIC(c)

the information criteria requested for each model (AICc, AICc, QAIC, QAICc).
Delta_(Q)AIC(c)

the appropriate delta AIC component depending on the information criteria selected.
ModelLik

the relative likelihood of the model given the data (exp(-0.5*delta[i])). This is not to be confused with the likelihood of the parameters given the data. The relative likelihood can then be normalized across all models to get the model probabilities.
(Q)AIC(c)Wt

the Akaike weights, also termed "model probabilities" sensu Burnham and Anderson (2002) and Anderson (2008). These measures indicate the level of support (i.e., weight of evidence) in favor of any given model being the most parsimonious among the candidate model set.
PiWt

the relative frequencies of model selection from the bootstrap.
c.hat

if c.hat was specified as an argument, it is included in the table.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

See Also

AICc, confset, c_hat, evidence, importance, modavg, modavgShrink, modavgPred

Examples

##Mazerolle (2006) frog water loss example
data(dry.frog)

##setup a subset of models of Table 1
Cand.models <- list( )
Cand.models[[1]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2,
                       data = dry.frog)
Cand.models[[2]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2 +
                       Shade:Substrate, data = dry.frog)
Cand.models[[3]] <- lm(log_Mass_lost ~ cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[4]] <- lm(log_Mass_lost ~ Shade + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[5]] <- lm(log_Mass_lost ~ Substrate + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)

##create a vector of names to trace back models in set
Modnames <- paste("mod", 1:length(Cand.models), sep = " ")

##generate AICc table with bootstrapped relative
##frequencies of model selection
boot.wt(cand.set = Cand.models, modnames = Modnames, sort = TRUE,
        nsim = 10) #number of iterations should be much higher


##Burnham and Anderson (2002) flour beetle data
## Not run: 
data(beetle)
##models as suggested by Burnham and Anderson p. 198          
Cand.set <- list( )
Cand.set[[1]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link = "logit"), weights = Number_tested,
                     data = beetle)
Cand.set[[2]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link = "probit"), weights = Number_tested,
                     data = beetle)
Cand.set[[3]] <- glm(Mortality_rate ~ Dose, family =
                     binomial(link ="cloglog"), weights = Number_tested,
                     data = beetle)

##create a vector of names to trace back models in set
Modnames <- paste("Mod", 1:length(Cand.set), sep = " ")

##model selection table with bootstrapped
##relative frequencies
boot.wt(cand.set = Cand.set, modnames = Modnames)

## End(Not run)

Bullfrog Occupancy and Common Reed Invasion

Description

This is a data set from Mazerolle et al. (2014) on the occupancy of Bullfrogs (Lithobates catesbeianus) in 50 wetlands sampled in 2009 in the area of Montreal, QC.

Usage

data(bullfrog)

Format

A data frame with 50 observations on the following 23 variables.

Location

a factor with a unique identifier for each wetland.

Reed.presence

a binary variable, either 1 (reed present) or 0 (reed absent).

V1

a binary variable for detection (1) or non detection (0) of bullfrogs during the first survey.

V2

a binary variable for detection (1) or non detection (0) of bullfrogs during the second survey.

V3

a binary variable for detection (1) or non detection (0) of bullfrogs during the third survey.

V4

a binary variable for detection (1) or non detection (0) of bullfrogs during the fourth survey.

V5

a binary variable for detection (1) or non detection (0) of bullfrogs during the fifth survey.

V6

a binary variable for detection (1) or non detection (0) of bullfrogs during the sixth survey.

V7

a binary variable for detection (1) or non detection (0) of bullfrogs during the seventh survey.

Effort1

a numeric variable for the centered number of sampling stations during the first survey.

Effort2

a numeric variable for the centered number of sampling stations during the second survey.

Effort3

a numeric variable for the centered number of sampling stations during the third survey.

Effort4

a numeric variable for the centered number of sampling stations during the fourth survey.

Effort5

a numeric variable for the centered number of sampling stations during the fifth survey.

Effort6

a numeric variable for the centered number of sampling stations during the sixth survey.

Effort7

a numeric variable for the centered number of sampling stations during the seventh survey.

Type1

a binary variable to identify the survey type, either minnow trap (1) or call survey (0) during the first sampling occasion.

Type2

a binary variable to identify the survey type, either minnow trap (1) or call survey (0) during the second sampling occasion.

Type3

a binary variable to identify the survey type, either minnow trap (1) or call survey (0) during the third sampling occasion.

Type4

a binary variable to identify the survey type, either minnow trap (1) or call survey (0) during the fourth sampling occasion.

Type5

a binary variable to identify the survey type, either minnow trap (1) or call survey (0) during the fifth sampling occasion.

Type6

a binary variable to identify the survey type, either minnow trap (1) or call survey (0) during the sixth sampling occasion.

Type7

a binary variable to identify the survey type, either minnow trap (1) or call survey (0) during the seventh sampling occasion.

Details

This data set is used to illustrate single-species single-season occupancy models (MacKenzie et al. 2002) in Mazerolle (2015). The average number of sampling stations on each visit was 8.665714, and was used to center Effort on each visit.

Source

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002). Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

Mazerolle, M. J., Perez, A., Brisson, J. (2014) Common reed (Phragmites australis) invasion and amphibian distribution in freshwater wetlands. Wetlands Ecology and Management 22, 325–340.

Mazerolle, M. J. (2015) Estimating detectability and biological parameters of interest with the use of the R environment. Journal of Herpetology 49, 541–559.

Examples

data(bullfrog)
str(bullfrog)

Estimate Dispersion for Poisson and Binomial GLM's and GLMM's

Description

Functions to compute an estimate of c-hat for binomial or Poisson GLM's and GLMM's using different estimators of overdispersion.

Usage

c_hat(mod, method = "pearson", ...)

## S3 method for class 'glm'
c_hat(mod, method = "pearson", ...)

## S3 method for class 'glmmTMB'
c_hat(mod, method = "pearson", ...)

## S3 method for class 'merMod'
c_hat(mod, method = "pearson", ...)

## S3 method for class 'vglm'
c_hat(mod, method = "pearson", ...)

Arguments

mod

an object of class glm, glmmTMB, merMod, or vglm for which a c-hat estimate is required.

method

this argument defines the estimator used. The default "pearson" uses the Pearson chi-square divided by the residual degrees of freedom. Other methods include "deviance" consisting of the residual deviance divided by the residual degrees of freedom, "farrington" for the estimator suggested by Farrington (1996), and "fletcher" for the estimator suggested by Fletcher (2012).

...

additional arguments passed to the function.

Details

Poisson and binomial GLM's do not have a parameter for the variance and it is usually held fixed to 1 (i.e., mean = variance). However, one must check whether this assumption is appropriate by estimating the overdispersion parameter (c-hat). Though one can obtain an estimate of c-hat by dividing the residual deviance by the residual degrees of freedom (i.e., method = "deviance"), McCullagh and Nelder (1989) and Venables and Ripley (2002) recommend using Pearson's chi-square divided by the residual degrees of freedom (method = "pearson"). An estimator based on Farrington (1996) is also implemented by the function using the argument method = "farrington". Recent work by Fletcher (2012) suggests that an alternative estimator performs better than the above-mentioned methods in the presence of sparse data and is now implemented with method = "fletcher". For GLMM's, only the Pearson chi-square estimator of overdispersion is currently implemented.

Note that values of c-hat > 1 indicate overdispersion (variance > mean), but that values much higher than 1 (i.e., > 4) probably indicate lack-of-fit. In cases of moderate overdispersion, one usually multiplies the variance-covariance matrix of the estimates by c-hat. As a result, the SE's of the estimates are inflated (c-hat is also known as a variance inflation factor).

In model selection, c-hat should be estimated from the global model of the candidate model set and the same value of c-hat applied to the entire model set. Specifically, a global model is the most complex model which can be simplified to obtain all the other (nested) models of the set. When no single global model exists in the set of models considered, such as when sample size does not allow a complex model, one can estimate c-hat from 'subglobal' models. Here, 'subglobal' models denote models from which only a subset of the models of the candidate set can be derived. In such cases, one can use the smallest value of c-hat for model selection (Burnham and Anderson 2002).

Note that c-hat counts as an additional parameter estimated and should be added to K. All functions in package AICcmodavg automatically add 1 when the c.hat argument > 1 and apply the same value of c-hat for the entire model set. When c.hat > 1, functions compute quasi-likelihood information criteria (either QAICc or QAIC, depending on the value of the second.ord argument) by scaling the log-likelihood of the model by c.hat. The value of c.hat can influence the ranking of the models: as c-hat increases, QAIC or QAICc will favor models with fewer parameters. As an additional check against this potential problem, one can create several model selection tables by incrementing values of c-hat to assess the model selection uncertainty. If ranking changes little up to the c-hat value observed, one can be confident in making inference.

In cases of underdispersion (c-hat < 1), it is recommended to keep the value of c.hat to 1. However, note that values of c-hat << 1 can also indicate lack-of-fit and that an alternative model (and distribution) should be investigated.

Note that c_hat only supports the estimation of c-hat for binomial models with trials > 1 (i.e., success/trial or cbind(success, failure) syntax) or with Poisson GLM's or GLMM's.

Value

c_hat returns an object of class c_hat with the estimated c-hat value and an attribute for the type of estimator used.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Farrington, C. P. (1996) On assessing goodness of fit of generalized linear models to sparse data. Journal of the Royal Statistical Society B 58, 349–360.

Fletcher, D. J. (2012) Estimating overdispersion when fitting a generalized linear model to sparse data. Biometrika 99, 230–237.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

McCullagh, P., Nelder, J. A. (1989) Generalized Linear Models. Second edition. Chapman and Hall: New York.

Venables, W. N., Ripley, B. D. (2002) Modern Applied Statistics with S. Second edition. Springer: New York.

See Also

AICc, confset, evidence, modavg, importance, modavgPred, mb.gof.test, Nmix.gof.test, anovaOD, summaryOD

Examples

#binomial glm example
set.seed(seed = 10)
resp <- rbinom(n = 60, size = 1, prob = 0.5)
set.seed(seed = 10)
treat <- as.factor(sample(c(rep(x = "m", times = 30), rep(x = "f",
                                           times = 30))))
age <- as.factor(c(rep("young", 20), rep("med", 20), rep("old", 20)))
#each invidual has its own response (n = 1)
mod1 <- glm(resp ~ treat + age, family = binomial)
## Not run: 
c_hat(mod1) #gives an error because model not appropriate for
##computation of c-hat

## End(Not run)

##computing table to summarize successes
table(resp, treat, age)
dat2 <- as.data.frame(table(resp, treat, age)) #not quite what we need
data2 <- data.frame(success = c(9, 4, 2, 3, 5, 2),
                    sex = c("f", "m", "f", "m", "f", "m"),
                    age = c("med", "med", "old", "old", "young",
                      "young"), total = c(13, 7, 10, 10, 7, 13))
data2$prop <- data2$success/data2$total
data2$fail <- data2$total - data2$success

##run model with success/total syntax using weights argument
mod2 <- glm(prop ~ sex + age, family = binomial, weights = total,
            data = data2)
c_hat(mod2)

##run model with other syntax cbind(success, fail)
mod3 <- glm(cbind(success, fail) ~ sex + age, family = binomial,
            data = data2) 
c_hat(mod3)

Blood Calcium Concentration in Birds

Description

This data set features calcium concentration in the plasma of birds of both sexes following a hormonal treatment.

Usage

data(calcium)

Format

A data frame with 20 rows and 3 variables.

Calcium

calcium concentration in mg/100 ml in the blood of birds.

Hormone

a factor with two levels indicating whether the bird received a hormonal treatment or not.

Sex

a factor with two levels coding for the sex of birds.

Details

Zar (1984, p. 206) illustrates a two-way ANOVA with interaction with this data set.

Source

Zar, J. H. (1984) Biostatistical analysis. Second edition. Prentice Hall: Englewood Cliffs, New Jersey.

Examples

data(calcium)
str(calcium)

Heat Expended Following Hardening of Portland Cement

Description

This data set illustrates the heat expended (calories) from mixtures of four different ingredients of Portland cement expressed as a percentage by weight.

Usage

data(cement)

Format

A data frame with 13 observations on the following 5 variables.

x1

calcium aluminate.

x2

tricalcium silicate.

x3

tetracalcium alumino ferrite.

x4

dicalcium silicate.

y

calories of heat per gram of cement following 180 days of hardening.

Details

Burnham and Anderson (2002, p. 101) use this data set originally from Woods et al. (1932) to select among a set of multiple regression models.

Source

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Woods, H., Steinour, H. H., Starke, H. R. (1932) Effect of composition of Portland cement on heat evolved during hardening. Industrial and Engineering Chemistry 24, 1207–1214.

Examples

data(cement)
## maybe str(cement) ; plot(cement) ...

Check Convergence of Fitted Model

Description

This function checks the convergence information contained in models of various classes.

Usage

checkConv(mod, ...)

## S3 method for class 'betareg'
checkConv(mod, ...)

## S3 method for class 'clm'
checkConv(mod, ...)

## S3 method for class 'clmm'
checkConv(mod, ...)

## S3 method for class 'glm'
checkConv(mod, ...)

## S3 method for class 'glmmTMB'
checkConv(mod, ...)

## S3 method for class 'hurdle'
checkConv(mod, ...)

## S3 method for class 'lavaan'
checkConv(mod, ...)

## S3 method for class 'maxlikeFit'
checkConv(mod, ...)

## S3 method for class 'merMod'
checkConv(mod, ...)

## S3 method for class 'lmerModLmerTest'
checkConv(mod, ...)

## S3 method for class 'multinom'
checkConv(mod, ...)

## S3 method for class 'nls'
checkConv(mod, ...)

## S3 method for class 'polr'
checkConv(mod, ...)

## S3 method for class 'unmarkedFit'
checkConv(mod, ...)

## S3 method for class 'zeroinfl'
checkConv(mod, ...)

Arguments

mod

an object containing the output of a model of the classes mentioned above.

...

additional arguments passed to the function.

Details

This function checks the element of a model object that contains the convergence information from the optimization function. The function is currently implemented for models of classes betareg, clm, clmm, glm, glmmTMB, hurdle, lavaan, maxlikeFit, merMod, lmerModLmerTest, multinom, nls, polr, unmarkedFit, and zeroinfl. The function is particularly useful for functions with several groups of parameters, such as those of the unmarked package (Fiske and Chandler, 2011).

Value

checkConv returns a list with the following components:

converged

a logical value indicating whether the algorithm converged or not.
message

a string containing the message from the optimization function.

Author(s)

Marc J. Mazerolle

References

Fiske, I., Chandler, R. (2011) unmarked: An R Package for fitting hierarchical models of wildlife occurrence and abundance. Journal of Statistical Software 43, 1–23.

See Also

checkParms, covDiag, mb.gof.test, Nmix.gof.test

Examples

##example modified from ?pcount
## Not run: 
if(require(unmarked)){
##Simulate data
set.seed(3)
nSites <- 100
nVisits <- 3
##covariate
x <- rnorm(nSites)               
beta0 <- 0
beta1 <- 1
##expected counts
lambda <- exp(beta0 + beta1*x)   
N <- rpois(nSites, lambda)      
y <- matrix(NA, nSites, nVisits)
p <- c(0.3, 0.6, 0.8)           
for(j in 1:nVisits) {
  y[,j] <- rbinom(nSites, N, p[j])
}
## Organize data
visitMat <- matrix(as.character(1:nVisits),
                   nSites, nVisits, byrow=TRUE)
     
umf <- unmarkedFramePCount(y=y, siteCovs=data.frame(x=x),
                           obsCovs=list(visit=visitMat))
## Fit model
fm1 <- pcount(~ visit ~ 1, umf, K=50)
checkConv(fm1)
}

## End(Not run)

Identify Parameters with Large Standard Errors

Description

This function identifies parameter estimates with large standard errors in a model. It is particularly useful for complex models with different parameter types such as those of unmarkedFit classes implemented in package unmarked (Fiske and Chandler, 2011), as well as other types of regression models.

Usage

checkParms(mod, se.max = 25, simplify = TRUE, ...)

## S3 method for class 'betareg'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'clm'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'clmm'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'coxme'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'coxph'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'glm'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'glmmTMB'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'gls'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'gnls'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'hurdle'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'lm'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'lme'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'lmekin'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'maxlikeFit'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'merMod'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'lmerModLmerTest'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'multinom'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'nlme'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'nls'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'polr'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'rlm'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'survreg'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'unmarkedFit'
checkParms(mod, se.max = 25, simplify = TRUE,
...)

## S3 method for class 'vglm'
checkParms(mod, se.max = 25, ...)

## S3 method for class 'zeroinfl'
checkParms(mod, se.max = 25, ...)

Arguments

mod

a model of unmarkedFit classes or other regression model. This model is checked to determine the occurrence of large standard errors for parameter estimates.

se.max

specifies the value beyond which standard errors are deemed high for the model at hand. The function will determine the number of estimates with standard errors that exceed se.max.

simplify

this argument is only valid for models of unmarkedFit classes which consist of several parameter types for detection probability and demographic parameters (e.g., abundance, occupancy, extinction). If TRUE, the function returns a matrix with a single row identifying the parameter type and estimate with the highest standard error. If FALSE, the function returns a matrix with as many rows as there are parameter types in the model. In the latter case, the estimate with the highest standard error for each parameter type is presented.

...

additional arguments passed to the function.

Details

In some complex models such as certain hierarchical models (Royle and Dorazio 2008, Kéry and Royle 2015), issues in estimating parameters and their standard errors can occur. Large standard errors can be indicative of problems in estimating certain parameters due to sparse data, parameters on the boundary, or model misspecification. The checkParms function computes the number of parameter estimates with standard errors larger than se.max and identifies the parameter estimate with the largest standard error across all parameter types (simplify = TRUE) or for each parameter type (simplify = FALSE).

To help identify large standard errors, users can standardize numeric explanatory variables to zero mean and unit variance. The checkParms function can also be useful to identify boundary estimates in classic generalized models or their extensions (Venables and Ripley 2002).

Value

checkParms returns a list of class checkParms with the following components:

model.class

the class of the model for which diagnostics are requested.
se.max

the value of SE used as a threshold in diagnostics. The function reports the number of parameter estimates with SE > se.max.
result

a matrix consisting of three columns, namely, the identity of the parameter estimate with the highest SE (variable), its standard error (max.se), and the number of parameter estimates with SE larger than se.max (n.high.se). For classical regression models with a single response variable, the row name is labeled beta. For unmarkedFit models, the matrix either consists of a single row (simplify = TRUE) labeled with the name of the parameter type (e.g., psi, gam, eps, p) where the highest SE occurs, or consists of as many rows as there are parameter types (simplify = FALSE).

Author(s)

Marc J. Mazerolle

References

Agresti, A. (2002) Categorical data analysis. John Wiley and Sons, Inc.: Hoboken.

Fiske, I., Chandler, R. (2011) unmarked: An R Package for fitting hierarchical models of wildlife occurrence and abundance. Journal of Statistical Software 43, 1–23.

Kéry, M., Royle, J. A. (2015) Applied hierarchical modeling in ecology: analysis of distribution, abundance and species richness in R and BUGS. Academic Press, New York, USA.

Royle, J. A., Dorazio, R. M. (2008) Hierarchical modeling and inference in ecology: the analysis of data from populations, metapopulations and communities. Academic Press: New York.

Venables, W. N., Ripley, B. D. (2002) Modern applied statistics with S, 2nd edition. Springer-Verlag: New York.

See Also

c_hat, detHist, checkConv, countDist, countHist, extractCN, mb.gof.test, Nmix.gof.test, parboot

Examples

##example with multiple-season occupancy model modified from ?colext
## Not run: 
require(unmarked)
data(frogs)
umf <- formatMult(masspcru)
obsCovs(umf) <- scale(obsCovs(umf))
siteCovs(umf) <- rnorm(numSites(umf))
yearlySiteCovs(umf) <- data.frame(year = factor(rep(1:7,
                                    numSites(umf))))

##model with with year-dependent transition rates
fm.yearly <- colext(psiformula = ~ 1, gammaformula = ~ year,
                    epsilonformula = ~ year,
                    pformula = ~ JulianDate + I(JulianDate^2),
                    data = umf)

##check for high SE's and report highest
##across all parameter types
checkParms(fm.yearly, simplify = TRUE)

##check for high SE's and report highest
##for each parameter type
checkParms(fm.yearly, simplify = FALSE)
detach(package:unmarked)

## End(Not run)

##example from Agresti 2002 of logistic regression
##with parameters estimated at the boundary (complete separation)
## Not run: 
x <- c(10, 20, 30, 40, 60, 70, 80, 90)
y <- c(0, 0, 0, 0, 1, 1, 1, 1)

m1 <- glm(y ~ x, family = binomial)
checkParms(m1)

## End(Not run)

Computing Confidence Set for the Kullback-Leibler Best Model

Description

This function computes the confidence set on the best model given the data and model set. confset implements three different methods proposed by Burnham and Anderson (2002).

Usage

confset(cand.set, modnames = NULL, second.ord = TRUE, nobs = NULL,
        method = "raw", level = 0.95, delta = 6, c.hat = 1)

Arguments

cand.set

a list storing each of the models in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

method

a character value, either as raw, ordinal, or ratio, indicating the method for determining the confidence set for the best model (see 'Description' above for details).

level

the level of confidence (i.e., sum of model probabilities) used to determine the confidence set on the best model when using the raw method. Note that the argument is not used for the other methods of determining the confidence set on the best model.

delta

the delta (Q)AIC(c) value associated with the cutoff point to determine the confidence set for the best model. Note that the argument is only used when method = ratio.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, confset will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

Details

The first and simplest (method = 'raw'), relies on summing the Akaike weights (i.e., model probabilities) of the ranked models until we reach a given cutpoint (e.g., 0.95 for a 95 percent set).

The second method (method = 'ordinal') suggested is based on the classification of the models on an ordinal scale based on the delta (Q)AIC(c). The models are grouped in different classes based on their weight of support as determined by the delta (Q)AIC(c) values: substantial support (delta (Q)AIC(c) <= 2), some support (2 < delta (Q)AIC(c) <= 7), little support (7 < delta (Q)AIC(c) <= 10), no support (delta (Q)AIC(c) > 10).

The third method (method = 'ratio') is based on identifying the ratios of model likelihoods (i.e., exp(-delta_(Q)AIC(c)/2) ) that exceed a cutpoint, similar to the building of profile likelihood intervals. An evidence ratio of each model relative to the top-ranked model is computed and the ratios exceeding the cutpoint determine which models are included in the confidence set. Note here that small cutoff points are suggested (e.g., 0.125, 0.050). The cutoff point is linked to delta (Q)AIC(c) by the following relationship: cutoff=exp(−1∗δ(Q)AIC(c)/2)cutoff = exp(-1*\delta_{(Q)AIC(c)}/2).

Value

confset returns an object of class confset as a list with the following components, depending on which method is used:

when method = 'raw':

method

identifies the method of determining the confidence set on the best model.
level

the confidence level used to determine the confidence set on the best model.
table

a reduced table with the models included in the confidence set.

when method = 'ordinal':

method

identifies the method of determining the confidence set on the best model.
substantial

a reduced table with the models included in the confidence set for which delta (Q)AIC(c) <= 2.
some

a reduced table with the models included in the confidence set for which 2 < delta (Q)AIC(c) <= 7.
little

a reduced table with the models included in the confidence set for which 7 < delta (Q)AIC(c) <= 10.
none

a reduced table with the models included in the confidence set for which delta (Q)AIC(c) > 10.

when method = 'ratio':

method

identifies the method of determining the confidence set on the best model.
cutoff

the cutoff value for the ratios used to determine the confidence set on the best model.
delta

the delta (Q)AIC(c) used to compute the cutoff value for ratios to determine the confidence set on the best model.
table

a reduced table with the models included in the confidence set.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICc, aictab, c_hat, evidence, importance, modavg, modavgShrink, modavgPred

Examples

##anuran larvae example from Mazerolle (2006)
data(min.trap)
##assign "UPLAND" as the reference level as in Mazerolle (2006)          
min.trap$Type <- relevel(min.trap$Type, ref = "UPLAND") 

##set up candidate models          
Cand.mod <- list()
##global model          
Cand.mod[[1]] <- glm(Num_anura ~ Type + log.Perimeter + Num_ranatra,
                     family = poisson, offset = log(Effort),
                     data = min.trap) 
Cand.mod[[2]] <- glm(Num_anura ~ Type + log.Perimeter, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[3]] <- glm(Num_anura ~ Type + Num_ranatra, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[4]] <- glm(Num_anura ~ Type, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[5]] <- glm(Num_anura ~ log.Perimeter + Num_ranatra,
                     family = poisson, offset = log(Effort),
                     data = min.trap) 
Cand.mod[[6]] <- glm(Num_anura ~ log.Perimeter, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[7]] <- glm(Num_anura ~ Num_ranatra, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[8]] <- glm(Num_anura ~ 1, family = poisson,
                     offset = log(Effort), data = min.trap) 
          
##check c-hat for global model
c_hat(Cand.mod[[1]]) #uses Pearson's chi-square/df
##note the very low overdispersion: in this case, the analysis could be
##conducted without correcting for c-hat as its value is reasonably close
##to 1  

##assign names to each model
Modnames <- c("type + logperim + invertpred", "type + logperim",
              "type + invertpred", "type", "logperim + invertpred",
              "logperim", "invertpred", "intercept only") 

##compute confidence set based on 'raw' method
confset(cand.set = Cand.mod, modnames = Modnames, second.ord = TRUE,
        method = "raw")  


##example with linear mixed model
## Not run: 
require(nlme)

##set up candidate model list for Orthodont data set shown in Pinheiro
##and Bates (2000:  Mixed-effect models in S and S-PLUS. Springer Verlag:
##New York.)
Cand.models <- list()
Cand.models[[1]] <- lme(distance ~ age, random = ~age | Subject,
                        data = Orthodont, method = "ML")
Cand.models[[2]] <- lme(distance ~ age + Sex, data = Orthodont,
                        random = ~ 1 | Subject, method = "ML")
Cand.models[[3]] <- lme(distance ~ 1, data = Orthodont,
                        random = ~ 1 | Subject, method = "ML")

##create a vector of model names
Modnames <- paste("mod", 1:length(Cand.models), sep = "")

##compute confidence set based on 'raw' method
confset(cand.set = Cand.models, modnames = Modnames, second.ord = TRUE,
        method = "raw")
##round to 4 digits after decimal point
print(confset(cand.set = Cand.models, modnames = Modnames,
              second.ord = TRUE, method = "raw"), digits = 4)

confset(cand.set = Cand.models, modnames = Modnames, second.ord = TRUE,
        level = 0.9, method = "raw")

##compute confidence set based on 'ordinal' method
confset(cand.set = Cand.models, modnames = Modnames, second.ord = TRUE,
        method = "ordinal")

##compute confidence set based on 'ratio' method
confset(cand.set = Cand.models, modnames = Modnames, second.ord = TRUE,
        method = "ratio", delta = 4)

confset(cand.set = Cand.models, modnames = Modnames, second.ord = TRUE,
        method = "ratio", delta = 8)
detach(package:nlme)

## End(Not run)

Compute Summary Statistics from Distance Sampling Data

Description

This function extracts various summary statistics from distance sampling data of various unmarkedFrame and unmarkedFit classes.

Usage

countDist(object, plot.freq = TRUE, plot.distance = TRUE, cex.axis = 1,
          cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFrameDS'
countDist(object, plot.freq = TRUE,
                   plot.distance = TRUE, cex.axis = 1, cex.lab = 1,
                   cex.main = 1, ...)

## S3 method for class 'unmarkedFitDS'
countDist(object, plot.freq = TRUE,
                   plot.distance = TRUE, cex.axis = 1, cex.lab = 1,
                   cex.main = 1, ...)

## S3 method for class 'unmarkedFrameGDS'
countDist(object, plot.freq = TRUE,
                   plot.distance = TRUE, cex.axis = 1,
                   cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFitGDS'
countDist(object, plot.freq = TRUE,
                   plot.distance = TRUE, cex.axis = 1,
                   cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFrameDSO'
countDist(object, plot.freq = TRUE,
                   plot.distance = TRUE, cex.axis = 1,
                   cex.lab = 1, cex.main = 1, plot.seasons = FALSE,
                   ...)

## S3 method for class 'unmarkedFitDSO'
countDist(object, plot.freq = TRUE,
                   plot.distance = TRUE, cex.axis = 1,
                   cex.lab = 1, cex.main = 1, plot.seasons = FALSE,
                   ...)

Arguments

object

an object of various unmarkedFrame or unmarkedFit classes containing distance sampling data.

plot.freq

logical. Specifies if the count data (pooled across seasons and distance classes) should be plotted.

plot.distance

logical. Specifies if the counts in each distance class (pooled across seasons) should be plotted.

cex.axis

expansion factor influencing the size of axis annotations on plots produced by the function.

cex.lab

expansion factor influencing the size of axis labels on plots produced by the function.

cex.main

expansion factor influencing the size of the main title above plots produced by the function.

plot.seasons

logical. Specifies if the count data should be plotted for each distance class and season separately. This argument is only relevant for data collected across more than a single season.

...

additional arguments passed to the function.

Details

This function computes a number of summary statistics in data sets used for the distance sampling models of Royle et al. (2004), Chandler et al. (2011), and distance-sampling versions of models of Dail and Madsen (2011) and Hostetler and Chandler (2015) based on Sollmann et al. (2015).

countDist can take data frames of the unmarkedFrameDS, unmarkedFrameGDS, unmarkedFrameDSO classes as input. For convenience, the function can also extract the raw data from model objects of classes unmarkedFitDS, unmarkedFitGDS, and unmarkedFitDSO. Note that different model objects using the same data set will have identical values.

Value

countDist returns a list with the following components:

count.table.full

a table with the frequency of each observed count pooled across distances classes.
count.table.seasons

a list of tables with the frequency of each season-specific count pooled across distance classes.
dist.sums.full

a table with the frequency of counts in each distance class across the entire sampling seasons.
dist.table.seasons

a list of tables with the frequency of counts in each distance class for each primary period.
dist.names

a character string of labels for the distance classes.
n.dist.classes

the number of distance classes.
out.freqs

a matrix where the rows correspond to each sampling season and where columns consist of the number of sites sampled in season tt (sampled) and the number of sites with at least one detection in season tt (detected). For multiseason data, the matrix includes the number of sites sampled in season t−1t - 1 with colonizations observed in season tt (colonized), the number of sites sampled in season t−1t - 1 with extinctions observed in season tt (extinct), the number of sites sampled in season t−1t - 1 without changes observed in season tt (static), and the number of sites sampled in season tt that were also sampled in season t−1t - 1 (common).

out.props

a matrix where the rows correspond to each sampling season and where columns consist of the proportion of sites in season t with at least one detection (naive.occ). For multiseason data, the matrix includes the proportion of sites sampled in season t−1t - 1 with colonizations observed in season tt (naive.colonization), the proportion of sites sampled in season t−1t - 1 with extinctions observed in season tt (naive.extinction), and the proportion of sites sampled in season t−1t - 1 with no changes observed in season tt.

n.seasons

the number of seasons (primary periods) in the data set.

n.visits.season

the maximum number of visits per season in the data set.

missing.seasons

logical vector indicating whether data were collected or not during a given season (primary period), where TRUE indicates that no data were collected during the season.

Author(s)

Marc J. Mazerolle

References

Chandler, R. B., Royle, J. A., King, D. I. (2011) Inference about density and temporary emigration in unmarked populations. Ecology 92, 1429–1435.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Hostetler, J. A., Chandler, R. B. (2015) Improved state-space models for inference about spatial and temporal variation in abundance from count data. Ecology 96, 1713–1723.

Royle, J. A., Dawson, D. K., Bates, S. (2004) Modeling abundance effects in distance sampling. Ecology 85, 1591–1597.

Sollmann, R., Gardner, B., Chandler, R. B., Royle, J. A., Sillett, T. S. (2015) An open-population hierarchical distance sampling model. Ecology 96, 325–331.

See Also

covDiag, detHist, detTime, countHist, Nmix.chisq, Nmix.gof.test

Examples

##modified example from ?distsamp
## Not run: 
if(require(unmarked)){
data(linetran)
##format data
ltUMF <- with(linetran, {
              unmarkedFrameDS(y = cbind(dc1, dc2, dc3, dc4),
              siteCovs = data.frame(Length, area, habitat),
              dist.breaks = c(0, 5, 10, 15, 20),
              tlength = linetran$Length * 1000, survey = "line",
              unitsIn = "m") 
             })

##compute descriptive stats from data object
countDist(ltUMF)
     
##Half-normal detection function
fm1 <- distsamp(~ 1 ~ 1, ltUMF)
##compute descriptive stats from model object
countDist(fm1)
}

## End(Not run)

Compute Summary Statistics from Count Histories

Description

This function extracts various summary statistics from count data of various unmarkedFrame and unmarkedFit classes.

Usage

countHist(object, plot.freq = TRUE, cex.axis = 1, cex.lab = 1,
          cex.main = 1, ...) 

## S3 method for class 'unmarkedFramePCount'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFitPCount'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFrameGPC'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFitGPC'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFrameMPois'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFitMPois'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFramePCO'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1,
                   plot.seasons = FALSE, ...)

## S3 method for class 'unmarkedFitPCO'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1,
                   plot.seasons = FALSE, ...)

## S3 method for class 'unmarkedFrameGMM'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1,
                   plot.seasons = FALSE, ...)

## S3 method for class 'unmarkedFitGMM'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1,
                   plot.seasons = FALSE, ...)

## S3 method for class 'unmarkedFrameMMO'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1,
                   plot.seasons = FALSE, ...)

## S3 method for class 'unmarkedFitMMO'
countHist(object, plot.freq = TRUE,
                   cex.axis = 1, cex.lab = 1, cex.main = 1,
                   plot.seasons = FALSE, ...)

Arguments

object

an object of various unmarkedFrame or unmarkedFit classes containing count history data.

plot.freq

logical. Specifies if the count data (pooled across seasons) should be plotted.

cex.axis

expansion factor influencing the size of axis annotations on plots produced by the function.

cex.lab

expansion factor influencing the size of axis labels on plots produced by the function.

cex.main

expansion factor influencing the size of the main title above plots produced by the function.

plot.seasons

logical. Specifies if the count data should be plotted for each season separately. This argument is only relevant for data collected across more than a single season.

...

additional arguments passed to the function.

Details

This function computes a number of summary statistics in data sets used for various N-mixture models including those of Royle (2004a, b), Dail and Madsen (2011), and Chandler et al. (2011).

countHist can take data frames of the unmarkedFramePCount, unmarkedFrameGPC, unmarkedFrameMPois, unmarkedFramePCO, unmarkedFrameGMM, unmarkedFrameMMO classes as input. For convenience, the function can also extract the raw data from model objects of classes unmarkedFitPCount, unmarkedFitGPC, unmarkedFitMPois, unmarkedFitPCO, unmarkedFitGMM, and unmarkedFitMMO. Note that different model objects using the same data set will have identical values.

Value

countHist returns a list with the following components:

count.table.full

a table with the frequency of each observed count.
count.table.seasons

a list of tables with the frequency of each season-specific count.
hist.table.full

a table with the frequency of each count history across the entire sampling period.
hist.table.seasons

a list of tables with the frequency of each count history for each primary period (season).
out.freqs

a matrix where the rows correspond to each sampling season and where columns consist of the number of sites sampled in season tt (sampled) and the number of sites with at least one detection in season tt (detected). For multiseason data, the matrix includes the number of sites sampled in season t−1t - 1 with colonizations observed in season tt (colonized), the number of sites sampled in season t−1t - 1 with extinctions observed in season tt (extinct), the number of sites sampled in season t−1t - 1 without changes observed in season tt (static), and the number of sites sampled in season tt that were also sampled in season t−1t - 1 (common).

out.props

a matrix where the rows correspond to each sampling season and where columns consist of the proportion of sites in season t with at least one detection (naive.occ). For multiseason data, the matrix includes the proportion of sites sampled in season t−1t - 1 with colonizations observed in season tt (naive.colonization), the proportion of sites sampled in season t−1t - 1 with extinctions observed in season tt (naive.extinction), and the proportion of sites sampled in season t−1t - 1 with no changes observed in season tt.

n.seasons

the number of seasons (primary periods) in the data set.

n.visits.season

the maximum number of visits per season in the data set.

missing.seasons

logical vector indicating whether data were collected or not during a given season (primary period), where TRUE indicates that no data were collected during the season.

Author(s)

Marc J. Mazerolle

References

Chandler, R. B., Royle, J. A., King, D. I. (2011) Inference about density and temporary emigration in unmarked populations. Ecology 92, 1429–1435.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Royle, J. A. (2004a) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Royle, J. A. (2004b) Generalized estimators of avian abundance from count survey data. Animal Biodiversity and Conservation 27, 375–386.

See Also

covDiag, detHist, detTime, countDist, Nmix.chisq, Nmix.gof.test

Examples

##modified example from ?pcount
## Not run: 
if(require(unmarked)){
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
##compute descriptive stats from data object
countHist(mallardUMF)

##run single season model
fm.mallard <- pcount(~ ivel+ date + I(date^2) ~ length + elev +
                     forest, mallardUMF, K=30)
##compute descriptive stats from model object
countHist(fm.mallard)
}

## End(Not run)

Compute Covariance Diagnostic for Lambda in N-mixture Models

Description

This function extracts the covariance diagnostic of Dennis et al. (2015) for lambda in N-mixture models (Royle 2004) of the unmarkedFitPCount class as well as in data frames of the unmarkedFramePcount class.

Usage

covDiag(object, ...)

## S3 method for class 'unmarkedFitPCount'
covDiag(object, ...)

## S3 method for class 'unmarkedFramePCount'
covDiag(object, ...)

Arguments

object

an object of class unmarkedFitPCount or unmarkedFramePCount.

...

additional arguments passed to the function.

Details

This function extracts the covariance diagnostic developed by Dennis et al. (2015) for lambda in N-mixture models. Values <= 0 suggest sparse data and potential problems during model fitting. covDiag can take data frames of the unmarkedFramePcount class as input. For convenience, the function also takes the repeated count model object as input, extracts the raw data, and computes the covariance diagnostic. Thus, different models on the same data set will have identical values for this covariance diagnostic.

Value

covDiag returns a list with the following components:

cov.diag

the value of the covariance diagnostic.
message

a string indicating whether a warning was issued (i.e., "Warning: lambda is infinite, data too sparse") or not (i.e., NULL).

Author(s)

Marc J. Mazerolle

References

Dennis, E. B., Morgan, B. J. T., Ridout, M. S. (2015) Computational aspects of N-mixture models. Biometrics 71, 237–246.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

modavg, modavgPred, Nmix.chisq, Nmix.gof.test, predictSE, pcount

Examples

##modified example from ?pcount
## Not run: 
if(require(unmarked)){
##Simulate data
set.seed(3)
nSites <- 100
nVisits <- 3
##covariate
x <- rnorm(nSites)               
beta0 <- 0
beta1 <- 1
##expected counts
lambda <- exp(beta0 + beta1*x)   
N <- rpois(nSites, lambda)      
y <- matrix(NA, nSites, nVisits)
p <- c(0.3, 0.6, 0.8)           
for(j in 1:nVisits) {
  y[,j] <- rbinom(nSites, N, p[j])
}
## Organize data
visitMat <- matrix(as.character(1:nVisits),
                   nSites, nVisits, byrow=TRUE)
     
umf <- unmarkedFramePCount(y=y, siteCovs=data.frame(x=x),
                           obsCovs=list(visit=visitMat))
## Fit model
fm1 <- pcount(~ visit ~ 1, umf, K=50)
covDiag(fm1)

##sparser data
p <- c(0.01, 0.001, 0.01)           
for(j in 1:nVisits) {
  y[,j] <- rbinom(nSites, N, p[j])
}
## Organize data
visitMat <- matrix(as.character(1:nVisits),
                   nSites, nVisits, byrow=TRUE)
     
umf <- unmarkedFramePCount(y=y, siteCovs=data.frame(x=x),
                           obsCovs=list(visit=visitMat))
## Fit model
fm.sparse <- pcount(~ visit ~ 1, umf, K=50)
covDiag(fm.sparse)
}

## End(Not run)

Compute Summary Statistics from Detection Histories

Description

This function extracts various summary statistics from detection history data of various unmarkedFrame and unmarkedFit classes.

Usage

detHist(object, ...)

## S3 method for class 'unmarkedFitColExt'
detHist(object, ...)

## S3 method for class 'unmarkedFitOccu'
detHist(object, ...)

## S3 method for class 'unmarkedFitOccuFP'
detHist(object, ...)

## S3 method for class 'unmarkedFitOccuRN'
detHist(object, ...)

## S3 method for class 'unmarkedFitOccuMulti'
detHist(object, ...)

## S3 method for class 'unmarkedFitOccuMS'
detHist(object, ...)

## S3 method for class 'unmarkedFrameOccu'
detHist(object, ...)

## S3 method for class 'unmarkedFrameOccuFP'
detHist(object, ...)

## S3 method for class 'unmarkedMultFrame'
detHist(object, ...)

## S3 method for class 'unmarkedFrameOccuMulti'
detHist(object, ...)

## S3 method for class 'unmarkedFrameOccuMS'
detHist(object, ...)

Arguments

object

an object of various unmarkedFrame or unmarkedFit classes containing detection history data.

...

additional arguments passed to the function.

Details

This function computes a number of summary statistics in data sets used for single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), Royle-Nichols models (Royle and Nichols 2003), false-positive occupancy models (Royle and Link 2006, Miller et al. 2011), multispecies occupancy models (Rota et al. 2016), and multistate occupancy models (Nichols et al. 2007, MacKenzie et al. 2009).

detHist can take data frames of the unmarkedFrameOccu, unmarkedFrameOccuFP, unmarkedMultFrame, unmarkedFrameOccuMulti, unmarkedFrameOccuMS classes as input. For convenience, the function can also extract the raw data from model objects of classes unmarkedFitColExt, unmarkedFitOccu, unmarkedFitOccuFP, unmarkedFitOccuRN, unmarkedFrameOccuMulti, and unmarkedFrameOccuMS. Note that different model objects using the same data set will have identical values.

Value

For objects of classes unmarkedFitOccu, unmarkedFitOccuRN, unmarkedFitOccuFP, unmarkedFitColExt, unmarkedFitOccuMS, unmarkedFrameOccu, unmarkedFrameOccuFP, unmarkedMultFrame, and unmarkedFrameOccuMS, detHist returns a list with the following components:

hist.table.full

a table with the frequency of each observed detection history.
hist.table.seasons

a list of tables with the frequency of each season-specific detection history.
out.freqs

a matrix where the rows correspond to each sampling season and where columns consist of the number of sites sampled in season tt (sampled) and the number of sites with at least one detection in season tt (detected). For multiseason data, the matrix includes the number of sites sampled in season t−1t - 1 with colonizations observed in season tt (colonized), the number of sites sampled in season t−1t - 1 with extinctions observed in season tt (extinct), the number of sites sampled in season t−1t - 1 without changes observed in season tt (static), and the number of sites sampled in season tt that were also sampled in season t−1t - 1 (common). For multispecies data, out.freqs presents for each species the number of sites sampled and the number of sites with at least one detection.

out.props

a matrix where the rows correspond to each sampling season and where columns consist of the proportion of sites in season t with at least one detection (naive.occ). For multiseason data, the matrix includes the proportion of sites sampled in season t−1t - 1 with colonizations observed in season tt (naive.colonization), the proportion of sites sampled in season t−1t - 1 with extinctions observed in season tt (naive.extinction), and the proportion of sites sampled in season t−1t - 1 with no changes observed in season tt. For multispecies data, out.props presents the proportion of sites with a least one detection for each species.

n.seasons

the number of seasons (primary periods) in the data set.

n.visits.season

the maximum number of visits per season in the data set.

n.species

the number of species in the data set.

missing.seasons

logical vector indicating whether data were collected or not during a given season (primary period), where TRUE indicates that no data were collected during the season.

For objects of classes unmarkedFitOccuMulti and unmarkedFrameOccuMulti, detHist returns a list with the following components:

hist.table.full

a table with the frequency of each observed detection history. The species are coded with letters and follow the same order of presentation as in the other parts of the output.
hist.table.species

a list of tables with the frequency of each species-specific detection history. The last element of hist.table.species features the number of sites with co-occurrence of the different species (coOcc).
out.freqs

a matrix where the rows correspond to each species and where columns consist of the number of sites sampled during the season (sampled) and the number of sites with at least one detection (detected).
out.props

a matrix where the rows correspond to each species and where columns consist of the proportion of sites with at least one detection during the season (naive.occ).

n.seasons

the number of seasons (primary periods) in the data set.

n.visits.season

the maximum number of visits per season in the data set.

n.species

the number of species in the data set.

missing.seasons

logical vector indicating whether data were collected or not during a given season (primary period), where TRUE indicates that no data were collected during the season.

Author(s)

Marc J. Mazerolle

References

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

MacKenzie, D. I., Nichols, J. D., Seamans, M. E., Gutierrez, R. J. (2009) Modeling species occurrence dynamics with multiple states and imperfect detection. Ecology 90, 823–835.

Mazerolle, M. J. (2015) Estimating detectability and biological parameters of interest with the use of the R environment. Journal of Herpetology 49, 541–559.

Miller, D. A. W., Nichols, J. D., McClintock, B. T., Campbell Grant, E. H., Bailey, L. L. (2011) Improving occupancy estimation when two types of observational error occur: non-detection and species misidentification. Ecology 92, 1422–1428.

Nichols, J. D., Hines, J. E., Mackenzie, D. I., Seamans, M. E., Gutierrez, R. J. (2007) Occupancy estimation and modeling with multiple states and state uncertainty. Ecology 88, 1395–1400.

Rota, C. T., Ferreira, M. A. R., Kays, R. W., Forrester, T. D., Kalies, E. L., McShea, W. J., Parsons, A. W., Millspaugh, J. J. (2016) A multispecies occupancy model for two or more interacting species. Methods in Ecology and Evolution 7, 1164–1173.

Royle, J. A., Link, W. A. (2006) Generalized site occupancy models allowing for false positive and false negative errors. Ecology 87, 835–841.

Royle, J. A., Nichols, J. D. (2003) Estimating abundance from repeated presence-absence data or point counts. Ecology 84, 777–790.

See Also

covDiag, countHist, countDist, detTime, mb.chisq, mb.gof.test

Examples

##data from Mazerolle (2015)
## Not run: 
data(bullfrog)

##detection data
detections <- bullfrog[, 3:9]

##load unmarked package
if(require(unmarked)){

##assemble in unmarkedFrameOccu
bfrog <- unmarkedFrameOccu(y = detections)

##compute descriptive stats from data object
detHist(bfrog)

##run model
fm <- occu(~ 1 ~ 1, data = bfrog)
##compute descriptive stats from model object
detHist(fm)
}

## End(Not run)

Compute Summary Statistics from Time to Detection Data

Description

This function extracts various summary statistics from time to detection data of various unmarkedFrame and unmarkedFit classes.

Usage

detTime(object, plot.time = TRUE, plot.seasons = FALSE,
        cex.axis = 1, cex.lab = 1, cex.main = 1, ...)

## S3 method for class 'unmarkedFrameOccuTTD'
detTime(object, plot.time = TRUE,
                 plot.seasons = FALSE, cex.axis = 1, cex.lab = 1,
                 cex.main = 1, ...)

## S3 method for class 'unmarkedFitOccuTTD'
detTime(object, plot.time = TRUE,
                   plot.seasons = FALSE, cex.axis = 1, cex.lab = 1,
                   cex.main = 1, ...)

Arguments

object

an object of various unmarkedFrame or unmarkedFit classes containing time to detection data.

plot.time

logical. Specifies if the time to detection data (pooled across seasons) should be plotted.

plot.seasons

logical. Specifies if the time to detection data should be plotted for each season separately. This argument is only relevant for data collected across more than a single season.

cex.axis

expansion factor influencing the size of axis annotations on plots produced by the function.

cex.lab

expansion factor influencing the size of axis labels on plots produced by the function.

cex.main

expansion factor influencing the size of the main title above plots produced by the function.

...

additional arguments passed to the function.

Details

This function computes a number of summary statistics in data sets used for the time to detection models of Garrard et al. (2008, 2013).

detTime can take data frames of the unmarkedFrameOccuTTD class as input, or can also extract the raw data from model objects of the unmarkedFitOccuTTD class. Note that different model objects using the same data set will have identical values.

Value

detTime returns a list with the following components:

time.table.full

a table with the quantiles of time to detection data pooled across seasons, but excluding censored observations.
time.table.seasons

a list of tables with the quantiles of season-specific time to detection data, but excluding censored observations.
out.freqs

a matrix where the rows correspond to each sampling season and where columns consist of the number of sites sampled in season tt (sampled) and the number of sites with at least one detection in season tt (detected). For multiseason data, the matrix includes the number of sites sampled in season t−1t - 1 with colonizations observed in season tt (colonized), the number of sites sampled in season t−1t - 1 with extinctions observed in season tt (extinct), the number of sites sampled in season t−1t - 1 without changes observed in season tt (static), and the number of sites sampled in season tt that were also sampled in season t−1t - 1 (common).

out.props

a matrix where the rows correspond to each sampling season and where columns consist of the proportion of sites in season t with at least one detection (naive.occ). For multiseason data, the matrix includes the proportion of sites sampled in season t−1t - 1 with colonizations observed in season tt (naive.colonization), the proportion of sites sampled in season t−1t - 1 with extinctions observed in season tt (naive.extinction), and the proportion of sites sampled in season t−1t - 1 with no changes observed in season tt.

n.seasons

the number of seasons (primary periods) in the data set.

n.visits.season

the maximum number of visits per season in the data set.

missing.seasons

logical vector indicating whether data were collected or not during a given season (primary period), where TRUE indicates that no data were collected during the season.

Author(s)

Marc J. Mazerolle

References

Garrard, G. E., Bekessy, S. A., McCarthy, M. A., Wintle, B. A. (2008) When have we looked hard enough? A novel method for setting minimum survey effort protocols for flora surveys. Austral Ecology 33, 986–998.

Garrard, G. E., McCarthy, M. A., Williams, N. S., Bekessy, S. A., Wintle, B. A. (2013) A general model of detectability using species traits. Methods in Ecology and Evolution 4, 45–52.

See Also

countDist, countHist, detHist, Nmix.chisq, Nmix.gof.test

Examples

##example from ?occuTTD
## Not run: 
if(require(unmarked)){
N <- 500; J <- 1
     
##Simulate occupancy
scovs <- data.frame(elev=c(scale(runif(N, 0,100))),
                    forest=runif(N,0,1),
                    wind=runif(N,0,1))
     
beta_psi <- c(-0.69, 0.71, -0.5)
psi <- plogis(cbind(1, scovs$elev, scovs$forest) 
z <- rbinom(N, 1, psi)
     
##Simulate detection
Tmax <- 10 #Same survey length for all observations
beta_lam <- c(-2, -0.2, 0.7)
rate <- exp(cbind(1, scovs$elev, scovs$wind) 
ttd <- rexp(N, rate)
ttd[z==0] <- Tmax #Censor unoccupied sites
ttd[ttd>Tmax] <- Tmax #Censor when ttd was greater than survey length
     
##Build unmarkedFrame
umf <- unmarkedFrameOccuTTD(y=ttd, surveyLength=Tmax, siteCovs=scovs)

##compute descriptive stats from data object
detTime(umf)
     
##Fit model
fit.occuTTD <- occuTTD(psiformula=~elev+forest, detformula=~elev+wind, data=umf)
##extract info from model object
detTime(fit.occuTTD)     
     
}

## End(Not run)

Computing DIC

Description

Functions to extract deviance information criterion (DIC).

Usage

DIC(mod, return.pD = FALSE, ...) 

## S3 method for class 'bugs'
DIC(mod, return.pD = FALSE, ...)

## S3 method for class 'rjags'
DIC(mod, return.pD = FALSE, ...)

## S3 method for class 'jagsUI'
DIC(mod, return.pD = FALSE, ...)

Arguments

mod

an object of class bugs, rjags, or jagsUI containing the output of a model.

return.pD

logical. If FALSE, the function returns the DIC. If TRUE, the function returns the effective number of estimated parameters (pD) for a given model.

...

additional arguments passed to the function.

Details

DIC is implemented for bugs, rjags, and jagsUI classes. The function extracts the deviance information criterion (DIC, Spiegelhalter et al. 2002) or the effective number of parameters (pD).

Value

DIC the DIC or pD depending on the values of the arguments.

Note

The actual DIC values are not really interesting in themselves, as they depend directly on the data, parameters estimated, and likelihood function. Furthermore, a single value does not tell much about model fit. Information criteria become relevant when compared to Yone another for a given data set and set of candidate models. Model selection with hierarchical models is problematic as the classic DIC is not appropriate for such types of models (Millar 2009).

Author(s)

Marc J. Mazerolle

References

Millar, R. B. (2009) Comparison of hierarchical Bayesian models for overdispersed count data using DIC and Bayes' factors. Biometrics, 65, 962–969.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P., van der Linde, A. (2002). Bayesian measures of complexity and fit. Journal of the Royal Statistical Society, Series B 64, 583–639.

See Also

AICcCustom, aictab, dictab, confset, evidence

Examples

##from ?jags example in R2jags package
## Not run: 
require(R2jags)
##example model file
model.file <- system.file(package="R2jags", "model", "schools.txt")
file.show(model.file)     

##data
J <- 8.0
y <- c(28.4,7.9,-2.8,6.8,-0.6,0.6,18.0,12.2)
sd <- c(14.9,10.2,16.3,11.0,9.4,11.4,10.4,17.6)
     
##arrange data in list     
jags.data <- list (J = J, y = y, sd = sd)

##initial values
jags.inits <- function(){
  list(theta=rnorm(J, 0, 100), mu=rnorm(1, 0, 100),
       sigma=runif(1, 0, 100))
}

##parameters to be monitored
jags.parameters <- c("theta", "mu", "sigma")
     
##run model
schools.sim <- jags(data = jags.data, inits = jags.inits,
                    parameters = jags.parameters,
                    model.file = model.file,
                    n.chains = 3, n.iter = 10)
##note that n.iter should be higher

##extract DIC
DIC(schools.sim)
##extract pD
DIC(schools.sim, return.pD = TRUE)
detach(package:R2jags)

## End(Not run)

Create Model Selection Tables from Bayesian Analyses

Description

This function creates a model selection table based on the deviance information criterion (DIC). The table ranks the models based on the DIC and also provides delta DIC and DIC weights. dictab selects the appropriate function to create the model selection table based on the object class. The current version works with objects of bugs, rjags, jagsUI classes.

Usage

dictab(cand.set, modnames = NULL, sort = TRUE, ...)

## S3 method for class 'AICbugs'
dictab(cand.set, modnames = NULL, sort = TRUE, ...)

## S3 method for class 'AICrjags'
dictab(cand.set, modnames = NULL, sort = TRUE, ...)

## S3 method for class 'AICjagsUI'
dictab(cand.set, modnames = NULL, sort = TRUE, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

sort

logical. If TRUE, the model selection table is ranked according to the DIC values.

...

additional arguments passed to the function.

Details

dictab internally creates a new class for the cand.set list of candidate models, according to the contents of the list. The current function is implemented for bugs, jags, jagsUI classes. The function constructs a model selection table based on the DIC (Spiegelhalter et al. 2002). Note that DIC might not be appropriate to select among a set of hierarchical models and that modifications to the information criterion have been proposed (Millar 2009).

Value

dictab creates an object of class dictab with the following components:

Modname

the name of each model of the candidate model set.
pD

the effective number of estimated parameters for each model.
DIC

the deviance information criterion for each model.
Delta_DIC

the delta DIC of each model, measuring the difference in DIC between each model and the top-ranked model.
ModelLik

the relative likelihood of the model given the data (exp(-0.5*delta[i])). This is not to be confused with the likelihood of the parameters given the data. The relative likelihood can then be normalized across all models to get the model probabilities.
DICWt

the DIC weights, sensu Burnham and Anderson (2002) and Anderson (2008). These measures indicate the level of support (i.e., weight of evidence) in favor of any given model being the most parsimonious among the candidate model set.
Cum.Wt

the cumulative DIC weights. These are only meaningful if results in table are sorted in decreasing order of DIC weights (i.e., sort = TRUE).
Deviance

the deviance of each model.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P., van der Linde, A. (2002). Bayesian measures of complexity and fit. Journal of the Royal Statistical Society, Series B 64, 583–639.

See Also

aictabCustom, aictab, confset, DIC, evidence

Examples

##from ?jags example in R2jags package
## Not run: 
require(R2jags)
model.file <- system.file(package="R2jags", "model", "schools.txt")
file.show(model.file)     

##data
J <- 8.0
y <- c(28.4,7.9,-2.8,6.8,-0.6,0.6,18.0,12.2)
sd <- c(14.9,10.2,16.3,11.0,9.4,11.4,10.4,17.6)
     
     
jags.data <- list (J = J, y = y, sd = sd)
jags.inits <- function(){
  list(theta=rnorm(J, 0, 100), mu=rnorm(1, 0, 100),
       sigma=runif(1, 0, 100))
}
jags.parameters <- c("theta", "mu", "sigma")
     
##run model
schools.sim <- jags(data = jags.data, inits = jags.inits,
                    parameters = jags.parameters,
                    model.file = model.file,
                    n.chains = 3, n.iter = 10)
#note that n.iter should be higher

##set up in list
Cand.mods <- list(schools.sim)
Model.names <- "hierarchical model"
##other models can be added to Cand.mods
##to compare them to the top model

##model selection table
dictab(cand.set = Cand.mods, modnames = Model.names)
detach(package:R2jags)

## End(Not run)

Frog Dehydration Experiment on Three Substrate Types

Description

This is a data set modified from Mazerolle and Desrochers (2005) on the mass lost by frogs after spending two hours on one of three substrates that are encountered in some landscape types.

Usage

data(dry.frog)

Format

A data frame with 121 observations on the following 16 variables.

Individual

a numeric identifier unique to each individual.

Species

a factor with levels Racla.

Shade

a numeric vector, either 1 (shade) or 0 (no shade).

SVL

the snout-vent length of the individual.

Substrate

the substrate type, a factor with levels PEAT, SOIL, and SPHAGNUM.

Initial_mass

the initial mass of individuals.

Mass_lost

the mass lost in g.

Airtemp

the air temperature in degrees C.

Wind_cat

the wind intensity, either 0 (no wind), 1 (low wind), 2 (moderate wind), or 3 (strong wind).

Cloud

cloud cover expressed as a percentage.

cent_Initial_mass

centered inital mass.

Initial_mass2

initial mass squared.

cent_Air

centered air temperature.

Perc.cloud

proportion of cloud cover

Wind

wind intensity, either 1 (no or low wind) or 1 (moderate to strong wind).

log_Mass_lost

log of mass lost.

Details

Note that the original analysis in Mazerolle and Desrochers (2005) consisted of generalized estimating equations for three mass measurements: mass at time 0, 1 hour, and 2 hours following exposure on the substrate.

Source

Mazerolle, M. J., Desrochers, A. (2005) Landscape resistance to frog movements. Canadian Journal of Zoology 83, 455–464.

Examples

data(dry.frog)
## maybe str(dry.frog) ; plot(dry.frog) ...

Compute Evidence Ratio Between Two Models

Description

This function compares two models of a candidate model set based on their evidence ratio (i.e., ratio of model weights). The default computes the evidence ratio of the model weights between the top-ranked model and the second-ranked model. You must supply a model selection table of class aictab, bictab, boot.wt, dictab, ictab as the first argument.

Usage

evidence(aic.table, model.high = "top", model.low = "second.ranked")

Arguments

aic.table

a model selection table of class aictab such as that produced by aictab or of classes bictab, boot.wt, dictab, or ictab. The table may be sorted or not, as the function sorts the table internally.

model.high

the top-ranked model (default), or alternatively, the name of another model as it appears in the model selection table.

model.low

the second-ranked model (default), or alternatively, the name of a lower-ranked model such as it appears in the model selection table.

Details

The default compares the model weights of the top-ranked model to the second-ranked model in the candidate model set. The evidence ratio can be interpreted as the number of times a given model is more parsimonious than a lower-ranked model. If one desires an evidence ratio that does not involve a comparison with the top-ranking model, the label of the required model must be specified in the model.high argument as it appears in the model selection table.

Value

evidence produces an object of class evidence with the following components:

Model.high

the model specified in model.high.

Model.low

the model specified in model.low.

Ev.ratio

the evidence ratio between the two models compared.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

See Also

AICc, aictab, bictab, c_hat, confset, importance, modavg, modavgShrink, modavgPred

Examples

##run example from Burnham and Anderson (2002, p. 183) with two
##non-nested models
data(pine)
Cand.set <- list( )
Cand.set[[1]] <- lm(y ~ x, data = pine)
Cand.set[[2]] <- lm(y ~ z, data = pine)

##assign model names
Modnames <- c("raw density", "density corrected for resin content")

##compute model selection table
aicctable.out <- aictab(cand.set = Cand.set, modnames = Modnames)

##compute evidence ratio
evidence(aic.table = aicctable.out, model.low = "raw density")
evidence(aic.table = aicctable.out) #gives the same answer
##round to 4 digits after decimal point
print(evidence(aic.table = aicctable.out, model.low = "raw density"),
      digits = 4)


##example with bictab
## Not run: 
##compute model selection table
bictable.out <- bictab(cand.set = Cand.set, modnames = Modnames)
##compute evidence ratio
evidence(bictable.out, model.low = "raw density")

## End(Not run)


##run models for the Orthodont data set in nlme package
## Not run: 
require(nlme)

##set up candidate model list
Cand.models <- list()
Cand.models[[1]] <- lme(distance ~ age, data = Orthodont, method = "ML")
##random is ~ age | Subject
Cand.models[[2]] <- lme(distance ~ age + Sex, data = Orthodont,
                        random = ~ 1, method = "ML")
Cand.models[[3]] <- lme(distance ~ 1, data = Orthodont, random = ~ 1,
                        method = "ML")

##create a vector of model names
Modnames <- paste("mod", 1:length(Cand.models), sep = " ")

##compute AICc table
aic.table.1 <- aictab(cand.set = Cand.models, modnames = Modnames,
                      second.ord = TRUE)

##compute evidence ratio between best model and second-ranked model
evidence(aic.table = aic.table.1)

##compute the same value but from an unsorted model selection table
evidence(aic.table = aictab(cand.set = Cand.models,
           modnames = Modnames, second.ord = TRUE, sort = FALSE))

##compute evidence ratio between second-best model and third-ranked
##model  
evidence(aic.table = aic.table.1, model.high = "mod1",
         model.low = "mod3")
detach(package:nlme)

## End(Not run)

Compute Condition Number

Description

This function computes the condition number for models of unmarkedFit classes as the ratio of the largest eigenvalue of the Hessian matrix to the smallest eigenvalue of the Hessian matrix.

Usage

extractCN(mod, method = "svd", ...)

## S3 method for class 'unmarkedFit'
extractCN(mod, method = "svd", ...)

Arguments

mod

a model of one the unmarkedFit classes for which a condition number is requested.

method

specifies the method used to extract the singular values or eigenvalues from the Hessian matrix using singular value decomposition (method = "svd") or eigenvalue decomposition (method = "eigen").

...

additional arguments passed to the function.

Details

The condition number (κ\kappa) is a measure of the transfer of error to the solution in response to small changes in the input (Cheney and Kincaid 2008). In this implementation, the condition number is computed on the Hessian matrix of models of unmarkedFit classes from the optim results stored in the model object. The condition number is defined as the ratio of the largest to the smallest non-negative singular values of a given matrix (Cline et al. 1979, Dixon 1983). In the special case of positive semi-definite matrices, the singular values are equal to the eigenvalues (Ruhe 1975).

Large values of the condition number may indicate problems in estimating parameters or their variance (ill-conditioning), possibly due to a model having too many parameters for the given data set. Cheney and Ward (2008) suggest using the log⁡10(κ)\log_{10}(\kappa) of the condition number as a crude estimate of the number of digits of precision lost.

Value

extractCN returns a list of class extractCN with the following components:

CN

the condition number (κ\kappa) of the model.

log10

the log base 10 of the condition number.

method

the method used to extract the singular values or eigenvalues.

Author(s)

Marc J. Mazerolle

References

Cheney, W., Kincaid, D. (2008) Numerical mathematics and computing. Sixth edition. Thomson Brooks/Cole: Belmont.

Cline, A. K., Moler, C. B., Stewart, G. W., Wilkinson, J. H. (1979) An estimate for the condition number of a matrix. SIAM Journal on Numerical Analysis 16, 368–375.

Dixon, J. D. (1983) Estimating extremal eigenvalues and condition numbers of matrices. SIAM Journal on Numerical Analysis 20, 812–814.

Ruhe, A. (1975) On the closeness of eigenvalues and singular values for almost normal matrices. Linear Algebra and its Applications 11, 87–94.

See Also

c_hat, mb.gof.test, Nmix.gof.test, parboot, kappa, rcond

Examples

##N-mixture model example modified from ?pcount
## Not run: 
require(unmarked)
##single season
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
##run model
fm.mallard <- pcount(~ ivel+ date + I(date^2) ~ length + elev + forest,
                     mallardUMF, K=30)

##compute condition number
extractCN(fm.mallard)

##compare against 'kappa'
kappa(fm.mallard@opt$hessian, exact = TRUE)
detach(package:unmarked)

## End(Not run)

Extract Log-Likelihood of Model

Description

This function extracts the log-likelihood from an object of coxme, coxph, lmekin, maxlikeFit, vglm, or various unmarkedFit classes.

Usage

extractLL(mod, ...)

## S3 method for class 'coxme'
extractLL(mod, type = "Integrated", ...)

## S3 method for class 'coxph'
extractLL(mod, ...)

## S3 method for class 'lmekin'
extractLL(mod, ...)

## S3 method for class 'maxlikeFit'
extractLL(mod, ...)

## S3 method for class 'unmarkedFit'
extractLL(mod, ...)

## S3 method for class 'vglm'
extractLL(mod, ...)

Arguments

mod

an object of coxme, coxph, lmekin, maxlikeFit, vglm, or unmarkedFit class resulting from the fit of distsamp, gdistsamp, gmultmix, multinomPois, gpcount, occu, occuRN, colext, pcount, or pcountOpen.
...

additional arguments passed to the function.
type

a character string indicating whether the integrated partial likelihood ("Integrated") or penalized likelihood ("Penalized") is to be used for a coxme object.

Details

This utility function extracts the information from a coxme, coxph, lmekin, maxlikeFit, vglm, or unmarkedFit object resulting from distsamp, gdistsamp, gmultmix, multinomPois, gpcount, occu, occuRN, colext, pcount, or pcountOpen.

Value

These functions return the value of the log-likelihood of the model and associated degrees of freedom.

Author(s)

Marc J. Mazerolle

See Also

AICc, aictab, coxme, coxph, lmekin, maxlike, distsamp, gdistsamp, occu, occuRN, colext, pcount, pcountOpen

Examples

##single-season occupancy model example modified from ?occu
## Not run: 
require(unmarked)
##single season
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
## add some fake covariates for illustration
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)),
                                sitevar2 = rnorm(numSites(pferUMF))) 
     
## observation covariates are in site-major, observation-minor order
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) *
                                 obsNum(pferUMF))) 

##run model set
fm1 <- occu(~ obsvar1 ~ sitevar1, pferUMF)

##extract log-likelihood
extractLL(fm1)
detach(package:unmarked)

## End(Not run)

Extract SE of Fixed Effects

Description

This function extracts the standard errors (SE) of the fixed effects of a mixed model fit with coxme, glmer, lmer, lmerModLmerTest, and lmekin and adds the appropriate labels.

Usage

extractSE(mod, ...)

## S3 method for class 'coxme'
extractSE(mod, ...)

## S3 method for class 'lmekin'
extractSE(mod, ...)

## S3 method for class 'mer'
extractSE(mod, ...)

## S3 method for class 'merMod'
extractSE(mod, ...)

## S3 method for class 'lmerModLmerTest'
extractSE(mod, ...)

Arguments

mod

an object of coxme, lmekin, mer, merMod, or lmerModTest class.

...

additional arguments passed to the function.

Details

These extractor functions use vcov.coxme, vcov.lmekin, vcov.mer, and vcov.merMod. Some of these functions are called by modavg and modavgShrink, depending on the class of the objects.

Value

Returns the SE's of the fixed effects with the appropriate labels for each.

Author(s)

Marc J. Mazerolle

See Also

modavg, glmer, lmer, coxme, lmekin

Examples

##modified example from ?glmer
## Not run: 
if(require(lme4)) {
##create proportion of incidence
cbpp$prop <- cbpp$incidence/cbpp$size
gm1 <- glmer(prop ~ period + (1 | herd), family = binomial,
             weights = size, data = cbpp)
##print summary
summary(gm1)
##extract variance-covariance matrix of fixed effects
vcov(gm1)
##extract SE's of fixed effects - no labels
sqrt(diag(vcov(gm1))) #no labels
extractSE(gm1)  #with labels
detach(package:lme4)
}

## End(Not run)

Extract Predictors from Candidate Model List

Description

This function extracts the predictors used in candidate models. The function is currently implemented for glm, glmmTMB, gls, lm, lme, merMod, lmerModLmerTest, rlm, survreg object classes that are stored in a list as well as various models of unmarkedFit classes.

Usage

extractX(cand.set, ...)

## S3 method for class 'AICaov.lm'
extractX(cand.set, ...) 

## S3 method for class 'AICglm.lm'
extractX(cand.set, ...)

## S3 method for class 'AICglmmTMB'
extractX(cand.set, ...)

## S3 method for class 'AIClm'
extractX(cand.set, ...) 

## S3 method for class 'AICgls'
extractX(cand.set, ...) 

## S3 method for class 'AIClme'
extractX(cand.set, ...)

## S3 method for class 'AICglmerMod'
extractX(cand.set, ...)

## S3 method for class 'AIClmerMod'
extractX(cand.set, ...)

## S3 method for class 'AIClmerModLmerTest'
extractX(cand.set, ...)

## S3 method for class 'AICrlm.lm'
extractX(cand.set, ...) 

## S3 method for class 'AICsurvreg'
extractX(cand.set, ...) 

## S3 method for class 'AICunmarkedFitOccu'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitColExt'
extractX(cand.set,
        parm.type = NULL, ...)  

## S3 method for class 'AICunmarkedFitOccuRN'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCount'
extractX(cand.set,
        parm.type = NULL, ...) 

## S3 method for class 'AICunmarkedFitPCO'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDS'
extractX(cand.set,
        parm.type = NULL, ...) 

## S3 method for class 'AICunmarkedFitGDS'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMPois'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGMM'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGPC'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMMO'
extractX(cand.set,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDSO'
extractX(cand.set,
        parm.type = NULL, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

parm.type

this argument specifies the parameter type on which the predictors will be extracted and is only relevant for models of unmarkedFit classes. The character strings supported vary with the type of model fitted. For unmarkedFitOccu and unmarkedFitOccuMulti objects, either psi or detect can be supplied to indicate whether the parameter is on occupancy or detectability, respectively. For unmarkedFitColExt and unmarkedFitOccuTTD, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and detectability, respectively. For unmarkedFitOccuFP objects, one can specify psi, detect, falsepos, and certain, for occupancy, detectability, probability of assigning false-positives, and probability detections are certain, respectively. For unmarkedFitOccuMS objects, possible values are psi, phi, or detect, denoting occupancy, transition, and detection probabilities, respectively. For unmarkedFitOccuRN objects, either lambda or detect can be entered for abundance and detectability parameters, respectively. For unmarkedFitPCount and unmarkedFitMPois objects, lambda or detect denote parameters on abundance and detectability, respectively. For unmarkedFitPCO, unmarkedFitMMO, and unmarkedFitDSO objects, one can enter lambda, gamma, omega, iota, or detect, to specify parameters on abundance, recruitment, apparent survival, immigration, and detectability, respectively. For unmarkedFitDS objects, lambda and detect are supported. For unmarkedFitGDS, lambda, phi, and detect denote abundance, availability, and detection probability, respectively. For unmarkedFitGMM and unmarkedFitGPC objects, lambda, phi, and detect denote abundance, availability, and detectability, respectively.

...

additional arguments passed to the function.

Details

The candidate models must be stored in a list. The results of extractX are useful in preparing a newdata data frame to use in computing model-averaged predictions with modavgPred or differences between groups with modavgEffect (Burnham and Anderson 2002, Anderson 2008, Burnham et al. 2011).

Value

extractX returns an object of class extractX with the following components:

predictors

a character vector of the names of the predictors included in the model, excluding the intercept term.

data

a data frame or, in the case of unmarkedFit objects, a list of data frames (e.g., obsCovs, siteCovs, yearlySiteCovs).

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R., Huyvaert, K. P. (2011) AIC model selection and multimodel inference in behaviorial ecology: some background, observations and comparisons. Behavioral Ecology and Sociobiology 65, 23–25.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Pinheiro, J. C., Bates, D. M. (2000). Mixed-effects Models in S and S-PLUS. Springer Verlag: New York.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

extractCN, extractSE, modavgPred, modavgCustom, modavgEffect, predict, predictSE

Examples

##example from subset of models in Table 1 in Mazerolle (2006)
data(dry.frog)

Cand.models <- list( )
Cand.models[[1]] <- lm(log_Mass_lost ~ Shade + Substrate +
                           cent_Initial_mass + Initial_mass2,
                       data = dry.frog)
Cand.models[[2]] <- lm(log_Mass_lost ~ Shade + Substrate +
                           cent_Initial_mass + Initial_mass2 +
                           Shade:Substrate, data = dry.frog)
Cand.models[[3]] <- lm(log_Mass_lost ~ cent_Initial_mass +
                           Initial_mass2, data = dry.frog)
Cand.models[[4]] <- lm(log_Mass_lost ~ Shade + cent_Initial_mass +
                           Initial_mass2, data = dry.frog)
Cand.models[[4]] <- lm(log_Mass_lost ~ Shade + cent_Initial_mass +
                           Initial_mass2, data = dry.frog)
Cand.models[[5]] <- lm(log_Mass_lost ~ Substrate + cent_Initial_mass +
                           Initial_mass2, data = dry.frog)
##assign names
names(Cand.models) <- paste(1:length(Cand.models))

##extract predictors from candidate model set
orig.data <- extractX(cand.set = Cand.models)
orig.data
str(orig.data)

## Not run: 
##model-averaged prediction with original variables
modavgPred(Cand.models, newdata = orig.data$data)

## End(Not run)


##example of model-averaged predictions from N-mixture model (e.g., Royle 2004)
##modified from ?pcount
##each variable appears twice on lambda in the models
## Not run: 
require(unmarked)    
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
##set up models so that each variable on abundance appears twice
fm.mall.one <- pcount(~ ivel + date  ~ length + forest, mallardUMF,
                      K = 30)
fm.mall.two <- pcount(~ ivel + date  ~ elev + forest, mallardUMF,
                      K = 30)
fm.mall.three <- pcount(~ ivel + date  ~ length + elev, mallardUMF,
                        K = 30)
fm.mall.four <- pcount(~ ivel + date  ~ 1, mallardUMF, K = 30)

##model list
Cands <- list(fm.mall.one, fm.mall.two, fm.mall.three, fm.mall.four)
names(Cands) <- c("length + forest", "elev + forest", "length + elev",
                  "null")

##extract predictors on lambda
lam.dat <- extractX(cand.set = Cands, parm.type = "lambda")
lam.dat
str(lam.dat)

##extract predictors on detectability
extractX(cand.set = Cands, parm.type = "detect")

##model-averaged predictions on lambda
##extract data
siteCovs <- lam.dat$data$siteCovs
##create vector of forest values
forest <- seq(min(siteCovs$forest),
              max(siteCovs$forest),
              length.out = 40)
dframe <- data.frame(forest = forest,
                     length = mean(siteCovs$length),
                     elev = mean(siteCovs$elev))
modavgPred(Cands, parm.type = "lambda",
           newdata = dframe)
detach(package:unmarked)

## End(Not run)


##example of model-averaged abundance from distance model
## Not run: 
require(unmarked)
data(linetran) #example from ?distsamp
     
ltUMF <- with(linetran, {
  unmarkedFrameDS(y = cbind(dc1, dc2, dc3, dc4),
                  siteCovs = data.frame(Length, area, habitat),
                  dist.breaks = c(0, 5, 10, 15, 20),
                  tlength = linetran$Length * 1000, survey = "line",
                  unitsIn = "m")
})
     
## Half-normal detection function. Density output (log scale). No covariates.
fm1 <- distsamp(~ 1 ~ 1, ltUMF)
     
## Halfnormal. Covariates affecting both density and and detection.
fm2 <- distsamp(~area + habitat ~ habitat, ltUMF)

## Hazard function. Covariates affecting both density and and detection.
fm3 <- distsamp(~area + habitat ~ habitat, ltUMF, keyfun="hazard")

##assemble model list
Cands <- list(fm1, fm2, fm3)

##model-average predictions on abundance
extractX(cand.set = Cands, parm.type = "lambda")
detach(package:unmarked)

## End(Not run)



##example using Orthodont data set from Pinheiro and Bates (2000)
## Not run: 
require(nlme)

##set up candidate models
m1 <- gls(distance ~ age, correlation = corCompSymm(value = 0.5, form = ~ 1 | Subject),
          data = Orthodont, method = "ML")

m2 <- gls(distance ~ 1, correlation = corCompSymm(value = 0.5, form = ~ 1 | Subject),
          data = Orthodont, method = "ML")

##assemble in list
Cand.models <- list("age effect" = m1, "null model" = m2)

##model-averaged predictions
extractX(cand.set = Cand.models)
detach(package:nlme)

## End(Not run)

Extract Distribution Family and Link Function

Description

This function extracts the distribution family and link function of a generalized linear mixed model fit with glmer or lmer.

Usage

fam.link.mer(mod)

Arguments

mod

an object of mer or merMod class resulting from the fit of glmer or lmer.

Details

This utility function extracts the information from an mer or merMod object resulting from glmer or lmer. The function is called by modavg, modavgEffect, modavgPred, and predictSE.

Value

fam.link.mer returns a list with the following components:

family

the family of the distribution of the model.
link

the link function of the model.
supp.link

a character value indicating whether the link function used is supported by predictSE and modavgPred.

Author(s)

Marc J. Mazerolle

See Also

modavg, modavgPred, predictSE, glmer, lmer

Examples

##modified example from ?glmer
## Not run: 
if(require(lme4)){
##create proportion of incidence
cbpp$prop <- cbpp$incidence/cbpp$size
gm1 <- glmer(prop ~ period + (1 | herd), family = binomial,
             weights = size, data = cbpp)
fam.link.mer(gm1)
gm2 <- glmer(prop ~ period + (1 | herd),
             family = binomial(link = "cloglog"), weights = size,
             data = cbpp)
fam.link.mer(gm2)
}

## End(Not run)


##example with linear mixed model with Orthodont data from
##Pinheiro and Bates (2000)
## Not run: 
data(Orthodont, package = "nlme")
m1 <- lmer(distance ~ Sex + (1 | Subject), data = Orthodont,
           REML = FALSE)
fam.link.mer(m1)
m2 <- glmer(distance ~ Sex + (1 | Subject),
            family = gaussian(link = "log"), data = Orthodont,
            REML = FALSE)
fam.link.mer(m2)
detach(package:lme4)

## End(Not run)

Fat Data and Body Measurements

Description

This data set illustrates the relationship between body measurements and body fat in 252 males aged between 21 and 81 years.

Usage

data(fat)

Format

A data frame with 252 rows and 26 variables.

Obs

observation number.

Perc.body.fat.Brozek

percent body fat using Brozek's equation, i.e., 457/Density−414.2457/Density - 414.2.

Perc.body.fat.Siri

percent body fat using Siri's equation, i.e., 495/Density−450495/Density - 450.

Density

density (gcm3\frac{g}{cm^3}).

Age

age (years).

Weight

weight (lbs).

Height

height (inches).

Adiposity.index

adiposity index computed as Weight/Height2Weight/Height^2 (kgm2\frac{kg}{m^2}).

Fat.free.weight

fat free weight computed as (1−Brozek′spercentbodyfat)∗Weight(1 - Brozek's percent body fat) * Weight (lbs).

Neck.circ

neck circumference (cm).

Chest.circ

chest circumference (cm).

Abdomen.circ

abdomen circumference (cm) measured at the umbilicus and level with the iliac crest.

Hip.circ

hip circumference (cm).

Thigh.circ

thigh circumference (cm).

Knee.circ

knee circumference (cm).

Ankle.circ

ankle circumference (cm).

Biceps.circ

extended biceps circumference (cm).

Forearm.circ

forearm circumference (cm).

Wrist.circ

wrist circumference (cm).

inv.Density

inverse of density (cm3g\frac{cm^3}{g}).

z1

log of weight divided by log of height (allometric measure).

z2

abdomen circumference divided by chest circumference (beer gut factor).

z3

index based on knee, wrist, and ankle circumference relative to height ((Knee.circ∗Wrist.circ∗Ankle.circ)(1/3)Height\frac{(Knee.circ * Wrist.circ * Ankle.circ)^(1/3)}{Height}).

z4

fleshiness index based on biceps, thigh, forearm, knee, wrist, and ankle circumference (Biceps.circ∗Thigh.circ∗Forearm.circKnee.circ∗Wrist.circ∗Ankle.circ\frac{Biceps.circ * Thigh.circ * Forearm.circ}{Knee.circ * Wrist.circ * Ankle.circ}).

z5

age standardized to zero mean and unit variance.

z6

square of standardized age.

Details

Burnham and Anderson (2002, p. 268) use this data set to show model selection uncertainty in the context of all possible combinations of explanatory variables. The data are originally from Penrose et al. (1985) who used only the first 143 cases of the 252 observations in the data set. Johnson (1996) later used these data as an example of multiple regression. Note that observation number 42 originally had an erroneous height of 29.5 inches and that this value was changed to 69.5 inches.

Burnham and Anderson (2002, p. 274) created six indices based on the original measurements (i.e., z1 – z6). Although Burnham and Anderson (2002) indicate that the fleshiness index (z4) involved the cubic root in the equation, the result table for the full model on p. 276 suggests that the index did not include the cubic root for z4. The latter is the version of z4 used in the data set here.

Source

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Johnson, J. W. (1996). Fitting percentage of body fat to simple body measurements. Journal of Statistics Education 4 [online].

Penrose, K., Nelson, A., Fisher, A. (1985) Generalized body composition prediction equation for men using simple measurement techniques Medicine and Science in Sports and Exercise 17, 189.

Examples

data(fat)
str(fat)

GPA Data and Standardized Test Scores

Description

This data set features the first-year college GPA and four standardized tests conducted before matriculation.

Usage

data(gpa)

Format

A data frame with 20 rows and 5 variables.

gpa.y

first-year GPA.

sat.math.x1

SAT math score.

sat.verb.x2

SAT verbal score.

hs.math.x3

high school math score.

hs.engl.x4

high school English score.

Details

Burnham and Anderson (2002, p. 225) use this data set originally from Graybill and Iyer (1994) to show model selection for all subsets regression.

Source

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Graybill, F. A., Iyer, H. K. (1994) Regression analysis: concepts and applications. Duxbury Press: Belmont.

Examples

data(gpa)
str(gpa)

Create Model Selection Tables from User-supplied Information Criterion

Description

This function creates a model selection table from information criterion values supplied by the user. The table ranks the models based on the values of the information criterion and also displays delta values and information criterion weights.

Usage

ictab(ic, K, modnames = NULL, sort = TRUE, ic.name = NULL)

Arguments

ic

a vector of information criterion values for each model in the candidate model set.

K

a vector containing the number of estimated parameters for each model in the candidate model set.

modnames

a character vector of model names to identify each model in the model selection table. If NULL, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as the information criterion values.

sort

logical. If TRUE, the model selection table is ranked according to the values of the information criterion.

ic.name

a character string denoting the name of the information criterion input by the user. This character string will appear in certain column labels of the model selection table.

Details

ictab constructs a model selection table based on the information criterion values supplied by the user. This function is most useful for information criterion other than AIC, AICc, QAIC, and QAICc (e.g., WAIC: Watanabe 2010) or for classes not supported by aictab or bictab.

Value

ictab creates an object of class ictab with the following components:

Modname

the name of each model of the candidate model set.
K

the number of estimated parameters for each model.
IC

the values of the information criterion input by the user. If a value for ic.name is provided, the table modifies the labels of the table.
Delta_IC

the delta information criterion component comparing each model to the top-ranked model.
ModelLik

the relative likelihood of the model given the data (exp(-0.5*delta[i])). This is not to be confused with the likelihood of the parameters given the data. The relative likelihood can then be normalized across all models to get the model probabilities.
ICWt

the information criterion weights, also termed "model probabilities" sensu Burnham and Anderson (2002) and Anderson (2008). These measures indicate the level of support (i.e., weight of evidence) in favor of any given model being the most parsimonious among the candidate model set.
Cum.Wt

the cumulative information criterion weights. These are only meaningful if results in table are sorted in decreasing order of Akaike weights (i.e., sort = TRUE).

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Watanabe, S. (2010) Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research 11, 3571–3594.

See Also

aictabCustom, confset, evidence, modavgCustom, modavgIC

Examples

##create a vector of names to trace back models in set
Modnames <- c("global model", "interactive model",
              "additive model", "invertpred model")

##WAIC values
waic <- c(105.74, 107.36, 108.24, 100.57)
##number of effective parameters
effK <- c(7.45, 5.61, 6.14, 6.05)

##generate WAIC table
ictab(ic = waic, K = effK, modnames = Modnames,
      sort = TRUE, ic.name = "WAIC")

Compute Importance Values of Variable

Description

This function calculates the relative importance of variables (w+) based on the sum of Akaike weights (model probabilities) of the models that include the variable. Note that this measure of evidence is only appropriate when the variable appears in the same number of models as those that do not include the variable.

Usage

importance(cand.set, parm, modnames = NULL, second.ord = TRUE,
           nobs = NULL, ...)

## S3 method for class 'AICaov.lm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICbetareg'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICsclm.clm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICclm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICclmm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICclogit.coxph'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICcoxme'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICcoxph'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICglm.lm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, ...)

## S3 method for class 'AICglmerMod'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AIClmerModLmerTest'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICglmmTMB'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, ...)

## S3 method for class 'AICgls'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AIClm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AIClme'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AIClmekin'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICmaxlikeFit.list'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, ...)

## S3 method for class 'AICmer'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICmultinom.nnet'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, ...)

## S3 method for class 'AICnegbin.glm.lm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICnlmerMod'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICpolr'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICrlm.lm'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICsurvreg'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, ...)

## S3 method for class 'AICunmarkedFitColExt'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccu'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuRN'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCount'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCO'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDS'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGDS'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMPois'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGMM'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGPC'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL,
       ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL,
       ...)

## S3 method for class 'AICunmarkedFitMMO'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL,
       ...)

## S3 method for class 'AICunmarkedFitDSO'
importance(cand.set, parm, modnames = NULL,
           second.ord = TRUE, nobs = NULL, c.hat = 1, parm.type = NULL,
       ...)

## S3 method for class 'AICvglm'
importance(cand.set, parm, modnames = NULL, 
           second.ord = TRUE, nobs = NULL, c.hat = 1, ...)

## S3 method for class 'AICzeroinfl'
importance(cand.set, parm, modnames = NULL, 
           second.ord = TRUE, nobs = NULL, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

parm

the parameter of interest for which a measure of relative importance is required.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, importance will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

parm.type

this argument specifies the parameter type on which the variable of interest will be extracted and is only relevant for models of unmarkedFit classes. The character strings supported vary with the type of model fitted. For unmarkedFitOccu and unmarkedFitOccuMulti objects, either psi or detect can be supplied to indicate whether the parameter is on occupancy or detectability, respectively. For unmarkedFitColExt objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and detectability, respectively. For unmarkedFitOccuTTD objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and time-to-dection (lambda rate parameter), respectively. For unmarkedFitOccuFP objects, one can specify psi, detect, falsepos, and certain, for occupancy, detectability, probability of assigning false-positives, and probability detections are certain, respectively. For unmarkedFitOccuMS objects, possible values are psi, phi, or detect, denoting occupancy, transition, and detection probabilities, respectively. For unmarkedFitOccuRN objects, either lambda or detect can be entered for abundance and detectability parameters, respectively. For unmarkedFitPCount and unmarkedFitMPois objects, lambda or detect denote parameters on abundance and detectability, respectively. For unmarkedFitPCO, unmarkedFitMMO, and unmarkedFitDSO objects, one can enter lambda, gamma, omega, iota, or detect, to specify parameters on abundance, recruitment, apparent survival, immigration, and detectability, respectively. For unmarkedFitDS objects, lambda and detect are supported. For unmarkedFitGDS, lambda, phi, and detect denote abundance, availability, and detection probability, respectively. For unmarkedFitGMM and unmarkedFitGPC objects, lambda, phi, and detect denote abundance, availability, and detectability, respectively.

...

additional arguments passed to the function.

Value

importance returns an object of class importance consisting of the following components:

parm

the parameter for which an importance value is required.
w.plus

the sum of Akaike weights for the models that include the parameter of interest.
w.minus

the sum of Akaike weights for the models that exclude the parameter of interest.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., and Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICc, aictab, c_hat, confset, evidence, modavg, modavgShrink, modavgPred

Examples

##example on Orthodont data set in nlme
## Not run: 
require(nlme)

##set up candidate model list
Cand.models <- list( )
Cand.models[[1]] <- lme(distance ~ age, data = Orthodont, method = "ML")
##random is ~ age | Subject
Cand.models[[2]] <- lme(distance ~ age + Sex, data = Orthodont,
                        random = ~ 1, method = "ML")
Cand.models[[3]] <- lme(distance ~ 1, data = Orthodont, random = ~ 1,
                        method = "ML") 
Cand.models[[4]] <- lme(distance ~ Sex, data = Orthodont, random = ~ 1,
                        method = "ML") 

##create a vector of model names
Modnames <- paste("mod", 1:length(Cand.models), sep = "")

importance(cand.set = Cand.models, parm = "age", modnames = Modnames,
           second.ord = TRUE, nobs = NULL)
##round to 4 digits after decimal point
print(importance(cand.set = Cand.models, parm = "age", modnames = Modnames,
                 second.ord = TRUE, nobs = NULL), digits = 4)
detach(package:nlme)

## End(Not run)


##single-season occupancy model example modified from ?occu
## Not run: 
require(unmarked)
##single season
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
## add some fake covariates for illustration
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)),
                                sitevar2 = rnorm(numSites(pferUMF))) 
     
## observation covariates are in site-major, observation-minor order
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) *
                                 obsNum(pferUMF))) 

##set up candidate model set
fm1 <- occu(~ obsvar1 ~ sitevar1, pferUMF)
fm2 <- occu(~ 1 ~ sitevar1, pferUMF)
fm3 <- occu(~ obsvar1 ~ sitevar2, pferUMF)
fm4 <- occu(~ 1 ~ sitevar2, pferUMF)
Cand.mods <- list(fm1, fm2, fm3, fm4)
Modnames <- c("fm1", "fm2", "fm3", "fm4")

##compute importance value for 'sitevar1' on occupancy
importance(cand.set = Cand.mods, modnames = Modnames, parm = "sitevar1",
           parm.type = "psi")
##compute importance value for 'obsvar1' on detectability
importance(cand.set = Cand.mods, modnames = Modnames, parm = "obsvar1",
           parm.type = "detect")


##example with multispecies occupancy modify from ?occuMulti
##Simulate 3 species data
N <- 80
nspecies <- 3
J <- 4
     
occ_covs <- as.data.frame(matrix(rnorm(N * 10),ncol=10))
names(occ_covs) <- paste('par',1:10,sep='')

det_covs <- list()
for (i in 1:nspecies){
    det_covs[[i]] <- matrix(rnorm(N*J),nrow=N)
}
names(det_covs) <- paste('par',1:nspecies,sep='')

##True vals
beta <- c(0.5,0.2,0.4,0.5,-0.1,-0.3,0.2,0.1,-1,0.1)
f1 <- beta[1] + beta[2]*occ_covs$par1
f2 <- beta[3] + beta[4]*occ_covs$par2
f3 <- beta[5] + beta[6]*occ_covs$par3
f4 <- beta[7]
f5 <- beta[8]
f6 <- beta[9]
f7 <- beta[10]
f <- cbind(f1,f2,f3,f4,f5,f6,f7)
z <- expand.grid(rep(list(1:0),nspecies))[,nspecies:1]
colnames(z) <- paste('sp',1:nspecies,sep='')
dm <- model.matrix(as.formula(paste0("~.^",nspecies,"-1")),z)

psi <- exp(f 
psi <- psi/rowSums(psi)

##True state
ztruth <- matrix(NA,nrow=N,ncol=nspecies)
for (i in 1:N){
    ztruth[i,] <- as.matrix(z[sample(8,1,prob=psi[i,]),])
}
     
p_true <- c(0.6,0.7,0.5)

## fake y data
y <- list()
     
for (i in 1:nspecies){
    y[[i]] <- matrix(NA,N,J)
    for (j in 1:N){
        for (k in 1:J){
            y[[i]][j,k] <- rbinom(1,1,ztruth[j,i]*p_true[i])
        }
    }
}
names(y) <- c('coyote','tiger','bear')
     
##Create the unmarked data object
data <- unmarkedFrameOccuMulti(y=y,siteCovs=occ_covs,obsCovs=det_covs)

## Formulas for state and detection processes
## Length should match number/order of columns in fDesign
occFormulas <- c('~par1 + par2','~par2','~par3','~1','~1','~1','~1')
occFormulas2 <- c('~par1 + par3','~par1 + par2','~par1 + par2 + par3',
                  "~ 1", "~1", "~ 1", "~1")

##Length should match number/order of species in data@ylist
detFormulas <- c('~1','~1','~1')
     
fit <- occuMulti(detFormulas,occFormulas,data)
fit2 <- occuMulti(detFormulas,occFormulas2,data)

##importance
importance(cand.set = list(fit, fit2), parm = "[coyote] par2",
           parm.type = "psi")


detach(package:unmarked)

## End(Not run)

Iron Content in Food

Description

This data set, originally from Adish et al. (1999), describes the iron content of food cooked in different pot types.

Usage

data(iron)

Format

A data frame with 36 rows and 3 variables.

Pot

pot type, one of "aluminium", "clay", or "iron".

Food

food type, one of "legumes", "meat", or "vegetables".

Iron

iron content measured in mg/100 g of food.

Details

Heiberger and Holland (2004, p. 378) use these data as an exercise on two-way ANOVA with interaction.

Source

Heiberger, R. M., Holland, B. (2004) Statistical Analysis and Data Display: an intermediate course with examples in S-Plus, R, and SAS. Springer: New York.

Adish, A. A., Esrey, S. A., Gyorkos, T. W., Jean-Baptiste, J., Rojhani, A. (1999) Effect of consumption of food cooked in iron pots on iron status and growth of young children: a randomised trial. The Lancet 353, 712–716.

Examples

data(iron)
str(iron)

Habitat Preference of Lizards

Description

This data set describes the habitat preference of two species of lizards, Anolis grahami and A. opalinus, on the island of Jamaica and is originally from Schoener (1970). McCullagh and Nelder (1989) and Burnham and Anderson (2002) reanalyzed the data. Note that a typo occurs in table 3.11 of Burnham and Anderson (2002).

Usage

data(lizards)

Format

A data frame with 48 rows and 6 variables.

Insolation

position of perch, either shaded or sunny.

Diameter

diameter of the perch, either < 2 in or >= 2 in.

Height

perch height, either < 5 or >= 5.

Time

time of day, either morning, midday, or afternoon.

Species

species observed, either grahami or opalinus.

Counts

number of individuals observed.

Details

Burnham and Anderson (2002, p. 137) use this data set originally from Schoener (1970) to illustrate model selection for log-linear models.

Source

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

McCullagh, P., Nelder, J. A. (1989) Generalized Linear Models. Second edition. Chapman and Hall: New York.

Schoener, T. W. (1970) Nonsynchronous spatial overlap of lizards in patchy habitats. Ecology 51, 408–418.

Examples

data(lizards)
## Not run: 
##log-linear model as in Burnham and Anderson 2002, p. 137
##main effects
m1 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species, 
          family = poisson, data = lizards)


##main effects and all second order interactions = base
m2 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species +
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Time +
          Diameter:Species + Height:Time + Height:Species +
          Time:Species, family = poisson, data = lizards)

##base - DT
m3 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species +
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Species +
          Height:Time + Height:Species + Time:Species, 
          family = poisson, data = lizards)

##base + HDI + HDT + HDS
m4 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species +
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Time +
          Diameter:Species + Height:Time + Height:Species +
          Time:Species + Height:Diameter:Insolation +
          Height:Diameter:Time + Height:Diameter:Species, 
          family = poisson, data = lizards)

##base + HDI + HDS + HIT + HIS + HTS + ITS
m5 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species + 
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Time +
          Diameter:Species + Height:Time + Height:Species +
          Time:Species + Height:Diameter:Insolation +
          Height:Diameter:Species + Height:Insolation:Time +
          Height:Insolation:Species + Height:Time:Species +
          Insolation:Time:Species, family = poisson, data = lizards)

##base + HIT + HIS + HTS + ITS
m6 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species + 
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Time +
          Diameter:Species + Height:Time + Height:Species +
          Time:Species + Height:Insolation:Time +
          Height:Insolation:Species +  Height:Time:Species +
          Insolation:Time:Species,  family = poisson, data = lizards)

##base + HIS + HTS + ITS
m7 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species +
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Time +
          Diameter:Species + Height:Time + Height:Species +
          Time:Species + Height:Insolation:Species +
          Height:Time:Species + Insolation:Time:Species, 
          family = poisson, data = lizards)

##base + HIT + HIS + HTS + ITS - DT
m8 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species +
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Species +
          Height:Time + Height:Species + Time:Species +
          Height:Insolation:Time + Height:Insolation:Species +
          Height:Time:Species + Insolation:Time:Species, 
          family = poisson, data = lizards)

##base + HIT + HIS + ITS - DT
m9 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species +
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Species +
          Height:Time + Height:Species + Time:Species +
          Height:Insolation:Time + Height:Insolation:Species +
          Insolation:Time:Species, 
          family = poisson, data = lizards)

##base + HIT + HIS - DT
m10 <- glm(Counts ~ Insolation + Diameter + Height + Time + Species +
          Insolation:Diameter + Insolation:Height + Insolation:Time +
          Insolation:Species + Diameter:Height + Diameter:Species +
          Height:Time + Height:Species + Time:Species +
          Height:Insolation:Time + Height:Insolation:Species,
          family = poisson, data = lizards)

##set up in list
Cands <- list(m1, m2, m3, m4, m5, m6, m7, m8, m9, m10)
Modnames <- paste("m", 1:length(Cands), sep = "")

##model selection
library(AICcmodavg)
aictab(Cands, Modnames)

## End(Not run)

Compute MacKenzie and Bailey Goodness-of-fit Test for Single Season, Dynamic, and Royle-Nichols Occupancy Models

Description

These functions compute the MacKenzie and Bailey (2004) goodness-of-fit test for single season occupancy models based on Pearson's chi-square and extend it to dynamic (multiple season) and Royle-Nichols (2003) occupancy models.

Usage

mb.chisq(mod, print.table = TRUE, ...)

## S3 method for class 'unmarkedFitOccu'
mb.chisq(mod, print.table = TRUE, ...)

## S3 method for class 'unmarkedFitColExt'
mb.chisq(mod, print.table = TRUE, ...)

## S3 method for class 'unmarkedFitOccuRN'
mb.chisq(mod, print.table = TRUE,
                  maxK = NULL, ...)

mb.gof.test(mod, nsim = 5, plot.hist = TRUE, report = NULL,
            parallel = TRUE, ncores, cex.axis = 1, cex.lab = 1,
            cex.main = 1, lwd = 1, ...)

## S3 method for class 'unmarkedFitOccu'
mb.gof.test(mod, nsim = 5, plot.hist = TRUE,
                     report = NULL, parallel = TRUE, ncores,
                     cex.axis = 1, cex.lab = 1, cex.main = 1,
                     lwd = 1, ...)

## S3 method for class 'unmarkedFitColExt'
mb.gof.test(mod, nsim = 5, plot.hist = TRUE,
                     report = NULL, parallel = TRUE,
                     ncores, cex.axis = 1, cex.lab = 1, cex.main = 1,
                     lwd = 1, plot.seasons = FALSE, ...) 

## S3 method for class 'unmarkedFitOccuRN'
mb.gof.test(mod, nsim = 5, plot.hist = TRUE,
                     report = NULL, parallel = TRUE, ncores,
                     cex.axis = 1, cex.lab = 1, cex.main = 1,
                     lwd = 1, maxK = NULL, ...)

Arguments

mod

the model for which a goodness-of-fit test is required.

print.table

logical. Specifies if the detailed table of observed and expected values is to be included in the output.

nsim

the number of bootstrapped samples.

plot.hist

logical. Specifies that a histogram of the bootstrapped test statistic is to be included in the output. For dynamic occupancy models, this produces a histogram of the sum of the season-specific chi-squares for each bootstrap sample.

report

If NULL, the test statistic for each iteration is not printed in the terminal. Otherwise, an integer indicating the number of values of the test statistic that should be printed on the same line. For example, if report = 3, the values of the test statistic for three iterations are reported on each line.

parallel

logical. If TRUE, requests that parboot use multiple cores to accelerate computations of the bootstrap.

ncores

integer indicating the number of cores to use when bootstrapping in parallel during the analysis of simulated data sets. If ncores is not specified, one less than the number of available cores on the computer is used.

cex.axis

expansion factor influencing the size of axis annotations on plots produced by the function.

cex.lab

expansion factor influencing the size of axis labels on plots produced by the function.

cex.main

expansion factor influencing the size of the main title above plots produced by the function.

lwd

expansion factor of line width on plots produced by the function.

plot.seasons

logical. For dynamic occupancy models, specifies that a histogram of the bootstrapped test statistic for each primary period (season) is to be included in the output.

maxK

the number of support points used as the summation index in the likelihood of the Royle-Nichols model (2003). If NULL, extracts the value of K used by occuRN to fit the model to the original data.

...

additional arguments passed to the function.

Details

MacKenzie and Bailey (2004) and MacKenzie et al. (2006) suggest using the Pearson chi-square to assess the fit of single season occupancy models (MacKenzie et al. 2002). Given low expected frequencies, the chi-square statistic will deviate from the theoretical distribution and it is recommended to use a parametric bootstrap approach to obtain P-values with the parboot function of the unmarked package. mb.chisq computes the table of observed and expected values based on the detection histories and single season occupancy model used. mb.gof.test calls internally mb.chisq and parboot to generate simulated data sets based on the model and compute the MacKenzie and Bailey test statistic. Missing values are accomodated by creating cohorts for each pattern of missing values.

It is also possible to obtain an estimate of the overdispersion parameter (c-hat) for the model at hand by dividing the observed chi-square statistic by the mean of the statistics obtained from simulation.

This test is extended to dynamic occupancy models of MacKenzie et al. (2003) by using the occupancy estimates for each season obtained from the model. These estimates are then used to compute the predicted and observed frequencies separately within each season. The chi-squares are then summed to be used as the test statistic for the dynamic occupancy model.

Note that values of c-hat > 1 indicate overdispersion (variance > mean), but that values much higher than 1 (i.e., > 4) probably indicate lack-of-fit. In cases of moderate overdispersion, one usually multiplies the variance-covariance matrix of the estimates by c-hat. As a result, the SE's of the estimates are inflated (c-hat is also known as a variance inflation factor).

In model selection, c-hat should be estimated from the global model and the same value of c-hat applied to the entire model set. Specifically, a global model is the most complex model from which all the other models of the set are simpler versions (nested). When no single global model exists in the set of models considered, such as when sample size does not allow a complex model, one can estimate c-hat from 'subglobal' models. Here, 'subglobal' models denote models from which only a subset of the models of the candidate set can be derived. In such cases, one can use the smallest value of c-hat for model selection (Burnham and Anderson 2002).

Note that c-hat counts as an additional parameter estimated and should be added to K. All functions in package AICcmodavg automatically add 1 when the c.hat argument > 1 and apply the same value of c-hat for the entire model set. When c-hat > 1, functions compute quasi-likelihood information criteria (either QAICc or QAIC, depending on the value of the second.ord argument) by scaling the log-likelihood of the model by c-hat. The value of c-hat can influence the ranking of the models: as c-hat increases, QAIC or QAICc will favor models with fewer parameters. As an additional check against this potential problem, one can generate several model selection tables by incrementing values of c-hat to assess the model selection uncertainty. If ranking changes little up to the c-hat value observed, one can be confident in making inference.

In cases of underdispersion (c-hat < 1), it is recommended to keep the value of c-hat to 1. However, note that values of c-hat << 1 can also indicate lack-of-fit and that an alternative model should be investigated.

Value

mb.chisq returns the following components for single-season and Royle-Nichols occupancy models:

chisq.table

the table of observed and expected values for each detection history and its chi-square component (if print.table = TRUE). Note that the table only shows the observed detection histories. Unobserved detection histories are not shown, but are included in the computation of the test statistic.

chi.square

the Pearson chi-square statistic. This test statistic should be compared against a bootstrap distribution instead of the theoretical chi-square distribution because low expected frequencies invalidate the chi-square assumption.

model.type

the model type, either single-season, royle-nichols, or dynamic.

mb.chisq returns the following additional components for dynamic occupancy models:

tables

a list containing the season-specific chi-square tables (if print.table = TRUE).

all.chisq

an element containing the season-specific chi-squares.

n.seasons

the number of primary periods (seasons).

n.seasons

the number of primary periods (seasons).

missing.seasons

logical vector indicating whether data were collected or not during a given season (primary period), where TRUE indicates that no data were collected during the season.

mb.gof.test returns the following components for single-season and Royle-Nichols occupancy models:

chisq.table

the table of observed and expected values for each detection history and its chi-square component.

chi.square

the Pearson chi-square statistic.

t.star

the bootstrapped chi-square test statistics (i.e., obtained for each of the simulated data sets).

p.value

the P-value assessed from the parametric bootstrap, computed as the proportion of the simulated test statistics greater than or equal to the observed test statistic.

c.hat.est

the estimate of the overdispersion parameter, c-hat, computed as the observed test statistic divided by the mean of the simulated test statistics.

nsim

the number of bootstrap samples. The recommended number of samples varies with the data set, but should be on the order of 1000 or 5000, and in cases with a large number of visits, even 10 000 samples, namely to reduce the effect of unusually small values of the test statistics.

mb.gof.test returns the following additional components for dynamic occupancy models:

chisq.table

a list including the table of observed and expected values for each detection history and its chi-square component for each primary period (season).

chi.square

the chi-square test statistic, as the sum of the chi-squares across the primary periods.

p.value

a list of the P-values for each of the primary periods, computed separately as the proportion of the simulated test statistics greater than or equal to the observed test statistic.

p.global

the P-value of the chi-square test statistic for the dynamic occupancy model. This P-value is computed as the proportion of the simulated sums of chi-squares greater than or equal to the observed sum of chi-squares across the primary periods.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

MacKenzie, D. I., Bailey, L. L. (2004) Assessing the fit of site-occupancy models. Journal of Agricultural, Biological, and Environmental Statistics 9, 300–318.

MacKenzie, D. I., Nichols, J. D., Royle, J. A., Pollock, K. H., Bailey, L. L., Hines, J. E. (2006) Occupancy estimation and modeling: inferring patterns and dynamics of species occurrence. Academic Press: New York.

Royle, J. A., Nichols, J. D. (2003) Estimating abundance from repeated presence-absence data or point counts. Ecology 84, 777–790.

See Also

AICc, c_hat, colext, evidence, modavg, importance, modavgPred, Nmix.gof.test, occu, parboot

Examples

##single-season occupancy model example modified from ?occu
## Not run: 
require(unmarked)
##single season
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
## add some fake covariates for illustration
siteCovs(pferUMF) <- data.frame(sitevar1 = rnorm(numSites(pferUMF)),
                                sitevar2 = rnorm(numSites(pferUMF))) 
     
## observation covariates are in site-major, observation-minor order
obsCovs(pferUMF) <- data.frame(obsvar1 = rnorm(numSites(pferUMF) *
                                 obsNum(pferUMF))) 

##run model
fm1 <- occu(~ obsvar1 ~ sitevar1, pferUMF)

##compute observed chi-square
obs <- mb.chisq(fm1)
obs
##round to 4 digits after decimal point
print(obs, digits.vals = 4)


##compute observed chi-square, assess significance, and estimate c-hat
obs.boot <- mb.gof.test(fm1, nsim = 3)
##note that more bootstrap samples are recommended
##(e.g., 1000, 5000, or 10 000)
obs.boot
print(obs.boot, digits.vals = 4, digits.chisq = 4)


##data with missing values
mat1 <- matrix(c(0, 0, 0), nrow = 120, ncol = 3, byrow = TRUE)
mat2 <- matrix(c(0, 0, 1), nrow = 23, ncol = 3, byrow = TRUE)
mat3 <- matrix(c(1, NA, NA), nrow = 42, ncol = 3, byrow = TRUE)
mat4 <- matrix(c(0, 1, NA), nrow = 33, ncol = 3, byrow = TRUE)
y.mat <- rbind(mat1, mat2, mat3, mat4)
y.sim.data <- unmarkedFrameOccu(y = y.mat)
m1 <- occu(~ 1 ~ 1, data = y.sim.data)

mb.gof.test(m1, nsim = 3)
##note that more bootstrap samples are recommended
##(e.g., 1000, 5000, or 10 000)

##modifying graphical parameters
mb.gof.test(m1, nsim = 3,
cex.axis = 1.2, #axis annotations are 1.2 the default size
cex.lab = 1.2,  #axis labels are 1.2 the default size
lwd = 2)        #line width is twice the default width
detach(package:unmarked)

## End(Not run)

Anuran Larvae Counts in Minnow Traps Across Pond Type

Description

This data set consists of counts of anuran larvae as a function of pond type, pond perimeter, and presence of water scorpions (Ranatra sp.).

Usage

data(min.trap)

Format

A data frame with 24 observations on the following 6 variables.

Type

pond type, denotes the location of ponds in either bog or upland environment

Num_anura

number of anuran larvae in minnow traps

Effort

number of trap nights (i.e., number of traps x days of trapping) in each pond

Perimeter

pond perimeter in meters

Num_ranatra

number of water scorpions trapped in minnow traps

log.Perimeter

natural log of perimeter

Details

Mazerolle (2006) uses this data set to illustrate model selection for Poisson regression with low overdispersion.

Source

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Examples

data(min.trap)
## maybe str(min.trap) ; plot(min.trap) ...

Compute Model-averaged Parameter Estimate (Multimodel Inference)

Description

This function model-averages the estimate of a parameter of interest among a set of candidate models, computes the unconditional standard error and unconditional confidence intervals as described in Buckland et al. (1997) and Burnham and Anderson (2002). This model-averaged estimate is also referred to as a natural average of the estimate by Burnham and Anderson (2002, p. 152).

Usage

modavg(cand.set, parm, modnames = NULL, second.ord = TRUE, nobs = NULL, 
       uncond.se = "revised", conf.level = 0.95, exclude = NULL, warn =
       TRUE, ...) 

## S3 method for class 'AICaov.lm'
modavg(cand.set, parm, modnames = NULL, second.ord =
        TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICbetareg'
modavg(cand.set, parm, modnames = NULL, second.ord =
        TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICsclm.clm'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICclm'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICclmm'
modavg(cand.set, parm, modnames = NULL, second.ord 
        = TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICcoxme'
modavg(cand.set, parm, modnames = NULL, second.ord
        = TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICcoxph'
modavg(cand.set, parm, modnames = NULL, second.ord
        = TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICglm.lm'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        gamdisp = NULL, ...)

## S3 method for class 'AICglmmTMB'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        ...)

## S3 method for class 'AICgls'
modavg(cand.set, parm, modnames = NULL, second.ord =
           TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
           exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AIChurdle'
modavg(cand.set, parm, modnames = NULL,
         second.ord = TRUE, nobs = NULL, uncond.se = "revised",
         conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AIClm'
modavg(cand.set, parm, modnames = NULL, second.ord =
        TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AIClme'
modavg(cand.set, parm, modnames = NULL, second.ord =
        TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AIClmekin'
modavg(cand.set, parm, modnames = NULL,
         second.ord = TRUE, nobs = NULL, uncond.se = "revised",
         conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICmaxlikeFit.list'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        ...)

## S3 method for class 'AICmer'
modavg(cand.set, parm, modnames = NULL, second.ord =
        TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AIClmerMod'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AIClmerModLmerTest'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICglmerMod'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICmultinom.nnet'
modavg(cand.set, parm, modnames = NULL, 
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        ...)

## S3 method for class 'AICnegbin.glm.lm'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICpolr'
modavg(cand.set, parm, modnames = NULL, second.ord
        = TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICrlm.lm'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICsurvreg'
modavg(cand.set, parm, modnames = NULL, second.ord =
        TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
        exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICvglm'
modavg(cand.set, parm, modnames = NULL, second.ord
         = TRUE, nobs = NULL, uncond.se = "revised", conf.level = 0.95,
         exclude = NULL, warn = TRUE, c.hat = 1, ...)

## S3 method for class 'AICzeroinfl'
modavg(cand.set, parm, modnames = NULL,
         second.ord = TRUE, nobs = NULL, uncond.se = "revised",
         conf.level = 0.95, exclude = NULL, warn = TRUE, ...)

## S3 method for class 'AICunmarkedFitOccu'
modavg(cand.set, parm, modnames = NULL, 
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitColExt'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuRN'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCount'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCO'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDS'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGDS'
modavg(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMPois'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGMM'
modavg(cand.set, parm, modnames =
       NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
       conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
       parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGPC'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMMO'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDSO'
modavg(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, exclude = NULL, warn = TRUE, c.hat = 1,
        parm.type = NULL, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

parm

the parameter of interest, enclosed between quotes, for which a model-averaged estimate is required. For a categorical variable, the label of the estimate must be included as it appears in the output (see 'Details' below).

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

uncond.se

either, "old", or "revised", specifying the equation used to compute the unconditional standard error of a model-averaged estimate. With uncond.se = "old", computations are based on equation 4.9 of Burnham and Anderson (2002), which was the former way to compute unconditional standard errors. With uncond.se = "revised", equation 6.12 of Burnham and Anderson (2002) is used. Anderson (2008, p. 111) recommends use of the revised version for the computation of unconditional standard errors and it is now the default. Note that versions of package AICcmodavg < 1.04 used the old method to compute unconditional standard errors.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of unconditional confidence intervals.

exclude

this argument excludes models based on the terms specified for the computation of a model-averaged estimate of parm. The exclude argument is set to NULL by default and does not exclude any models other than those without the parm. When parm is a main effect but is also involved in interactions/polynomial terms in some models, one should specify the interaction/polynomial terms as a list to exclude models with these terms from the computation of model-averaged estimate of the main effect (e.g., exclude = list("sex:mass", "mass2")). See 'Details' and 'Examples' below.

warn

logical. If TRUE, modavg performs a check and isssues a warning when the value in parm occurs more than once in any given model. This is a check for potential interaction/polynomial terms in the model when such terms are constructed with the usual operators (e.g., I( ) for polynomial terms, : for interaction terms).

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, modavg will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

gamdisp

if gamma GLM is used, the dispersion parameter should be specified here to apply the same value to each model.

parm.type

this argument specifies the parameter type on which the model-averaged estimate of a predictor will be computed and is only relevant for models of unmarkedFit classes. The character strings supported vary with the type of model fitted. For unmarkedFitOccu and unmarkedFitOccuMulti objects, either psi or detect can be supplied to indicate whether the parameter is on occupancy or detectability, respectively. For unmarkedFitColExt objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and detectability, respectively. For unmarkedFitOccuTTD objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and time-to-dection (lambda rate parameter), respectively. For unmarkedFitOccuFP objects, one can specify psi, detect, falsepos, and certain, for occupancy, detectability, probability of assigning false-positives, and probability detections are certain, respectively. For unmarkedFitOccuMS objects, possible values are psi, phi, or detect, denoting occupancy, transition, and detection probabilities, respectively. For unmarkedFitOccuRN objects, either lambda or detect can be entered for abundance and detectability parameters, respectively. For unmarkedFitPCount and unmarkedFitMPois objects, lambda or detect denote parameters on abundance and detectability, respectively. For unmarkedFitPCO, unmarkedFitMMO, and unmarkedFitDSO objects, one can enter lambda, gamma, omega, iota, or detect, to specify parameters on abundance, recruitment, apparent survival, immigration, and detectability, respectively. For unmarkedFitDS objects, lambda and detect are supported. For unmarkedFitGDS, lambda, phi, and detect denote abundance, availability, and detection probability, respectively. For unmarkedFitGMM and unmarkedFitGPC objects, lambda, phi, and detect denote abundance, availability, and detectability, respectively.

...

additional arguments passed to the function.

Details

The parameter for which a model-averaged estimate is requested must be specified with the parm argument and must be identical to its label in the model output (e.g., from summary). For factors, one must specify the name of the variable and the level of interest. modavg includes checks to find variations of interaction terms specified in the parm and exclude arguments. However, to avoid problems, one should specify interaction terms consistently for all models: e.g., either a:b or b:a for all models, but not a mixture of both.

You must exercise caution when some models include interaction or polynomial terms, because main effect terms do not have the same interpretation when they also appear in an interaction/polynomial term in the same model. In such cases, one should exclude models containing interaction terms where the main effect is involved with the exclude argument of modavg. Note that modavg checks for potential cases of multiple instances of a variable appearing more than once in a given model (presumably in an interaction) and issues a warning. To correctly compute the model-averaged estimate of a main effect involved in interaction/polynomial terms, specify the interaction terms(s) that should not appear in the same model with the exclude argument. This will effectively exclude models from the computation of the model-averaged estimate.

When warn = TRUE, modavg looks for matches among the labels of the estimates with identical. It then compares the results to partial matches with regexpr, and issues a warning whenever they are different. As a result, modavg may issue a warning when some variables or levels of categorical variables have nested names (e.g., treat, treat10; L, TL). When this warning is only due to the presence of similarly named variables in the models (and NOT due to interaction terms), you can suppress this warning by setting warn = FALSE.

The model-averaging estimator implemented in modavg is known to be biased away from 0 when there is substantial model selection uncertainty (Cade 2015). In such instances, it is recommended to use the model-averaging shrinkage estimator (i.e., modavgShrink) for inference on beta estimates or to focus on model-averaged effect sizes (modavgEffect) and model-averaged predictions (modavgPred).

modavg is implemented for a list containing objects of aov, betareg, clm, clmm, clogit, coxme, coxph, glm, glmmTMB, gls, hurdle, lm, lme, lmekin, maxlikeFit, mer, glmerMod, lmerMod, lmerModLmerTest, multinom, polr, rlm, survreg, vglm, zeroinfl classes as well as various models of unmarkedFit classes.

Value

modavg creates an object of class modavg with the following components:

Parameter

the parameter for which a model-averaged estimate was obtained.
Mod.avg.table

the reduced model selection table based on models including the parameter of interest.
Mod.avg.beta

the model-averaged estimate based on all models including the parameter of interest (see 'Details' above regarding the exclusion of models where parameter of interest is involved in an interaction).
Uncond.SE

the unconditional standard error for the model-averaged estimate (as opposed to the conditional SE based on a single model).
Conf.level

the confidence level used to compute the confidence interval.
Lower.CL

the lower confidence limit.
Upper.CL

the upper confidence limit.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Buckland, S. T., Burnham, K. P., Augustin, N. H. (1997) Model selection: an integral part of inference. Biometrics 53, 603–618.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Cade, B. S. (2015) Model averaging and muddled multimodel inferences. Ecology 96, 2370–2382.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67–118.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICc, aictab, c_hat, confset, evidence, importance, modavgCustom, modavgEffect, modavgShrink, modavgPred

Examples

##anuran larvae example modified from Mazerolle (2006)
##these are different models than in the paper
data(min.trap)
##assign "UPLAND" as the reference level as in Mazerolle (2006)          
min.trap$Type <- relevel(min.trap$Type, ref = "UPLAND") 

##set up candidate models          
Cand.mod <- list( )
##global model          
Cand.mod[[1]] <- glm(Num_anura ~ Type + log.Perimeter +
                     Type:log.Perimeter + Num_ranatra,
                     family = poisson, offset = log(Effort),
                     data = min.trap)
##interactive model
Cand.mod[[2]] <- glm(Num_anura ~ Type + log.Perimeter +
                     Type:log.Perimeter, family = poisson, 
                     offset = log(Effort), data = min.trap)
##additive model
Cand.mod[[3]] <- glm(Num_anura ~ Type + log.Perimeter, family = poisson,
                     offset = log(Effort), data = min.trap)
##Predator model
Cand.mod[[4]] <- glm(Num_anura ~ Type + Num_ranatra, family = poisson,
                     offset = log(Effort), data = min.trap) 
          
##check c-hat for global model
c_hat(Cand.mod[[1]]) #uses Pearson's chi-square/df
##note the very low overdispersion: in this case, the analysis could be
##conducted without correcting for c-hat as its value is reasonably close
##to 1  

##assign names to each model
Modnames <- c("global model", "interactive model",
              "additive model", "invertpred model") 

##model selection
aictab(Cand.mod, Modnames)

##compute model-averaged estimates for parameters appearing in top
##models
modavg(parm = "Num_ranatra", cand.set = Cand.mod, modnames = Modnames)
##round to 4 digits after decimal point
print(modavg(parm = "Num_ranatra", cand.set = Cand.mod,
             modnames = Modnames), digits = 4)

##model-averaging a variable involved in an interaction
##the following produces an error - because the variable is involved
##in an interaction in some candidate models
## Not run: modavg(parm = "TypeBOG", cand.set = Cand.mod,
         modnames = Modnames)
## End(Not run)


##exclude models where the variable is involved in an interaction
##to get model-averaged estimate of main effect
modavg(parm = "TypeBOG", cand.set = Cand.mod, modnames = Modnames,
       exclude = list("Type:log.Perimeter"))

##to get model-averaged estimate of interaction
modavg(parm = "TypeBOG:log.Perimeter", cand.set = Cand.mod,
       modnames = Modnames)



##beware of variables that have similar names
set.seed(seed = 4)
resp <- rnorm(n = 40, mean = 3, sd = 1)
size <- rep(c("small", "medsmall", "high", "medhigh"), times = 10)
set.seed(seed = 4)
mass <- rnorm(n = 40, mean = 2, sd = 0.1)
mass2 <- mass^2
age <- rpois(n = 40, lambda = 3.2)
agecorr <- rpois(n = 40, lambda = 2) 
sizecat <- rep(c("a", "ab"), times = 20)
data1 <- data.frame(resp = resp, size = size, sizecat = sizecat,
                    mass = mass, mass2 = mass2, age = age,
                    agecorr = agecorr)

##set up models in list
Cand <- list( )
Cand[[1]] <- lm(resp ~ size + agecorr, data = data1)
Cand[[2]] <- lm(resp ~ size + mass + agecorr, data = data1)
Cand[[3]] <- lm(resp ~ age + mass, data = data1)
Cand[[4]] <- lm(resp ~ age + mass + mass2, data = data1)
Cand[[5]] <- lm(resp ~ mass + mass2 + size, data = data1)
Cand[[6]] <- lm(resp ~ mass + mass2 + sizecat, data = data1)
Cand[[7]] <- lm(resp ~ sizecat, data = data1)
Cand[[8]] <- lm(resp ~ sizecat + mass + sizecat:mass, data = data1)
Cand[[9]] <- lm(resp ~ agecorr + sizecat + mass + sizecat:mass,
                 data = data1) 

##create vector of model names
Modnames <- paste("mod", 1:length(Cand), sep = "")

aictab(cand.set = Cand, modnames = Modnames, sort = TRUE) #correct

##as expected, issues warning as mass occurs sometimes with "mass2" or
##"sizecatab:mass" in some of the models
## Not run: modavg(cand.set = Cand, parm = "mass", modnames = Modnames)

##no warning issued, because "age" and "agecorr" never appear in same model
modavg(cand.set = Cand, parm = "age", modnames = Modnames)

##as expected, issues warning because warn=FALSE, but it is a very bad
##idea in this example since "mass" occurs with "mass2" and "sizecat:mass"
##in some of the models - results are INCORRECT
## Not run: modavg(cand.set = Cand, parm = "mass", modnames = Modnames,
                warn = FALSE)
## End(Not run)

##correctly excludes models with quadratic term and interaction term
##results are CORRECT
modavg(cand.set = Cand, parm = "mass", modnames = Modnames,
       exclude = list("mass2", "sizecat:mass")) 

##correctly computes model-averaged estimate because no other parameter
##occurs simultaneously in any of the models
modavg(cand.set = Cand, parm = "sizesmall", modnames = Modnames) #correct

##as expected, issues a warning because "sizecatab" occurs sometimes in
##an interaction in some models
## Not run: modavg(cand.set = Cand, parm = "sizecatab",
                modnames = Modnames) 
## End(Not run)

##exclude models with "sizecat:mass" interaction - results are CORRECT
modavg(cand.set = Cand, parm = "sizecatab", modnames = Modnames,
       exclude = list("sizecat:mass"))



##example with multiple-season occupancy model modified from ?colext
##this is a bit longer
## Not run: 
require(unmarked)
data(frogs)
umf <- formatMult(masspcru)
obsCovs(umf) <- scale(obsCovs(umf))
siteCovs(umf) <- rnorm(numSites(umf))
yearlySiteCovs(umf) <- data.frame(year = factor(rep(1:7,
                                    numSites(umf))))

##set up model with constant transition rates
fm <- colext(psiformula = ~ 1, gammaformula = ~ 1, epsilonformula = ~ 1,
             pformula = ~ JulianDate + I(JulianDate^2), data = umf,
             control = list(trace=1, maxit=1e4))

##model with with year-dependent transition rates
fm.yearly <- colext(psiformula = ~ 1, gammaformula = ~ year,
                    epsilonformula = ~ year,
                    pformula = ~ JulianDate + I(JulianDate^2),
                    data = umf)

##store in list and assign model names
Cand.mods <- list(fm, fm.yearly)
Modnames <- c("psi1(.)gam(.)eps(.)p(Date + Date2)",
              "psi1(.)gam(Year)eps(Year)p(Date + Date2)")

##compute model-averaged estimate of occupancy in the first year
modavg(cand.set = Cand.mods, modnames = Modnames, parm = "(Intercept)",
       parm.type = "psi")

##compute model-averaged estimate of Julian Day squared on detectability
modavg(cand.set = Cand.mods, modnames = Modnames,
       parm = "I(JulianDate^2)", parm.type = "detect")

## End(Not run)


##example of model-averaged estimate of area from distance model
##this is a bit longer
## Not run: 
data(linetran) #example modified from ?distsamp
     
ltUMF <- with(linetran, {
  unmarkedFrameDS(y = cbind(dc1, dc2, dc3, dc4),
                  siteCovs = data.frame(Length, area, habitat),
                  dist.breaks = c(0, 5, 10, 15, 20),
                  tlength = linetran$Length * 1000, survey = "line", unitsIn = "m")
})
     
## Half-normal detection function. Density output (log scale). No covariates.
fm1 <- distsamp(~ 1 ~ 1, ltUMF)
     
## Halfnormal. Covariates affecting both density and detection.
fm2 <- distsamp(~ area + habitat ~ area + habitat, ltUMF)

## Hazard function. Covariates affecting both density and detection.
fm3 <- distsamp(~ habitat ~ area + habitat, ltUMF, keyfun="hazard")

##assemble model list
Cands <- list(fm1, fm2, fm3)
Modnames <- paste("mod", 1:length(Cands), sep = "")

##model-average estimate of area on abundance
modavg(cand.set = Cands, modnames = Modnames, parm = "area", parm.type = "lambda")
detach(package:unmarked)

## End(Not run)

Various Utility Functions

Description

reverse.parm and reverse.exclude reverse the order of variables in an interaction term.

formatCands creates new classes for lists containing candidate models.

formulaShort prints a succinct formula from an unmarkedFit object.

Usage

reverse.parm(parm)
reverse.exclude(exclude)
formatCands(cand.set)
formulaShort(mod, unmarked.type = NULL)

Arguments

parm

a parameter to be model-averaged, enclosed between quotes, as it appears in the output of some models.

exclude

a list of interaction or polynomial terms appearing in some models, as they would appear in the call to the model function (i.e., A*B, A:B). Models containing elements from the list will be excluded to obtain a model-averaged estimate.

cand.set

a list storing each of the models in the candidate model set.

mod

an object storing the result of an unmarkedFit.

unmarked.type

a character string specifying the type of parameter in an unmarkedFit for which a formula is requested. This argument uses the character string for each parameter group as defined by unmarked.

Details

These utility functions are used internally by aictab, modavg, and other related functions.

reverse.parm and reverse.exclude enable the user to specify differently interaction terms (e.g., A:B, B:A) across models for model averaging. These functions have been added to avoid problems when users are not consistent in the specification of interaction terms across models.

formatCands creates new classes for the list of candidate models based on the contents of the list. These new classes are used for method dispatch.

formulaShort is used by anovaOD.

Value

reverse.parm returns all possible combinations of an interaction term to identify models that include the parm of interest and find the corresponding estimate and standard error in the model object.

reverse.exclude returns a list of all possible combinations of exclude to identify models that should be excluded when computing a model-averaged estimate.

formatCands adds a new class to the list of candidate models based on the classes of the models.

formulaShort creates a character string for the formula related to a given parameter type from an unmarkedFit object.

Author(s)

Marc J. Mazerolle

See Also

aictab, anovaOD, modavg, modavgShrink, modavgPred

Examples

##a main effect
reverse.parm(parm = "Ageyoung") #does not return anything


##an interaction term as it might appear in the output
reverse.parm(parm = "Ageyoung:time") #returns the reverse


##exclude two interaction terms
reverse.exclude(exclude = list("Age*time", "A:B"))
##returns all combinations
reverse.exclude(exclude = list("Age:time", "A*B"))
##returns all combinations


##Mazerolle (2006) frog water loss example
data(dry.frog)

##setup a subset of models of Table 1
Cand.models <- list( )
Cand.models[[1]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2,
                       data = dry.frog)
Cand.models[[2]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2 +
                       Shade:Substrate, data = dry.frog)
Cand.models[[3]] <- lm(log_Mass_lost ~ cent_Initial_mass +
                       Initial_mass2, data = dry.frog)

formatCands(Cand.models)

## Not run: 
require(unmarked)
data(bullfrog)
bfrog <- unmarkedFrameOccu(y = bullfrog[, c("V1", "V2", "V3", "V4")],
                           siteCovs = bullfrog[, 1:2])
fm1 <- occu(~ 1 ~ Reed.presence, data = bfrog)
formulaShort(fm1, unmarked.type = "state")
formulaShort(fm1, unmarked.type = "det")

## End(Not run)

Compute Model-averaged Parameter Estimate from User-supplied Input Based on (Q)AIC(c)

Description

This function model-averages the estimate of a parameter of interest among a set of candidate models, and computes the unconditional standard error and unconditional confidence intervals as described in Buckland et al. (1997) and Burnham and Anderson (2002).

Usage

modavgCustom(logL, K, modnames = NULL, estimate, se, second.ord = TRUE,
             nobs = NULL, uncond.se = "revised", conf.level = 0.95,
             c.hat = 1, useBIC = FALSE)

Arguments

logL

a vector of log-likelihood values for the models in the candidate model set.

K

a vector containing the number of estimated parameters for each model in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

estimate

a vector of estimates for each of the models in the candidate model set. Estimates can be either beta estimates for a parameter of interest or a single prediction from each model.

se

a vector of standard errors for each of the estimates appearing in the estimate vector.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc). This argument is ignored if useBIC = TRUE.

nobs

the sample size required to compute the AICc, QAICc, BIC, or QBIC.

uncond.se

either, "old", or "revised", specifying the equation used to compute the unconditional standard error of a model-averaged estimate. With uncond.se = "old", computations are based on equation 4.9 of Burnham and Anderson (2002), which was the former way to compute unconditional standard errors. With uncond.se = "revised", equation 6.12 of Burnham and Anderson (2002) is used. Anderson (2008, p. 111) recommends use of the revised version for the computation of unconditional standard errors and it is now the default.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of unconditional confidence intervals.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season and dynamic occupancy models (MacKenzie et al. 2002, 2003), N-mixture models (Royle 2004, Dail and Madsen 2011), or capture-mark-recapture models (e.g., Lebreton et al. 1992). If c.hat > 1, modavgCustom will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)).

useBIC

logical. If TRUE, the function returns the Bayesian information criterion (BIC) when c.hat = 1 or the quasi-likelihood BIC (QBIC) when c.hat > 1.

Details

modavgCustom computes a model-averaged estimate from the vector of parameter estimates specified in estimate. Estimates and their associated standard errors must be specified in the same order as the log-likelihood, number of estimated parameters, and model names. Estimates provided may be for a parameter of interest (i.e., beta estimates) or predictions from each model. This function is most useful when model input is imported into R from other software (e.g., Program MARK, PRESENCE) or for model classes that are not yet supported by the other model averaging functions such as modavg or modavgPred.

Value

modavgCustom creates an object of class modavgCustom with the following components:

Mod.avg.table

the model selection table
Mod.avg.est

the model-averaged estimate
Uncond.SE

the unconditional standard error for the model-averaged estimate
Conf.level

the confidence level used to compute the confidence interval
Lower.CL

the lower confidence limit
Upper.CL

the upper confidence limit

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Buckland, S. T., Burnham, K. P., Augustin, N. H. (1997) Model selection: an integral part of inference. Biometrics 53, 603–618.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67–118.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICcCustom, aictabCustom, bictabCustom, modavg, modavgIC, modavgShrink, modavgPred

Examples

## Not run: 
##model averaging parameter estimate (natural average)
##vector with model LL's
LL <- c(-38.8876, -35.1783, -64.8970)

##vector with number of parameters
Ks <- c(7, 9, 4)

##create a vector of names to trace back models in set
Modnames <- c("Cm1", "Cm2", "Cm3")

##vector of beta estimates for a parameter of interest
model.ests <- c(0.0478, 0.0480, 0.0478)

##vector of SE's of beta estimates for a parameter of interest
model.se.ests <- c(0.0028, 0.0028, 0.0034)

##compute model-averaged estimate and unconditional SE based on AICc
modavgCustom(logL = LL, K = Ks, modnames = Modnames, 
             estimate = model.ests, se = model.se.ests, nobs = 121)
##compute model-averaged estimate and unconditional SE based on BIC
modavgCustom(logL = LL, K = Ks, modnames = Modnames, 
             estimate = model.ests, se = model.se.ests, nobs = 121,
             useBIC = TRUE)


##model-averaging with shrinkage based on AICc
##set up candidate models
data(min.trap)
Cand.mod <- list( )
##global model          
Cand.mod[[1]] <- glm(Num_anura ~ Type + log.Perimeter,
                     family = poisson, offset = log(Effort),
                     data = min.trap) 
Cand.mod[[2]] <- glm(Num_anura ~ Type + Num_ranatra, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[3]] <- glm(Num_anura ~ log.Perimeter + Num_ranatra,
                     family = poisson, offset = log(Effort), data = min.trap)
Model.names <- c("Type + log.Perimeter", "Type + Num_ranatra",
                 "log.Perimeter + Num_ranatra")
##model-averaged estimate with shrinkage (glm model type is already supported)
modavgShrink(cand.set = Cand.mod, modnames = Model.names,
             parm = "log.Perimeter")

##equivalent manual version of model-averaging with shrinkage
##this is especially useful when model classes are not supported
##extract vector of LL
LLs <- sapply(Cand.mod, FUN = function(i) logLik(i)[1])
##extract vector of K
Ks <- sapply(Cand.mod, FUN = function(i) attr(logLik(i), "df"))
##extract betas
betas <- sapply(Cand.mod, FUN = function(i) coef(i)["log.Perimeter"])
##second model does not include log.Perimeter
betas[2] <- 0
##extract SE's
ses <- sapply(Cand.mod, FUN = function(i) sqrt(diag(vcov(i))["log.Perimeter"]))
ses[2] <- 0
##model-averaging with shrinkage based on AICc
modavgCustom(logL = LLs, K = Ks, modnames = Model.names,
             nobs = nrow(min.trap), estimate = betas, se = ses)
##model-averaging with shrinkage based on BIC
modavgCustom(logL = LLs, K = Ks, modnames = Model.names,
             nobs = nrow(min.trap), estimate = betas, se = ses,
             useBIC = TRUE)

## End(Not run)

Compute Model-averaged Effect Sizes (Multimodel Inference on Group Differences)

Description

This function model-averages the effect size between two groups defined by a categorical variable based on the entire model set and computes the unconditional standard error and unconditional confidence intervals as described in Buckland et al. (1997) and Burnham and Anderson (2002). This can be particularly useful when dealing with data from an experiment (e.g., ANOVA) and when the focus is to determine the effect of a given factor. This is an information-theoretic alternative to multiple comparisons (e.g., Burnham et al. 2011).

Usage

modavgEffect(cand.set, modnames = NULL, newdata, second.ord = TRUE,
             nobs = NULL, uncond.se = "revised", conf.level = 0.95,
             ...)

## S3 method for class 'AICaov.lm'
modavgEffect(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICglm.lm'
modavgEffect(cand.set, modnames = NULL, newdata, 
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1, gamdisp = NULL,
        ...)

## S3 method for class 'AICgls'
modavgEffect(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClm'
modavgEffect(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClme'
modavgEffect(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...) 

## S3 method for class 'AICmer'
modavgEffect(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", ...)

## S3 method for class 'AICglmerMod'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", ...)

## S3 method for class 'AIClmerMod'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClmerModLmerTest'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICnegbin.glm.lm'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", ...)

## S3 method for class 'AICrlm.lm'
modavgEffect(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICsurvreg'
modavgEffect(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", ...)

## S3 method for class 'AICunmarkedFitOccu'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitColExt'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuRN'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCount'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCO'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDS'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGDS'
modavgEffect(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMPois'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGMM'
modavgEffect(cand.set, modnames =
       NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
       "revised", conf.level = 0.95, type = "response",
       c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGPC'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMMO'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDSO'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
modavgEffect(cand.set, modnames =
        NULL, newdata, second.ord = TRUE, nobs = NULL, uncond.se =
        "revised", conf.level = 0.95, type = "response",
        c.hat = 1, parm.type = NULL, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

newdata

a data frame with two rows and where the columns correspond to the explanatory variables specified in the candidate models. Note that this data set must have the same structure as that of the original data frame for which we want to make predictions, specifically, the same variable type and names that appear in the original data set. Each row of the data set defines one of the two groups compared. The first row in newdata defines the first group, whereas the second row defines the second group. The effect size is computed as the prediction in the first row minus the prediction in the second row (first row - second row). Only the column relating to the grouping variable can change value and all others must be held constant for the comparison (see 'Details').

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows the specification of a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

uncond.se

either, "old", or "revised", specifying the equation used to compute the unconditional standard error of a model-averaged estimate. With uncond.se = "old", computations are based on equation 4.9 of Burnham and Anderson (2002), which was the former way to compute unconditional standard errors. With uncond.se = "revised", equation 6.12 of Burnham and Anderson (2002) is used. Anderson (2008, p. 111) recommends use of the revised version for the computation of unconditional standard errors and it is now the default. Note that versions of package AICcmodavg < 1.04 used the old method to compute unconditional standard errors.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of unconditional confidence intervals. To obtain confidence intervals corrected for multiple comparisons between pairs of treatments, it is possible to adjust the α\alpha level according to various strategies such as the Bonferroni correction (Dunn 1961).

type

the scale of prediction requested, one of "response" or "link" (only relevant for glm, mer, and unmarkedFit classes). Note that the value "terms" is not defined for modavgEffect).

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season and dynamic occupancy models (MacKenzie et al. 2002, 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, modavgEffect will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer class.

gamdisp

if gamma GLM is used, the dispersion parameter should be specified here to apply the same value to each model.

parm.type

this argument specifies the parameter type on which the effect size will be computed and is only relevant for models of unmarkedFit classes. The character strings supported vary with the type of model fitted. For unmarkedFitOccu objects, either psi or detect can be supplied to indicate whether the parameter is on occupancy or detectability, respectively. For unmarkedFitColExt objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and detectability, respectively. For unmarkedFitOccuTTD objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and time-to-dection (lambda rate parameter), respectively. For unmarkedFitOccuFP objects, one can specify psi, detect, falsepos, and certain, for occupancy, detectability, probability of assigning false-positives, and probability detections are certain, respectively. For unmarkedFitOccuRN objects, either lambda or detect can be entered for abundance and detectability parameters, respectively. For unmarkedFitPCount and unmarkedFitMPois objects, lambda or detect denote parameters on abundance and detectability, respectively. For unmarkedFitPCO, unmarkedFitMMO, and unmarkedFitDSO objects, one can enter lambda, gamma, omega, iota, or detect, to specify parameters on abundance, recruitment, apparent survival, immigration, and detectability, respectively. For unmarkedFitDS objects, lambda and detect are supported. For unmarkedFitGDS, lambda, phi, and detect denote abundance, availability, and detection probability, respectively. For unmarkedFitGMM and unmarkedFitGPC objects, lambda, phi, and detect denote abundance, availability, and detectability, respectively. For unmarkedFitOccuMS objects, psi, phi, and detect denote occupancy, transition, and detection probability, respectively. For unmarkedFitOccuMulti objects, psi and detect denote occupancy and detection probability, respectively.

...

additional arguments passed to the function.

Details

The strategy used here to compute effect sizes is to work from the newdata object to create two predictions from a given model and compute the differences and standard errors between both values. This step is executed for each model in the candidate model set, to obtain a model-averaged estimate of the effect size and unconditional standard error. As a result, the newdata argument is restricted to two rows, each for a given prediction. To specify each group, the values entered in the column for each explanatory variable can be identical, except for the grouping variable. In such a case, the function will identify the variable and the assign group names based on the values of the variable. If more than a single variable has different values in its respective column, the function will print generic names in the output to identify the two groups. A sensible choice of value for the explanatory variables to be held constant is the average of the variable.

Model-averaging effect sizes is most useful in true experiments (e.g., ANOVA-type designs), where one wants to obtain the best estimate of effect size given the support of each candidate model. This can be considered as a information-theoretic analog of traditional multiple comparisons, except that the information contained in the entire model set is used instead of being restricted to a single model. See 'Examples' below for applications.

modavgEffect calls the appropriate method depending on the class of objects in the list. The current classes supported include aov, glm, gls, lm, lme, mer, glmerMod, lmerMod, lmerModLmerTest, rlm, survreg, as well as models of unmarkedFitOccu, unmarkedFitColExt, unmarkedFitOccuFP, unmarkedFitOccuRN, unmarkedFitOccuTTD, unmarkedFitPCount, unmarkedFitPCO, unmarkedFitDS, unmarkedFitDSO, unmarkedFitGDS, unmarkedFitMPois, unmarkedFitGMM, unmarkedFitMMO, unmarkedFitGPC, unmarkedFitOccuMS, and unmarkedFitOccuMulti. classes.

Value

The result is an object of class modavgEffect with the following components:

Group.variable

the grouping variable defining the two groups compared.
Group1

the first group considered in the comparison.
Group2

the second group considered in the comparison.
Type

the scale on which the model-averaged effect size was computed (e.g., response or link).
Mod.avg.table

the full model selection table including the entire set of candidate models.
Mod.avg.eff

the model-averaged effect size based on the entire candidate model set.
Uncond.SE

the unconditional standard error for the model-averaged effect size.
Conf.level

the confidence level used to compute the confidence interval.
Lower.CL

the lower confidence limit.
Upper.CL

the upper confidence limit.
Matrix.output

a matrix containing the model-averaged effect size, the unconditional standard error, and the lower and upper confidence limits.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Buckland, S. T., Burnham, K. P., Augustin, N. H. (1997) Model selection: an integral part of inference. Biometrics 53, 603–618.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Burnham, K. P., Anderson, D. R., Huyvaert, K. P. (2011) AIC model selection and multimodel inference in behaviorial ecology: some background, observations and comparisons. Behavioral Ecology and Sociobiology 65, 23–25.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Dunn, O. J. (1961) Multiple comparisons among means. Journal of the American Statistical Association 56, 52–64.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICc, aictab, c_hat, confset, evidence, importance, modavgShrink, modavgPred

Examples

##heights (cm) of plants grown under two fertilizers, Ex. 9.5 from
##Zar (1984): Biostatistical Analysis. Prentice Hall: New Jersey.
heights <- data.frame(Height = c(48.2, 54.6, 58.3, 47.8, 51.4, 52.0,
                        55.2, 49.1, 49.9, 52.6, 52.3, 57.4, 55.6, 53.2,
                        61.3, 58.0, 59.8, 54.8),
                      Fertilizer = c(rep("old", 10), rep("new", 8)))

##run linear model hypothesizing an effect of fertilizer
m1 <- lm(Height ~ Fertilizer, data = heights)

##run null model (no effect of fertilizer)
m0 <- lm(Height ~ 1, data = heights)

##assemble models in list
Cands <- list(m1, m0)
Modnames <- c("Fert", "null")

##compute model selection table to compare
##both hypotheses
aictab(cand.set = Cands, modnames = Modnames)
##note that model with fertilizer effect is much better supported
##than the null

##compute model-averaged effect sizes: one model hypothesizes a
##difference of 0, whereas the other assumes a difference

##prepare newdata object from which differences between groups
##will be computed
##the first row of the newdata data.frame relates to the first group,
##whereas the second row corresponds to the second group
pred.data <- data.frame(Fertilizer = c("new", "old"))

##compute best estimate of effect size accounting for model selection
##uncertainty
modavgEffect(cand.set = Cands, modnames = Modnames,
              newdata = pred.data)


##classical one-way ANOVA type-design
## Not run: 
##generate data for two groups and control
set.seed(seed = 15)
y <- round(c(rnorm(n = 15, mean = 10, sd = 5),
       rnorm(n = 15, mean = 15, sd = 5),
       rnorm(n = 15, mean = 12, sd = 5)), digits = 2)
##groups
group <- c(rep("cont", 15), rep("trt1", 15), rep("trt2", 15))

##combine in data set
aov.data <- data.frame(Y = y, Group = group)
rm(y, group)

##run model with group effect
lm.eff <- lm(Y ~ Group, data = aov.data)
##null model
lm.0 <- lm(Y ~ 1, data = aov.data)

##compare both models
Cands <- list(lm.eff, lm.0)
Mods <- c("group effect", "no group effect")
aictab(cand.set = Cands, modnames = Mods)
##model with group effect has most of the weight

##compute model-averaged effect sizes
##trt1 - control
modavgEffect(cand.set = Cands, modnames = Modnames,
              newdata = data.frame(Group = c("trt1", "cont")))
##trt1 differs from cont

##trt2 - control
modavgEffect(cand.set = Cands, modnames = Modnames,
              newdata = data.frame(Group = c("trt2", "cont")))
##trt2 does not differ from cont

## End(Not run)


##two-way ANOVA type design, Ex. 13.1 (Zar 1984) of plasma calcium
##concentration (mg/100 ml) in birds as a function of sex and hormone
##treatment
## Not run: 
birds <- data.frame(Ca = c(16.87, 16.18, 17.12, 16.83, 17.19, 15.86,
                      14.92, 15.63, 15.24, 14.8, 19.07, 18.77, 17.63,
                      16.99, 18.04, 17.2, 17.64, 17.89, 16.78, 16.92,
                      32.45, 28.71, 34.65, 28.79, 24.46, 30.54, 32.41,
                      28.97, 28.46, 29.65),
                    Sex = c("M", "M", "M", "M", "M", "F", "F", "F", "F",
                      "F", "M", "M", "M", "M", "M", "F", "F", "F", "F",
                      "F", "M", "M", "M", "M", "M", "F", "F", "F", "F",
                      "F"),
                    Hormone = as.factor(c(1, 1, 1, 1, 1, 1, 1, 1, 1,
                      1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3,
                      3, 3, 3, 3, 3)))

##candidate models
##interactive effects
m.inter <- lm(Ca ~ Sex + Hormone + Sex:Hormone, data = birds)

##additive effects
m.add <- lm(Ca ~ Sex + Hormone, data = birds)

##Sex only
m.sex <- lm(Ca ~ Sex, data = birds)

##Hormone only
m.horm <- lm(Ca ~ Hormone, data = birds)

##null
m.0 <- lm(Ca ~ 1, data = birds)

##model selection
Cands <- list(m.inter, m.add, m.sex, m.horm, m.0)
Mods <- c("interaction", "additive", "sex only", "horm only", "null")
aictab(Cands, Mods)
##there is some support for a hormone only treatment, but also for
##additive effects

##compute model-averaged effects of sex, and set the other variable
##to a constant value
##M - F
sex.data <- data.frame(Sex = c("M", "F"), Hormone = c("1", "1"))
modavgEffect(Cands, Mods, newdata = sex.data)
##no support for a sex main effect

##hormone 1 - 3, but set Sex to a constant value
horm1.data <- data.frame(Sex = c("M", "M"), Hormone = c("1", "3"))
modavgEffect(Cands, Mods, newdata = horm1.data)

##hormone 2 - 3, but set Sex to a constant value
horm2.data <- data.frame(Sex = c("M", "M"), Hormone = c("2", "3"))
modavgEffect(Cands, Mods, newdata = horm2.data)

## End(Not run)


##Poisson regression with anuran larvae example from Mazerolle (2006)
## Not run: 
data(min.trap)
##assign "UPLAND" as the reference level as in Mazerolle (2006)          
min.trap$Type <- relevel(min.trap$Type, ref = "UPLAND") 

##set up candidate models          
Cand.mod <- list( )
##global model          
Cand.mod[[1]] <- glm(Num_anura ~ Type + log.Perimeter,
                     family = poisson, offset = log(Effort),
                     data = min.trap) 
Cand.mod[[2]] <- glm(Num_anura ~ log.Perimeter, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[3]] <- glm(Num_anura ~ Type, family = poisson,
                     offset = log(Effort), data = min.trap) 
Cand.mod[[4]] <- glm(Num_anura ~ 1, family = poisson,
                     offset = log(Effort), data = min.trap) 
          
##check c-hat for global model
vif.hat <- c_hat(Cand.mod[[1]]) #uses Pearson's chi-square/df

##assign names to each model
Modnames <- c("type + logperim", "type", "logperim", "intercept only") 

##compute model-averaged estimate of difference between abundance at bog
##pond and upland pond
##create newdata object to make predictions
pred.data <- data.frame(Type = c("BOG", "UPLAND"),
                        log.Perimeter = mean(min.trap$log.Perimeter),
                        Effort = mean(min.trap$Effort))
modavgEffect(Cand.mod, Modnames, newdata = pred.data, c.hat = vif.hat,
             type = "response")
##little suport for a pond type effect

## End(Not run)


##mixed linear model example from ?nlme
## Not run: 
library(nlme)
Cand.models <- list( )
Cand.models[[1]] <- lme(distance ~ age, data = Orthodont, method="ML")
Cand.models[[2]] <- lme(distance ~ age + Sex, data = Orthodont,
                        random = ~ 1, method="ML")
Cand.models[[3]] <-lme(distance ~ 1, data = Orthodont, random = ~ 1,
                       method="ML")
Cand.models[[4]] <-lme(distance ~ Sex, data = Orthodont, random = ~ 1,
                       method="ML")

Modnames <- c("age", "age + sex", "null", "sex")

data.other <- data.frame(age = mean(Orthodont$age),
                         Sex = factor(c("Male", "Female"))) 
modavgEffect(cand.set = Cand.models, modnames = Modnames,
             newdata = data.other, conf.level = 0.95, second.ord = TRUE,
             nobs = NULL, uncond.se = "revised")
detach(package:nlme)

## End(Not run)


##site occupancy analysis example
## Not run: 
library(unmarked)
##single season model
data(frogs)
pferUMF <- unmarkedFrameOccu(pfer.bin)
##create a bogus site group
site.group <- c(rep(1, times = nrow(pfer.bin)/2), rep(0, nrow(pfer.bin)/2))

## add some fake covariates for illustration
siteCovs(pferUMF) <- data.frame(site.group, sitevar1 =
                                rnorm(numSites(pferUMF)),
                                sitevar2 = runif(numSites(pferUMF)))
     
## observation covariates are in site-major, observation-minor order
obsCovs(pferUMF) <- data.frame(obsvar1 =
                               rnorm(numSites(pferUMF) * obsNum(pferUMF)))
     
fm1 <- occu(~ obsvar1 ~ site.group, pferUMF)
fm2 <- occu(~ obsvar1 ~ 1, pferUMF)

Cand.mods <- list(fm1, fm2)
Modnames <- c("fm1", "fm2")

##model selection table
aictab(cand.set = Cand.mods, modnames = Modnames, second.ord = TRUE)

##model-averaged effect sizes comparing site.group 1 - site.group 0
newer.dat <- data.frame(site.group = c(0, 1))

modavgEffect(cand.set = Cand.mods, modnames = Modnames, type = "response",
              second.ord = TRUE, newdata = newer.dat, parm.type = "psi")
##no support for an effect of site group

## End(Not run)


##single season N-mixture models
## Not run: 
data(mallard)
##this variable was created to illustrate the use of modavgEffect
##with detection variables
mallard.site$site.group <- c(rep(1, 119), rep(0, 120))
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
siteCovs(mallardUMF)
tmp.covs <- obsCovs(mallardUMF)
obsCovs(mallardUMF)$date2 <- tmp.covs$date^2
(fm.mall <- pcount(~ site.group ~ length + elev + forest, mallardUMF, K=30))
(fm.mallb <- pcount(~ 1 ~ length + elev + forest, mallardUMF, K=30))
     
Cands <- list(fm.mall, fm.mallb)
Modnames <- c("one", "null")

##model averaged effect size of site.group 1 - site.group 0 on response
##scale (point estimate)
modavgEffect(Cands, Modnames, newdata = data.frame(site.group = c(0, 1)),
              parm.type = "detect", type = "response")

##model averaged effect size of site.group 1 - site.group 0 on link
##scale (here, logit link)
modavgEffect(Cands, Modnames, newdata = data.frame(site.group = c(0, 1)),
              parm.type = "detect", type = "link")

detach(package:unmarked)

## End(Not run)

Compute Model-averaged Parameter Estimate from User-supplied Information Criterion

Description

This function model-averages the estimate of a parameter of interest among a set of candidate models, and computes the unconditional standard error and unconditional confidence intervals as described in Buckland et al. (1997) and Burnham and Anderson (2002). Computations are based on the values of the information criterion supplied manually by the user.

Usage

modavgIC(ic, K, modnames = NULL, estimate, se, uncond.se = "revised",
         conf.level = 0.95, ic.name = NULL)

Arguments

ic

a vector of information criterion values for each model in the candidate model set.

K

a vector containing the number of estimated parameters for each model in the candidate model set.

modnames

a character vector of model names to identify each model in the model selection table. If NULL, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as the information criterion values.

estimate

a vector of estimates for each of the models in the candidate model set. Estimates can be either beta estimates for a parameter of interest or a single prediction from each model.

se

a vector of standard errors for each of the estimates appearing in the estimate vector.

uncond.se

either, "old", or "revised", specifying the equation used to compute the unconditional standard error of a model-averaged estimate. With uncond.se = "old", computations are based on equation 4.9 of Burnham and Anderson (2002), which was the former way to compute unconditional standard errors. With uncond.se = "revised", equation 6.12 of Burnham and Anderson (2002) is used. Anderson (2008, p. 111) recommends use of the revised version for the computation of unconditional standard errors and it is now the default.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of unconditional confidence intervals.

ic.name

a character string denoting the name of the information criterion input by the user. This character string will appear in certain column labels of the model selection table.

Details

modavgIC computes a model-averaged estimate from the vector of parameter estimates specified in estimate. Estimates and their associated standard errors must be specified in the same order as the values of the information criterion, the number of estimated parameters, and the model names. Estimates provided may be for a parameter of interest (i.e., beta estimates) or predictions from each model. This function is most useful for information criterion other than AIC, AICc, QAIC, and QAICc (e.g., WAIC: Watanabe 2010) or for classes not supported by modavg, modavgCustom, or modavgPred.

Value

modavgIC creates an object of class modavgIC with the following components:

Mod.avg.table

the model selection table.
Mod.avg.est

the model-averaged estimate.
Uncond.SE

the unconditional standard error for the model-averaged estimate.
Conf.level

the confidence level used to compute the confidence interval.
Lower.CL

the lower confidence limit.
Upper.CL

the upper confidence limit.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Buckland, S. T., Burnham, K. P., Augustin, N. H. (1997) Model selection: an integral part of inference. Biometrics 53, 603–618.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Watanabe, S. (2010) Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research 11, 3571–3594.

See Also

aictabCustom, ictab, modavg, modavgCustom, modavgShrink, modavgPred

Examples

## Not run: 
##model averaging parameter estimate based on WAIC
##create a vector of names to trace back models in set
Modnames <- c("global model", "interactive model",
              "additive model", "invertpred model")

##WAIC values
waic <- c(105.74, 107.36, 108.24, 100.57)
##number of effective parameters
effK <- c(7.45, 5.61, 6.14, 6.05)

##vector of predictions
Preds <- c(0.106, 0.137, 0.067, 0.050)
##vector of SE's for prediction
Ses <- c(0.128, 0.159, 0.054, 0.039)

##compute model-averaged estimate and unconditional SE based on WAIC
modavgIC(ic = waic, K = effK, modnames = Modnames, 
         estimate = Preds, se = Ses,
         ic.name = "WAIC")

## End(Not run)

Compute Model-averaged Predictions

Description

This function computes the model-averaged predictions, unconditional standard errors, and confidence intervals based on the entire candidate model set. The function is currently implemented for glm, gls, lm, lme, mer, merMod, lmerModLmerTest, negbin, rlm, survreg object classes that are stored in a list as well as various models of unmarkedFit classes.

Usage

modavgPred(cand.set, modnames = NULL, newdata, second.ord = TRUE,
           nobs = NULL, uncond.se = "revised", conf.level = 0.95, ...)

## S3 method for class 'AICaov.lm'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...) 

## S3 method for class 'AICglm.lm'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        gamdisp = NULL, ...)

## S3 method for class 'AIClm'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...) 

## S3 method for class 'AICgls'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...) 

## S3 method for class 'AIClme'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICmer'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1, ...)

## S3 method for class 'AICglmerMod'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1, ...)

## S3 method for class 'AIClmerMod'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClmerModLmerTest'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICnegbin.glm.lm'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", ...)

## S3 method for class 'AICrlm.lm'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...) 

## S3 method for class 'AICsurvreg'
modavgPred(cand.set, modnames = NULL, newdata,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", ...) 

## S3 method for class 'AICunmarkedFitOccu'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised", 
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitColExt'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)  

## S3 method for class 'AICunmarkedFitOccuRN'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCount'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...) 

## S3 method for class 'AICunmarkedFitPCO'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDS'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...) 

## S3 method for class 'AICunmarkedFitGDS'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMPois'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGMM'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGPC'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMMO'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDSO'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
modavgPred(cand.set, modnames = NULL,
        newdata, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, type = "response", c.hat = 1,
        parm.type = NULL, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

newdata

a data frame with the same structure as that of the original data frame for which we want to make predictions.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

uncond.se

either, old, or revised, specifying the equation used to compute the unconditional standard error of a model-averaged estimate. With uncond.se = "old", computations are based on equation 4.9 of Burnham and Anderson (2002), which was the former way to compute unconditional standard errors. With uncond.se = "revised", equation 6.12 of Burnham and Anderson (2002) is used. Anderson (2008, p. 111) recommends use of the revised version for the computation of unconditional standard errors and it is now the default. Note that versions of package AICcmodavg < 1.04 used the old method to compute unconditional standard errors.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of unconditional confidence intervals.

type

the scale of prediction requested, one of response or link. The latter is only relevant for glm, mer, and unmarkedFit classes. Note that the value terms is not defined for modavgPred.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season and dynamic occupancy models (MacKenzie et al. 2002, 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, modavgPred will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer class.

gamdisp

the value of the gamma dispersion parameter.

parm.type

this argument specifies the parameter type on which the predictions will be computed and is only relevant for models of unmarkedFit classes. The character strings supported vary with the type of model fitted. For unmarkedFitOccu objects, either psi or detect can be supplied to indicate whether the parameter is on occupancy or detectability, respectively. For unmarkedFitColExt objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and detectability, respectively. For unmarkedFitOccuTTD objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and time-to-dection (lambda rate parameter), respectively. For unmarkedFitOccuFP objects, one can specify psi, detect, falsepos, and certain, for occupancy, detectability, probability of assigning false-positives, and probability detections are certain, respectively. For unmarkedFitOccuRN objects, either lambda or detect can be entered for abundance and detectability parameters, respectively. For unmarkedFitPCount and unmarkedFitMPois objects, lambda or detect denote parameters on abundance and detectability, respectively. For unmarkedFitPCO, unmarkedFitMMO, and unmarkedFitDSO objects, one can enter lambda, gamma, omega, iota, or detect, to specify parameters on abundance, recruitment, apparent survival, immigration, and detectability, respectively. For unmarkedFitDS objects, lambda and detect are supported. For unmarkedFitGDS, lambda, phi, and detect denote abundance, availability, and detection probability, respectively. For unmarkedFitGMM and unmarkedFitGPC objects, lambda, phi, and detect denote abundance, availability, and detectability, respectively. For unmarkedFitOccuMS objects, psi, phi, and detect denote occupancy, transition, and detection probability, respectively. For unmarkedFitOccuMulti objects, psi and detect denote occupancy and detection probability, respectively.

...

additional arguments passed to the function.

Details

The candidate models must be stored in a list. Note that a data frame from which to make predictions must be supplied with the newdata argument and that all variables appearing in the model set must appear in this data frame. Variables must be of the same type as in the original analysis (e.g., factor, numeric).

One can compute unconditional confidence intervals around the predictions from the elements returned by modavgPred. The classic computation based on asymptotic normality of the estimator is appropriate to estimate confidence intervals on the linear predictor (i.e., link scale). For predictions of some types of response variables such as counts or binary variables, the normal approximation may be inappropriate. In such cases, it is often better to compute the confidence intervals on the linear predictor scale and then back-transform the limits to the scale of the response variable. These are the confidence intervals returned by modavgPred. Burnham et al. (1987), Burnham and Anderson (2002, p. 164), and Williams et al. (2002) suggest alternative methods of computing confidence intervals for small degrees of freedom with profile likelihood intervals or bootstrapping, but these approaches are not yet implemented in modavgPred.

Value

modavgPred returns an object of class modavgPred with the following components:

type

the scale of predicted values (response or link) for glm, mer, merMod, or unmarkedFit classes.

mod.avg.pred

the model-averaged prediction over the entire candidate model set.

uncond.se

the unconditional standard error of each model-averaged prediction.

conf.level

the confidence level used to compute the confidence interval.

lower.CL

the lower confidence limit.

upper.CL

the upper confidence limit.

matrix.output

a matrix with rows consisting of the model-averaged predictions, the unconditional standard errors, and the confidence limits.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Burnham, K. P., Anderson, D. R., White, G. C., Brownie, C., Pollock, K. H. (1987) Design and analysis methods for fish survival experiments based on release-recapture. American Fisheries Society Monographs 5, 1–437.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Williams, B. K., Nichols, J. D., Conroy, M. J. (2002) Analysis and Management of Animal Populations. Academic Press: New York.

See Also

AICc, aictab, importance, c_hat, confset, evidence, modavg, modavgCustom, modavgEffect, modavgShrink, predict, predictSE

Examples

##example from subset of models in Table 1 in Mazerolle (2006)
data(dry.frog)

Cand.models <- list( )
Cand.models[[1]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2,
                       data = dry.frog)
Cand.models[[2]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2 +
                       Shade:Substrate, data = dry.frog)
Cand.models[[3]] <- lm(log_Mass_lost ~ cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[4]] <- lm(log_Mass_lost ~ Shade + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[4]] <- lm(log_Mass_lost ~ Shade + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Cand.models[[5]] <- lm(log_Mass_lost ~ Substrate + cent_Initial_mass +
                       Initial_mass2, data = dry.frog)

##setup model names
Modnames <- paste("mod", 1:length(Cand.models), sep = "")

##compute model-averaged value and unconditional SE of predicted log of
##mass lost for frogs of average mass in shade for each substrate type

##first create data set to use for predictions
new.dat <- data.frame(Shade = c(1, 1, 1),
                      cent_Initial_mass = c(0, 0, 0),
                      Initial_mass2 = c(0, 0, 0),
                      Substrate = c("SOIL", "SPHAGNUM", "PEAT")) 

##compare unconditional SE's using both methods
modavgPred(cand.set = Cand.models, modnames = Modnames,
           newdata = new.dat, type = "response", uncond.se = "old")
modavgPred(cand.set = Cand.models, modnames = Modnames,
           newdata = new.dat, type = "response", uncond.se = "revised")
##round to 4 digits after decimal point
print(modavgPred(cand.set = Cand.models, modnames = Modnames,
                 newdata = new.dat, type = "response",
                 uncond.se = "revised"), digits = 4)



##Gamma glm
## Not run: 
##clotting data example from 'gamma.shape' in MASS package of
##Venables and Ripley (2002, Modern applied statistics with
##S. Springer-Verlag: New York.)
clotting <- data.frame(u = c(5, 10, 15, 20, 30, 40, 60, 80, 100),
                       lot1 = c(118, 58, 42, 35, 27, 25, 21, 19, 18),
                       lot2 = c(69, 35, 26, 21, 18, 16, 13, 12, 12))
clot1 <- glm(lot1 ~ log(u), data = clotting, family = Gamma)

require(MASS)
gamma.dispersion(clot1) #dispersion parameter
gamma.shape(clot1) #reciprocal of dispersion parameter ==
##shape parameter 
summary(clot1, dispersion = gamma.dispersion(clot1))  #better

##create list with models
Cand <- list( )
Cand[[1]] <- glm(lot1 ~ log(u), data = clotting, family = Gamma)
Cand[[2]] <- glm(lot1 ~ 1, data = clotting, family = Gamma)

##create vector of model names
Modnames <- paste("mod", 1:length(Cand), sep = "")

##compute model-averaged predictions on scale of response variable for
##all observations
modavgPred(cand.set = Cand, modnames = Modnames, newdata = clotting,
           gamdisp = gamma.dispersion(clot1), type = "response") 

##compute model-averaged predictions on scale of linear predictor
modavgPred(cand.set = Cand, modnames = Modnames, newdata = clotting,
           gamdisp = gamma.dispersion(clot1), type = "link")

##compute model-averaged predictions on scale of linear predictor
modavgPred(cand.set = Cand, modnames = Modnames, newdata = clotting,
           gamdisp = gamma.dispersion(clot1), type = "terms") #returns an error
##because type = "terms" is not defined for 'modavgPred'

modavgPred(cand.set = Cand, modnames = Modnames, newdata = clotting,
           type = "terms") #returns an error because
##no gamma dispersion parameter was specified (i.e., 'gamdisp' missing) 

## End(Not run)


##example of model-averaged predictions from N-mixture model
##each variable appears twice in the models - this is a bit longer
## Not run: 
require(unmarked)
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
##set up models so that each variable on abundance appears twice
fm.mall.one <- pcount(~ ivel + date  ~ length + forest, mallardUMF,
                      K = 30)
fm.mall.two <- pcount(~ ivel + date  ~ elev + forest, mallardUMF,
                      K = 30)
fm.mall.three <- pcount(~ ivel + date  ~ length + elev, mallardUMF,
                        K = 30)
fm.mall.four <- pcount(~ ivel + date  ~ 1, mallardUMF, K = 30)

##model list
Cands <- list(fm.mall.one, fm.mall.two, fm.mall.three, fm.mall.four)
Modnames <- c("length + forest", "elev + forest", "length + elev",
              "null")

##compute model-averaged predictions of abundance for values of elev
modavgPred(cand.set = Cands, modnames = Modnames, newdata =
           data.frame(elev = seq(from = -1.4, to = 2.4, by = 0.1),
                      length = 0, forest = 0), parm.type = "lambda",
           type = "response")

##compute model-averaged predictions of detection for values of ivel
modavgPred(cand.set = Cands, modnames = Modnames, newdata =
           data.frame(ivel = seq(from = -1.75, to = 5.9, by = 0.5),
                      date = 0), parm.type = "detect",
           type = "response")
detach(package:unmarked)

## End(Not run)


##example of model-averaged abundance from distance model
## Not run: 
##this is a bit longer
data(linetran) #example from ?distsamp
     
ltUMF <- with(linetran, {
  unmarkedFrameDS(y = cbind(dc1, dc2, dc3, dc4),
                  siteCovs = data.frame(Length, area, habitat),
                  dist.breaks = c(0, 5, 10, 15, 20),
                  tlength = linetran$Length * 1000, survey = "line",
                  unitsIn = "m")
})
     
## Half-normal detection function. Density output (log scale). No covariates.
fm1 <- distsamp(~ 1 ~ 1, ltUMF)
     
## Halfnormal. Covariates affecting both density and and detection.
fm2 <- distsamp(~area + habitat ~ habitat, ltUMF)

## Hazard function. Covariates affecting both density and and detection.
fm3 <- distsamp(~area + habitat ~ habitat, ltUMF, keyfun="hazard")

##assemble model list
Cands <- list(fm1, fm2, fm3)
Modnames <- paste("mod", 1:length(Cands), sep = "")

##model-average predictions on abundance
modavgPred(cand.set = Cands, modnames = Modnames, parm.type = "lambda", type = "link",
           newdata = data.frame(area = mean(linetran$area), habitat = c("A", "B")))
detach(package:unmarked)

## End(Not run)



##example using Orthodont data set from Pinheiro and Bates (2000)
## Not run: 
require(nlme)

##set up candidate models
m1 <- gls(distance ~ age, correlation = corCompSymm(value = 0.5, form = ~ 1 | Subject),
          data = Orthodont, method = "ML")

m2 <- gls(distance ~ 1, correlation = corCompSymm(value = 0.5, form = ~ 1 | Subject),
          data = Orthodont, method = "ML")

##assemble in list
Cand.models <- list(m1, m2)
##model names
Modnames <- c("age effect", "null model")

##model selection table
aictab(cand.set = Cand.models, modnames = Modnames)

##model-averaged predictions
modavgPred(cand.set = Cand.models, modnames = Modnames, newdata =
data.frame(age = c(8, 10, 12, 14)))
detach(package:nlme)

## End(Not run)

Compute Model-averaged Parameter Estimate with Shrinkage (Multimodel Inference)

Description

This function computes an alternative version of model-averaging parameter estimates that consists in shrinking estimates toward 0 to reduce model selection bias as in Burnham and Anderson (2002, p. 152), Anderson (2008, pp. 130-132) and Lukacs et al. (2010). Specifically, models without the parameter of interest have an estimate and variance of 0. modavgShrink also returns unconditional standard errors and unconditional confidence intervals as described in Buckland et al. (1997) and Burnham and Anderson (2002).

Usage

modavgShrink(cand.set, parm, modnames = NULL, second.ord = TRUE,
              nobs = NULL, uncond.se = "revised", conf.level = 0.95,
              ...)
## S3 method for class 'AICaov.lm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICbetareg'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICsclm.clm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICclm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICclmm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICcoxme'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICcoxph'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICglm.lm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, gamdisp = NULL, ...)

## S3 method for class 'AICgls'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICglmmTMB'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, ...)

## S3 method for class 'AIChurdle'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClme'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClmekin'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICmer'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICglmerMod'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClmerMod'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AIClmerModLmerTest'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICmaxlikeFit.list'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, ...)

## S3 method for class 'AICmultinom.nnet'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, ...)

## S3 method for class 'AICnegbin.glm.lm'
modavgShrink(cand.set, parm, modnames = NULL, 
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICpolr'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICrlm.lm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICsurvreg'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICvglm'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, ...)

## S3 method for class 'AICzeroinfl'
modavgShrink(cand.set, parm, modnames = NULL,
        second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, ...)

## S3 method for class 'AICunmarkedFitOccu'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitColExt'
modavgShrink(cand.set, parm, modnames
           = NULL, second.ord = TRUE, nobs = NULL, uncond.se =
           "revised", conf.level = 0.95, c.hat = 1, parm.type = NULL,
           ...)

## S3 method for class 'AICunmarkedFitOccuRN'
modavgShrink(cand.set, parm, modnames
        = NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCount'
modavgShrink(cand.set, parm, modnames
        = NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitPCO'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDS'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGDS'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuFP'
modavgShrink(cand.set, parm, modnames
        = NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMPois'
modavgShrink(cand.set, parm, modnames
        = NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGMM'
modavgShrink(cand.set, parm, modnames
        = NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitGPC'
modavgShrink(cand.set, parm, modnames
        = NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMulti'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuMS'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitOccuTTD'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitMMO'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

## S3 method for class 'AICunmarkedFitDSO'
modavgShrink(cand.set, parm, modnames =
        NULL, second.ord = TRUE, nobs = NULL, uncond.se = "revised",
        conf.level = 0.95, c.hat = 1, parm.type = NULL, ...)

Arguments

cand.set

a list storing each of the models in the candidate model set.

parm

the parameter of interest, enclosed between quotes, for which a model-averaged estimate is required. For a categorical variable, the label of the estimate must be included as it appears in the output (see 'Details' below).

modnames

a character vector of model names to facilitate the identification of each model in the model selection table. If NULL, the function uses the names in the cand.set list of candidate models. If no names appear in the list, generic names (e.g., Mod1, Mod2) are supplied in the table in the same order as in the list of candidate models.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

uncond.se

either, "old", or "revised", specifying the equation used to compute the unconditional standard error of a model-averaged estimate. With uncond.se = "old", computations are based on equation 4.9 of Burnham and Anderson (2002), which was the former way to compute unconditional standard errors. With uncond.se = "revised", equation 6.12 of Burnham and Anderson (2002) is used. Anderson (2008, p. 111) recommends use of the revised version for the computation of unconditional standard errors and it is now the default. Note that versions of package AICcmodavg < 1.04 used the old method to compute unconditional standard errors.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of unconditional confidence intervals.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, modavgShrink will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

gamdisp

if gamma GLM is used, the dispersion parameter should be specified here to apply the same value to each model.

parm.type

this argument specifies the parameter type on which the model-averaged estimate of a predictor will be computed and is only relevant for models of unmarkedFit classes. The character strings supported vary with the type of model fitted. For unmarkedFitOccu objects, either psi or detect can be supplied to indicate whether the parameter is on occupancy or detectability, respectively. For unmarkedFitColExt objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and detectability, respectively. For unmarkedFitOccuTTD objects, possible values are psi, gamma, epsilon, and detect, for parameters on occupancy in the inital year, colonization, extinction, and time-to-dection (lambda rate parameter), respectively. For unmarkedFitOccuFP objects, one can specify psi, detect, falsepos, and certain, for occupancy, detectability, probability of assigning false-positives, and probability detections are certain, respectively. For unmarkedFitOccuRN objects, either lambda or detect can be entered for abundance and detectability parameters, respectively. For unmarkedFitPCount and unmarkedFitMPois objects, lambda or detect denote parameters on abundance and detectability, respectively. For unmarkedFitPCO, unmarkedFitMMO, and unmarkedFitDSO objects, one can enter lambda, gamma, omega, iota, or detect, to specify parameters on abundance, recruitment, apparent survival, immigration, and detectability, respectively. For unmarkedFitDS objects, lambda and detect are supported. For unmarkedFitGDS, lambda, phi, and detect denote abundance, availability, and detection probability, respectively. For unmarkedFitGMM and unmarkedFitGPC objects, lambda, phi, and detect denote abundance, availability, and detectability, respectively.

...

additional arguments passed to the function.

Details

The parameter for which a model-averaged estimate is requested must be specified with the parm argument and must be identical to its label in the model output (e.g., from summary). For factors, one must specify the name of the variable and the level of interest. The shrinkage version of model averaging is only appropriate for cases where each parameter is given an equal weighting in the model (i.e., each parameter must appear the same number of times in the models) and has the same interpretation across all models. As a result, models with interaction terms or polynomial terms are not supported by modavgShrink.

modavgShrink is implemented for a list containing objects of aov, betareg, clm, clmm, clogit, coxme, coxph, glm, glmmTMB, gls, hurdle, lm, lme, lmekin, maxlikeFit, mer, glmerMod, lmerMod, lmerModLmerTest, multinom, polr, rlm, survreg, vglm, zeroinfl classes as well as various models of unmarkedFit classes.

Value

modavgShrink creates an object of class modavgShrink with the following components:

Parameter

the parameter for which a model-averaged estimate with shrinkage was obtained.
Mod.avg.table

the model selection table based on models including the parameter of interest.
Mod.avg.beta

the model-averaged estimate based on all models.
Uncond.SE

the unconditional standard error for the model-averaged estimate (as opposed to the conditional SE based on a single model).
Conf.level

the confidence level used to compute the confidence interval.
Lower.CL

the lower confidence limit.
Upper.CL

the upper confidence limit.

Author(s)

Marc J. Mazerolle

References

Anderson, D. R. (2008) Model-based Inference in the Life Sciences: a primer on evidence. Springer: New York.

Buckland, S. T., Burnham, K. P., Augustin, N. H. (1997) Model selection: an integral part of inference. Biometrics 53, 603–618.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Burnham, K. P., Anderson, D. R. (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Lukacs, P. M., Burnham, K. P., Anderson, D. R. (2010) Model selection bias and Freedman's paradox. Annals of the Institute of Statistical Mathematics 62, 117–125.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

AICc, aictab, c_hat, importance, confset, evidence, modavg, modavgCustom, modavgPred

Examples

##cement example in Burnham and Anderson 2002
data(cement)
##setup same model set as in Table 3.2, p. 102         
Cand.models <- list( )
Cand.models[[1]] <- lm(y ~ x1 + x2, data = cement)
Cand.models[[2]] <- lm(y ~ x1 + x2 + x4, data = cement)          
Cand.models[[3]] <- lm(y ~ x1 + x2 + x3, data = cement)
Cand.models[[4]] <- lm(y ~ x1 + x4, data = cement)
Cand.models[[5]] <- lm(y ~ x1 + x3 + x4, data = cement)
Cand.models[[6]] <- lm(y ~ x2 + x3 + x4, data = cement)
Cand.models[[7]] <- lm(y ~ x1 + x2 + x3 + x4, data = cement)
Cand.models[[8]] <- lm(y ~ x3 + x4, data = cement)
Cand.models[[9]] <- lm(y ~ x2 + x3, data = cement)
Cand.models[[10]] <- lm(y ~ x4, data = cement)
Cand.models[[11]] <- lm(y ~ x2, data = cement)
Cand.models[[12]] <- lm(y ~ x2 + x4, data = cement)
Cand.models[[13]] <- lm(y ~ x1, data = cement)
Cand.models[[14]] <- lm(y ~ x1 + x3, data = cement)
Cand.models[[15]] <- lm(y ~ x3, data = cement)

##vector of model names
Modnames <- paste("mod", 1:15, sep="")

##AICc          
aictab(cand.set = Cand.models, modnames = Modnames)

##compute model-averaged estimate with shrinkage - each parameter
##appears 8 times in the models 
modavgShrink(cand.set = Cand.models, modnames = Modnames, parm = "x1")

##compare against classic model-averaging
modavg(cand.set = Cand.models, modnames = Modnames, parm = "x1")
##note that model-averaged estimate with shrinkage is closer to 0 than
##with the classic version

##remove a few models from the set and run again
Cand.unbalanced <- Cand.models[-c(3, 14, 15)]

##set up model names
Modnames <- paste("mod", 1:length(Cand.unbalanced), sep="")

##issues an error because some parameters appear more often than others
## Not run: modavgShrink(cand.set = Cand.unbalanced,
                       modnames = Modnames, parm = "x1")
## End(Not run)



##example on Orthodont data set in nlme
## Not run: 
require(nlme)

##set up candidate model list
##age and sex parameters appear in the same number of models
##same number of models with and without these parameters
Cand.models <- list( )
Cand.models[[1]] <- lme(distance ~ age, data = Orthodont, method = "ML") 
##random is ~ age | Subject as it is a grouped data frame
Cand.models[[2]] <- lme(distance ~ age + Sex, data = Orthodont,
                        random = ~ 1, method = "ML")
Cand.models[[3]] <- lme(distance ~ 1, data = Orthodont, random = ~ 1, 
                        method = "ML") 
Cand.models[[4]] <- lme(distance ~ Sex, data = Orthodont, random = ~ 1,
                        method = "ML")  

##create a vector of model names
Modnames <- paste("mod", 1:length(Cand.models), sep = "")

##compute importance values for age
imp.age <- importance(cand.set = Cand.models, parm = "age",
                      modnames = Modnames, second.ord = TRUE,
                      nobs = NULL)

##compute shrinkage version of model averaging on age
mod.avg.age.shrink <- modavgShrink(cand.set = Cand.models,
                                    parm = "age", modnames = Modnames,
                                    second.ord = TRUE, nobs = NULL)

##compute classic version of model averaging on age
mod.avg.age.classic <- modavg(cand.set = Cand.models, parm = "age",
                              modnames = Modnames, second.ord = TRUE,
                              nobs = NULL)

##correspondence between shrinkage version and classic version of
##model averaging 
mod.avg.age.shrink$Mod.avg.beta/imp.age$w.plus
mod.avg.age.classic$Mod.avg.beta
detach(package:nlme)

## End(Not run)


##example of N-mixture model modified from ?pcount
## Not run: 
require(unmarked)
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
##set up models so that each variable on abundance appears twice
fm.mall.one <- pcount(~ ivel + date  ~ length + forest, mallardUMF,
                      K = 30)
fm.mall.two <- pcount(~ ivel + date  ~ elev + forest, mallardUMF,
                      K = 30)
fm.mall.three <- pcount(~ ivel + date  ~ length + elev, mallardUMF,
                        K = 30)

##model list and names
Cands <- list(fm.mall.one, fm.mall.two, fm.mall.three)
Modnames <- c("length + forest", "elev + forest", "length + elev")

##compute model-averaged estimate with shrinkage for elev on abundance
modavgShrink(cand.set = Cands, modnames = Modnames, parm = "elev",
              parm.type = "lambda")
detach(package:unmarked)

## End(Not run)

Create Model Selection Tables based on Multiple Comparisons

Description

This function is an alternative to traditional multiple comparison tests in designed experiments. It creates a model selection table based on different grouping patterns of a factor and computes model-averaged predictions for each of the factor levels. The current version works with objects of aov, glm, gls, lm, lme, mer, merMod, lmerModLmerTest, negbin, and rlm, survreg classes.

Usage

multComp(mod, factor.id, letter.labels = TRUE, second.ord = TRUE,
         nobs = NULL, sort = TRUE, newdata = NULL, uncond.se = "revised", 
         conf.level = 0.95, correction = "none", ...)

## S3 method for class 'aov'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        ...) 

## S3 method for class 'lm'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        ...)

## S3 method for class 'gls'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        ...) 

## S3 method for class 'glm'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        type = "response", c.hat = 1, gamdisp = NULL, ...)

## S3 method for class 'lme'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        ...)

## S3 method for class 'negbin'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        type = "response", ...)

## S3 method for class 'rlm'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        ...)

## S3 method for class 'survreg'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        type = "response", ...)

## S3 method for class 'mer'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95, correction = "none",
        type = "response", ...)

## S3 method for class 'merMod'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95,
        correction = "none", type = "response", ...)

## S3 method for class 'lmerModLmerTest'
multComp(mod, factor.id, letter.labels = TRUE,
        second.ord = TRUE, nobs = NULL, sort = TRUE, newdata = NULL,
        uncond.se = "revised", conf.level = 0.95,
        correction = "none", ...)

Arguments

mod

a model of one of the above-mentioned classes that includes at least one factor as an explanatory variable.

factor.id

the factor of interest, on which the groupings (multiple comparisons) are based. The user must supply the name of the categorical variable between quotes as it appears in the model formula.

letter.labels

logical. If TRUE, letters are used as labels to denote the grouping structure. If FALSE, numbers are used as group labels.

second.ord

logical. If TRUE, the function returns the second-order Akaike information criterion (i.e., AICc), otherwise returns Akaike's Information Criterion (AIC).

nobs

this argument allows to specify a numeric value other than total sample size to compute the AICc (i.e., nobs defaults to total number of observations). This is relevant only for certain types of models such as mixed models where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

sort

logical. If TRUE, the model selection table is ranked according to the (Q)AIC(c) values.

newdata

a data frame with the same structure as that of the original data frame for which we want to make predictions. This data frame should hold all variables constant other than the factor.id variable. All levels of the factor.id variables should be included in the newdata data frame to get model-averaged predictions for each level. If NULL, model-averaged predictions are computed for each level of the factor.id variable while the values of the other explanatory variables are taken from the first row of the original data set.

uncond.se

either, "old", or "revised", specifying the equation used to compute the unconditional standard error of a model-averaged estimate. With uncond.se = "old", computations are based on equation 4.9 of Burnham and Anderson (2002), which was the former way to compute unconditional standard errors. With uncond.se = "revised", equation 6.12 of Burnham and Anderson (2002) is used. Anderson (2008, p. 111) recommends use of the revised version for the computation of unconditional standard errors and it is now the default. Note that versions of package AICcmodavg < 1.04 used the old method to compute unconditional standard errors.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of unconditional confidence intervals around predicted values for each level of factor.id.

correction

the type of correction applied to obtain confidence intervals for simultaneous inference (i.e., corrected for multiple comparisons). Current corrections include "none" for uncorrected unconditional confidence intervals, "bonferroni" for Bonferroni-adjusted confidence intervals (Dunn 1961), and "sidak" for Sidak-adjusted confidence intervals (Sidak 1967).

type

the scale of prediction requested, one of "response" or "link". The latter is only relevant for glm and mer classes. Note that the value "terms" is not defined for multComp.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax) or with Poisson GLM's. If c.hat > 1, multComp will return the quasi-likelihood analogue of the information criterion requested. This option is not supported for generalized linear mixed models of the mer class.

gamdisp

the value of the gamma dispersion parameter in a gamma GLM.

...

additional arguments passed to the function.

Details

A number of pairwise comparison tests are available for traditional experimental designs, some controlling for the experiment-wise error and others for comparison-wise errors (Day and Quinn 1991). With the advent of information-theoretic approaches, there has been a need for methods analogous to multiple comparison tests in a model selection framework. Dayton (1998) and Burnham et al. (2011) suggested using different parameterizations or grouping patterns of a factor to perform multiple comparisons with model selection. As such, it is possible to assess the support in favor of certain grouping patterns based on a factor.

For example, a factor with three levels has four possible grouping patterns: {abc} (all groups are different), {abb} (the first group differs from the other two), {aab} (the first two groups differ from the third), and {aaa} (all groups are equal). multComp implements such an approach by pooling groups of the factor variable in a model and updating the model, for each grouping pattern possible. The models are ranked according to one of four information criteria (AIC, AICc, QAIC, and QAICc), and the labels in the table correspond to the grouping pattern. Note that the factor levels are sorted according to their means for the response variable before being assigned to a group. The function also returns model-averaged predictions and unconditional standard errors for each level of the factor.id variable based on the support in favor of each model (i.e., grouping pattern).

The number of grouping patterns increases substantially with the number of factor levels, as 2k−12^{k - 1}, where kk is the number of factor levels. multComp supports factors with a maximum of 6 levels. Also note that multComp does not handle models where the factor.id variable is involved in an interaction. In such cases, one should create the interaction variable manually before fitting the model (see Examples).

multComp currently implements three methods of computing confidence intervals. The default unconditional confidence intervals do not account for multiple comparisons (correction = "none"). With a large number mm of potential pairwise comparisons among levels of factor.id, there is an increased risk of type I error. For mm pairwise comparisons and a given α\alpha level, correction = "bonferroni" computes the unconditional confidence intervals based on αcorr=αm\alpha_{corr} = \frac{\alpha}{m} (Dunn 1961). When correction = "sidak", multComp reports Sidak-adjusted confidence intervals, i.e., αcorr=1−(1−α)1m\alpha_{corr} = 1 - (1 - \alpha)^{\frac{1}{m}}.

Value

multComp creates a list of class multComp with the following components:

factor.id

the factor for which grouping patterns are investigated.
models

a list with the output of each model representing a different grouping pattern for the factor of interest.
model.names

a vector of model names denoting the grouping pattern for each level of the factor.
model.table

the model selection table for the models corresponding to each grouping pattern for the factor of interest.
ordered.levels

the levels of the factor ordered according to the mean of the response variable. The grouping patterns (and model names) in the model selection table are based on the same order.
model.avg.est

a matrix with the model-averaged prediction, unconditional standard error, and confidence intervals for each level of the factor.
conf.level

the confidence level used for the confidence intervals.
correction

the type of correction applied to the confidence intervals to account for potential pairwise comparisons.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., Anderson, D. R., Huyvaert, K. P. (2011) AIC model selection and multimodel inference in behaviorial ecology: some background, observations and comparisons. Behavioral Ecology and Sociobiology 65, 23–25.

Day, R. W., Quinn, G. P. (1989) Comparisons of treatments after an analysis of variance in ecology. Ecological Monographs 59, 433–463.

Dayton, C. M. (1998) Information criteria for the paired-comparisons problem. American Statistician, 52 144–151.

Dunn, O. J. (1961) Multiple comparisons among means. Journal of the American Statistical Association 56, 52–64.

Sidak, Z. (1967) Rectangular confidence regions for the means of multivariate normal distributions. Journal of the American Statistical Association 62, 626–633.

See Also

aictab, confset, c_hat, evidence, glht, fit.contrast

Examples

##one-way ANOVA example
data(turkey)

##convert diet to factor
turkey$Diet <- as.factor(turkey$Diet)
##run one-way ANOVA
m.aov <- lm(Weight.gain ~ Diet, data = turkey)

##compute models with different grouping patterns
##and also compute model-averaged group means
out <- multComp(m.aov, factor.id = "Diet", correction = "none")
##look at results
out

##look at grouping structure of a given model
##and compare with original variable
cbind(model.frame(out$models[[2]]), turkey$Diet)

##evidence ratio
evidence(out$model.table)

##compute Bonferroni-adjusted confidence intervals
multComp(m.aov, factor.id = "Diet", correction = "bonferroni")


##two-way ANOVA with interaction
## Not run: 
data(calcium)

m.aov2 <- lm(Calcium ~ Hormone + Sex + Hormone:Sex, data = calcium)

##multiple comparisons
multComp(m.aov2, factor.id = "Hormone")
##returns an error because 'Hormone' factor is
##involved in an interaction

##create interaction variable
calcium$inter <- interaction(calcium$Hormone, calcium$Sex)

##run model with interaction
m.aov.inter <- lm(Calcium ~ inter, data = calcium)

##compare both
logLik(m.aov2)
logLik(m.aov.inter)
##both are identical

##multiple comparisons
multComp(m.aov.inter, factor.id = "inter")

## End(Not run)


##Poisson regression
## Not run: 
##example from ?glm
##Dobson (1990) Page 93: Randomized Controlled Trial :
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
print(d.AD <- data.frame(treatment, outcome, counts))
glm.D93 <- glm(counts ~ outcome + treatment, data = d.AD, family = poisson)

multComp(mod = glm.D93, factor.id = "outcome")

## End(Not run)


##example specifying 'newdata'
## Not run: 
data(dry.frog)
m1 <- lm(log_Mass_lost ~ Shade + Substrate +
      cent_Initial_mass + Initial_mass2,
      data = dry.frog)

multComp(m1, factor.id = "Substrate",
          newdata = data.frame(
            Substrate = c("PEAT", "SOIL", "SPHAGNUM"),
            Shade = 0, cent_Initial_mass = 0,
            Initial_mass2 = 0))

## End(Not run)

Newt Capture-mark-recapture Data

Description

This is a capture-mark-recapture data set on adult male and female Red-spotted Newts (Notophthalmus viridescens) recorded by Gill (1985). A total of 1079 unique individuals were captured in pitfall traps at a breeding site (White Oak Flat pond, Virginia) between 1975 and 1983.

Usage

data(newt)

Format

A data frame with 78 observations on the following 11 variables.

T1975

a binary variable, either 1 (captured) or 0 (not captured) during the 1975 breeding season.

T1976

a binary variable, either 1 (captured) or 0 (not captured) during the 1976 breeding season.

T1977

a binary variable, either 1 (captured) or 0 (not captured) during the 1977 breeding season.

T1978

a binary variable, either 1 (captured) or 0 (not captured) during the 1978 breeding season.

T1979

a binary variable, either 1 (captured) or 0 (not captured) during the 1979 breeding season.

T1980

a binary variable, either 1 (captured) or 0 (not captured) during the 1980 breeding season.

T1981

a binary variable, either 1 (captured) or 0 (not captured) during the 1981 breeding season.

T1982

a binary variable, either 1 (captured) or 0 (not captured) during the 1982 breeding season.

T1983

a binary variable, either 1 (captured) or 0 (not captured) during the 1983 breeding season.

Males

a numeric variable indicating the total number of males with a given capture history.

Females

a numeric variable indicating the total number of females with a given capture history.

Details

A single cohort of individuals was followed throughout the study, as all individuals were marked in 1975 and no new individuals were added during the subsequent years. This data set is used to illustrate classic Cormack-Jolly-Seber and related models (Cormack 1964, Jolly 1965, Seber 1965, Lebreton et al. 1992, Mazerolle 2015).

Source

Cormack, R. M. (1964) Estimates of survival from the sighting of marked animals. Biometrika 51, 429–438.

Gill, D. E. (1985) Interpreting breeding patterns from census data: a solution to the Husting dilemma. Ecology 66, 344–354.

Jolly, G. M. (1965) Explicit estimates from capture-recapture data with both death and immigration: stochastic model. Biometrika 52, 225–247.

Laake, J. L. (2013) RMark: an R interface for analysis of capture-recapture data with MARK. Alaska Fisheries Science Center (AFSC), National Oceanic and Atmospheric Administration, National Marine Fisheries Service, AFSC Report 2013-01.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67-118.

Mazerolle, M. J. (2015) Estimating detectability and biological parameters of interest with the use of the R environment. Journal of Herpetology 49, 541–559.

Seber, G. A. F. (1965) A note on the multiple-recapture census. Biometrika 52, 249–259.

Examples

data(newt)
str(newt)

##convert raw capture data to capture histories
captures <- newt[, c("T1975", "T1976", "T1977", "T1978", "T1979",
                      "T1980", "T1981", "T1982", "T1983")]
newt.ch <- apply(captures, MARGIN = 1, FUN = function(i)
                  paste(i, collapse = ""))

##organize as a data frame readable by RMark package (Laake 2013)
##RMark requires at least one column called "ch" 
##and another "freq" if summarized captures are provided
newt.full <- data.frame(ch = rep(newt.ch, 2),
                        freq = c(newt$Males, newt$Females),
                        Sex = c(rep("male", length(newt.ch)),
                        rep("female", length(newt.ch))))
str(newt.full)
newt.full$ch <- as.character(newt.full$ch)

##delete rows with 0 freqs
newt.full.orig <- newt.full[which(newt.full$freq != 0), ]

Compute Chi-square Goodness-of-fit Test for N-mixture Models

Description

These functions compute a goodness-of-fit test for N-mixture models based on Pearson's chi-square.

Usage

##methods for 'unmarkedFitPCount', 'unmarkedFitPCO', 
##'unmarkedFitDS', 'unmarkedFitGDS', 'unmarkedFitGMM',
##'unmarkedFitGPC', and 'unmarkedFitMPois' classes
Nmix.chisq(mod, ...)

Nmix.gof.test(mod, nsim = 5, plot.hist = TRUE, report = NULL,
parallel = TRUE, ncores, cex.axis = 1, cex.lab = 1, cex.main = 1,
lwd = 1, ...)

Arguments

mod

the N-mixture model of unmarkedFitPCount, unmarkedFitPCO, unmarkedFitDS, unmarkedFitGDS, unmarkedFitGMM, unmarkedFitGPC, unmarkedFitMPois, unmarkedFitMMO, or unmarkedFitDSO classes for which a goodness-of-fit test is required.

nsim

the number of bootstrapped samples.

plot.hist

logical. Specifies that a histogram of the bootstrapped test statistic is to be included in the output.

report

If NULL, the test statistic for each iteration is not printed in the terminal. Otherwise, an integer indicating the number of values of the test statistic that should be printed on the same line. For example, if report = 3, the values of the test statistic for three iterations are reported on each line.

parallel

logical. If TRUE, requests that parboot use multiple cores to accelerate computations of the bootstrap.

ncores

integer indicating the number of cores to use when bootstrapping in parallel during the analysis of simulated data sets. If ncores is not specified, one less than the number of available cores on the computer is used.

cex.axis

expansion factor influencing the size of axis annotations on plots produced by the function.

cex.lab

expansion factor influencing the size of axis labels on plots produced by the function.

cex.main

expansion factor influencing the size of the main title above plots produced by the function.

lwd

expansion factor of line width on plots produced by the function.

...

additional arguments passed to the function.

Details

The Pearson chi-square can be used to assess the fit of N-mixture models. Instead of relying on the theoretical distribution of the chi-square, a parametric bootstrap approach is implemented to obtain P-values with the parboot function of the unmarked package. Nmix.chisq computes the observed chi-square statistic based on the observed and expected counts from the model. Nmix.gof.test calls internally Nmix.chisq and parboot to generate simulated data sets based on the model and compute the chi-square test statistic.

It is also possible to obtain an estimate of the overdispersion parameter (c-hat) for the model at hand by dividing the observed chi-square statistic by the mean of the statistics obtained from simulation (MacKenzie and Bailey 2004, McKenny et al. 2006). This method of estimating c-hat is similar to the one implemented for capture-mark-recapture models in program MARK (White and Burnham 1999).

Note that values of c-hat > 1 indicate overdispersion (variance > mean). Values much higher than 1 (i.e., > 4) probably indicate lack-of-fit. In cases of moderate overdispersion, one can multiply the variance-covariance matrix of the estimates by c-hat. As a result, the SE's of the estimates are inflated (c-hat is also known as a variance inflation factor).

In model selection, c-hat should be estimated from the global model and the same value of c-hat applied to the entire model set. Specifically, a global model is the most complex model which can be simplified to yield all the other (nested) models of the set. When no single global model exists in the set of models considered, such as when sample size does not allow a complex model, one can estimate c-hat from 'subglobal' models. Here, 'subglobal' models denote models from which only a subset of the models of the candidate set can be derived. In such cases, one can use the smallest value of c-hat for model selection (Burnham and Anderson 2002).

Note that c-hat counts as an additional parameter estimated and should be added to K. All functions in package AICcmodavg automatically add 1 when the c.hat argument > 1 and apply the same value of c-hat for the entire model set. When c-hat > 1, functions compute quasi-likelihood information criteria (either QAICc or QAIC, depending on the value of the second.ord argument) by scaling the log-likelihood of the model by c-hat. The value of c-hat can influence the ranking of the models: as c-hat increases, QAIC or QAICc will favor models with fewer parameters. As an additional check against this potential problem, one can generate several model selection tables by incrementing values of c-hat to assess the model selection uncertainty. If ranking changes only slightly up to the c-hat value observed, one can be confident in making inference.

In cases of underdispersion (c-hat < 1), it is recommended to keep the value of c-hat to 1. However, note that values of c-hat << 1 can also indicate lack-of-fit and that an alternative model should be investigated.

Value

Nmix.chisq returns two value:

chi.square

the Pearson chi-square statistic.

model.type

the class of the fitted model.

Nmix.gof.test returns the following components:

model.type

the class of the fitted model.

chi.square

the Pearson chi-square statistic.

t.star

the bootstrapped chi-square test statistics (i.e., obtained for each of the simulated data sets).

p.value

the P-value assessed from the parametric bootstrap, computed as the proportion of the simulated test statistics greater than or equal to the observed test statistic.

c.hat.est

the estimate of the overdispersion parameter, c-hat, computed as the observed test statistic divided by the mean of the simulated test statistics.

nsim

the number of bootstrap samples. The recommended number of samples varies with the data set, but should be on the order of 1000 or 5000, and in cases with a large number of visits, even 10 000 samples, namely to reduce the effect of unusually small values of the test statistics.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

MacKenzie, D. I., Bailey, L. L. (2004) Assessing the fit of site-occupancy models. Journal of Agricultural, Biological, and Environmental Statistics 9, 300–318.

McKenny, H. C., Keeton, W. S., Donovan, T. M. (2006). Effects of structural complexity enhancement on eastern red-backed salamander (Plethodon cinereus) populations in northern hardwood forests. Forest Ecology and Management 230, 186–196.

White, G. C., Burnham, K. P. (1999). Program MARK: Survival estimation from populations of marked animals. Bird Study 46 (Supplement), 120–138.

See Also

AICc, c_hat, evidence, modavg, importance, mb.gof.test, modavgPred, pcount, pcountOpen, parboot

Examples

##N-mixture model example modified from ?pcount
## Not run: 
require(unmarked)
##single season
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
##run model
fm.mallard <- pcount(~ ivel+ date + I(date^2) ~ length + elev + forest,
                     mallardUMF, K=30)

##compute observed chi-square
obs <- Nmix.chisq(fm.mallard)
obs

##round to 4 digits after decimal point
print(obs, digits.vals = 4)

##compute observed chi-square, assess significance, and estimate c-hat
obs.boot <- Nmix.gof.test(fm.mallard, nsim = 10)
##note that more bootstrap samples are recommended
##(e.g., 1000, 5000, or 10 000)
obs.boot
print(obs.boot, digits.vals = 4, digits.chisq = 4)
detach(package:unmarked)

## End(Not run)

Strength of Pine Wood Based on the Density Adjusted for Resin Content

Description

This data set consists of the strength of pine wood as a function of density or density adjusted for resin content.

Usage

data(pine)

Format

A data frame with 42 observations on the following 3 variables.

y

pine wood strength.

x

pine wood density.

z

pine wood density adjusted for resin content.

Details

Burnham and Anderson (2002, p. 183) use this data set originally from Carlin and Chib (1995) to illustrate model selection for two competing and non-nested models.

Source

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Carlin, B. P., Chib, S. (1995) Bayesian model choice via Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B 57, 473–484.

Examples

data(pine)
## maybe str(pine) ; plot(pine) ...

Computing Predicted Values and Standard Errors

Description

Function to compute predicted values based on linear predictor and associated standard errors from various fitted models.

Usage

predictSE(mod, newdata, se.fit = TRUE, print.matrix = FALSE, ...)

## S3 method for class 'gls'
predictSE(mod, newdata, se.fit = TRUE, print.matrix = 
        FALSE, ...) 

## S3 method for class 'lme'
predictSE(mod, newdata, se.fit = TRUE, print.matrix =
        FALSE, level = 0, ...)

## S3 method for class 'mer'
predictSE(mod, newdata, se.fit = TRUE, print.matrix =
        FALSE, level = 0, type = "response", ...)

## S3 method for class 'merMod'
predictSE(mod, newdata, se.fit = TRUE, print.matrix =
        FALSE, level = 0, type = "response", ...)

## S3 method for class 'lmerModLmerTest'
predictSE(mod, newdata, se.fit = TRUE, print.matrix =
        FALSE, level = 0, ...)

## S3 method for class 'unmarkedFitPCount'
predictSE(mod, newdata, se.fit = TRUE,
        print.matrix = FALSE, type = "response", c.hat = 1, parm.type =
        "lambda", ...)

## S3 method for class 'unmarkedFitPCO'
predictSE(mod, newdata, se.fit = TRUE, 
        print.matrix = FALSE, type = "response", c.hat = 1,
        parm.type = "lambda", ...)

Arguments

mod

an object of class gls, lme, mer, merMod, lmerModLmerTest, unmarkedFitPCount, or unmarkedFitPCO containing the output of a model.

newdata

a data frame with the same structure as that of the original data frame for which we want to make predictions.

se.fit

logical. If TRUE, compute standard errors on predictions.

print.matrix

logical. If TRUE, the output is returned as a matrix, with predicted values and standard errors in columns. If FALSE, the output is returned as a list.

level

the level for which predicted values and standard errors are to be computed. The current version of the function only supports predictions for the populations excluding random effects (i.e., level = 0).

type

specifies the type of prediction requested. This argument can take the value response or link, for predictions on the scale of the response variable or on the scale of the linear predictor, respectively.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from Nmix.gof.test. If c.hat > 1, predictSE will multiply the variance-covariance matrix of the predictions by this value (i.e., SE's are multiplied by sqrt(c.hat)). High values of c.hat (e.g., c.hat > 4) may indicate that model structure is inappropriate.

parm.type

the parameter for which predictions are made based on the N-mixture model of class unmarkedFitPCount or unmarkedFitPCO classes.

...

additional arguments passed to the function.

Details

predictSE computes predicted values and associated standard errors. Standard errors are approximated using the delta method (Oehlert 1992). Predictions and standard errors for objects of gls class and mixed models of lme, mer, merMod, lmerModLmerTest classes exclude the correlation or variance structure of the model.

predictSE computes predicted values on abundance and standard errors based on the estimates from an unmarkedFitPCount or unmarkedFitPCO object. Currently, only predictions on abundance (i.e., parm.type = "lambda") with the zero-inflated Poisson distribution is supported. For other parameters or distributions for models of unmarkedFit classes, use predict from the unmarked package.

Value

predictSE returns requested values either as a matrix (print.matrix = TRUE) or list (print.matrix = FALSE) with components:

fit

the predicted values.

se.fit

the standard errors of the predicted values (if se.fit = TRUE).

Note

For standard errors with better properties, especially for small samples, one can opt for simulations (see Gelman and Hill 2007), or nonparametric bootstrap (Efron and Tibshirani 1998).

Author(s)

Marc J. Mazerolle

References

Efron, B., Tibshirani, R. J. (1998) An Introduction to the Bootstrap. Chapman & Hall/CRC: New York.

Gelman, A., Hill, J. (2007) Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press: New York.

Oehlert, G. W. (1992) A note on the delta method. American Statistician 46, 27–29.

See Also

gls, lme, glmer, simulate.merMod, boot, parboot, nonparboot, pcount, pcountOpen, unmarkedFit-class

Examples

##Orthodont data from Pinheiro and Bates (2000) revisited
## Not run: 
require(nlme)
m1 <- gls(distance ~ age, correlation = corCompSymm(value = 0.5, form = ~ 1 | Subject),
          data = Orthodont, method= "ML")

##compare against lme fit
logLik(m1)
logLik(lme(distance ~ age, random = ~1 | Subject, data = Orthodont,
          method= "ML"))
##both are identical


##compute predictions and SE's for different ages
predictSE(m1, newdata = data.frame(age = c(8, 10, 12, 14)))
detach(package:nlme)

## End(Not run)


##example with mallard data set from unmarked package
## Not run: 
require(unmarked)
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
                                  obsCovs = mallard.obs)
##run model with zero-inflated Poisson abundance
fm.mall.one <- pcount(~ ivel + date  ~ length + forest, mallardUMF, K=30,
                      mixture = "ZIP")
##make prediction
predictSE(fm.mall.one, type = "response", parm.type = "lambda",
          newdata = data.frame(length = 0, forest = 0, elev = 0))
##compare against predict
predict(fm.mall.one, type = "state", backTransform = TRUE,
        newdata = data.frame(length = 0, forest = 0, elev = 0))

##add offset in model to scale abundance per transect length
fm.mall.off <- pcount(~ ivel + date  ~ forest + offset(length), mallardUMF, K=30,
                      mixture = "ZIP")
##make prediction
predictSE(fm.mall.off, type = "response", parm.type = "lambda",
          newdata = data.frame(length = 10, forest = 0, elev = 0))
##compare against predict
predict(fm.mall.off, type = "state", backTransform = TRUE,
        newdata = data.frame(length = 10, forest = 0, elev = 0))
detach(package:unmarked)

## End(Not run)

Salamander Capture-mark-recapture Data

Description

This is a capture-mark-recapture data set on male and female Spotted Salamanders (Ambystoma maculatum) recorded by Husting (1965). A total of 1244 unique individuals were captured in pitfall traps at a breeding site between 1959 and 1963.

Usage

data(salamander)

Format

A data frame with 36 observations on the following 7 variables.

T1959

a binary variable, either 1 (captured) or 0 (not captured) during the 1959 breeding season.

T1960

a binary variable, either 1 (captured) or 0 (not captured) during the 1960 breeding season.

T1961

a binary variable, either 1 (captured) or 0 (not captured) during the 1961 breeding season.

T1962

a binary variable, either 1 (captured) or 0 (not captured) during the 1962 breeding season.

T1963

a binary variable, either 1 (captured) or 0 (not captured) during the 1963 breeding season.

Males

a numeric variable indicating the total number of males with a given capture history. Negative values indicate losses on capture (animals not released on last capture).

Females

a numeric variable indicating the total number of females with a given capture history. Negative values indicate losses on capture (animals not released on last capture).

Details

This data set is used to illustrate classic Cormack-Jolly-Seber and related models (Cormack 1964, Jolly 1965, Seber 1965, Lebreton et al. 1992).

Source

Cormack, R. M. (1964) Estimates of survival from the sighting of marked animals. Biometrika 51, 429–438.

Husting, E. L. (1965) Survival and breeding structure in a population of Ambystoma maculatum. Copeia 1965, 352–362.

Jolly, G. M. (1965) Explicit estimates from capture-recapture data with both death and immigration: stochastic model. Biometrika 52, 225–247.

Laake, J. L. (2013) RMark: an R interface for analysis of capture-recapture data with MARK. Alaska Fisheries Science Center (AFSC), National Oceanic and Atmospheric Administration, National Marine Fisheries Service, AFSC Report 2013-01.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67-118.

Seber, G. A. F. (1965) A note on the multiple-recapture census. Biometrika 52, 249–259.

Examples

data(salamander)
str(salamander)

##convert raw capture data to capture histories
captures <- salamander[, c("T1959", "T1960", "T1961", "T1962", "T1963")]
salam.ch <- apply(captures, MARGIN = 1, FUN = function(i)
                  paste(i, collapse = ""))

##organize as a data frame readable by RMark package (Laake 2013)
##RMark requires at least one column called "ch" 
##and another "freq" if summarized captures are provided
salam.full <- data.frame(ch = rep(salam.ch, 2),
                         freq = c(salamander$Males, salamander$Females),
                         Sex = c(rep("male", length(salam.ch)),
                           rep("female", length(salam.ch))))
str(salam.full)
salam.full$ch <- as.character(salam.full$ch)

##delete rows with 0 freqs
salam.full.orig <- salam.full[which(salam.full$freq != 0), ]

Display Model Summary Corrected for Overdispersion

Description

This function displays the estimates of a model with standard errors corrected for overdispersion for a variety of model classes. The output includes either confidence intervals based on the normal approximation or Wald hypothesis tests corrected for overdispersion.

Usage

summaryOD(mod, c.hat = 1, conf.level = 0.95, 
          out.type = "confint", ...)

## S3 method for class 'glm'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitOccu'
summaryOD(mod, c.hat = 1, conf.level = 0.95,
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitColExt'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitOccuRN'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitPCount'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitPCO'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitDS'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitGDS'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitOccuFP'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitMPois'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitGMM'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitGPC'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitOccuMulti'
summaryOD(mod, c.hat = 1, conf.level = 0.95,
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitOccuMS'
summaryOD(mod, c.hat = 1, conf.level = 0.95,
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitOccuTTD'
summaryOD(mod, c.hat = 1, conf.level = 0.95,
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitMMO'
summaryOD(mod, c.hat = 1, conf.level = 0.95,
     out.type = "confint", ...)

## S3 method for class 'unmarkedFitDSO'
summaryOD(mod, c.hat = 1, conf.level = 0.95,
     out.type = "confint", ...)

## S3 method for class 'glmerMod'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'maxlikeFit'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'multinom'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

## S3 method for class 'vglm'
summaryOD(mod, c.hat = 1, conf.level = 0.95, 
     out.type = "confint", ...)

Arguments

mod

an object of class glm, glmmTMB, maxlikeFit, mer, merMod, multinom, vglm, and various unmarkedFit classes containing the output of a model.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat, mb.gof.test, or Nmix.gof.test.

conf.level

the confidence level (1−α1 - \alpha) requested for the computation of confidence intervals.

out.type

the type of summary requested for each parameter estimate. If out.type = "confint", computes confidence intervals corrected for overdispersion, whereas out.type = "nhst" conducts null-hypothesis statistical testing corrected for overdispersion.

...

additional arguments passed to the function.

Details

Overdispersion occurs when the variance in the data exceeds that expected from a theoretical distribution such as the Poisson or binomial (McCullagh and Nelder 1989, Burnham and Anderson 2002). When the model is correct, small values of c-hat (1 < c-hat < 4) can reflect minor deviations from model assumptions (Burnham and Anderson 2002). In such cases, it is possible to adjust standard errors of parameter estimates by multiplying with sqrt(c.hat) (McCullagh and Nelder 1989). This is the correction applied by summaryOD.

Depending on the type of summary requested, i.e., out.type = "confint" or out.type = "nhst", summaryOD will return either confidence intervals based on the normal approximation or Wald tests for each parameter estimate (Agresti 1990).

For binomial distributions, note that values of c.hat > 1 are only appropriate with trials > 1 (i.e., success/trial or cbind(success, failure) syntax). The function supports different model types such as Poisson GLM's and GLMM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011).

Value

summaryOD returns an object of class summaryOD as a list with the following components:

out.type

the type of output requested by the user.
c.hat

the c.hat estimate used to adjust standard errors.
conf.level

the confidence level used to compute confidence intervals around the estimates.
outMat

the output of the model corrected for overdispersion organized in a matrix.

Author(s)

Marc J. Mazerolle

References

Agresti, A. (2002) Categorical Data Analysis. Second edition. John Wiley and Sons: New Jersey.

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Mazerolle, M. J. (2006) Improving data analysis in herpetology: using Akaike's Information Criterion (AIC) to assess the strength of biological hypotheses. Amphibia-Reptilia 27, 169–180.

McCullagh, P., Nelder, J. A. (1989) Generalized Linear Models. Second edition. Chapman and Hall: New York.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

See Also

c_hat, mb.gof.test, Nmix.gof.test, anovaOD

Examples

##anuran larvae example from Mazerolle (2006)
data(min.trap)
##assign "UPLAND" as the reference level as in Mazerolle (2006)          
min.trap$Type <- relevel(min.trap$Type, ref = "UPLAND") 

##run model
m1 <- glm(Num_anura ~ Type + log.Perimeter + Num_ranatra,
          family = poisson, offset = log(Effort),
          data = min.trap) 

##check c-hat for global model
c_hat(m1) #uses Pearson's chi-square/df

##display results corrected for overdispersion
summaryOD(m1, c_hat(m1))
summaryOD(m1, c_hat(m1), out.type = "nhst")

##example with occupancy model
## Not run: 
##load unmarked package
if(require(unmarked)){
   
   data(bullfrog)
     
   ##detection data
   detections <- bullfrog[, 3:9]

   ##assemble in unmarkedFrameOccu
   bfrog <- unmarkedFrameOccu(y = detections)
     
   ##run model
   fm <- occu(~ 1 ~ 1, data = bfrog)

   ##check GOF
   ##GOF <- mb.gof.test(fm, nsim = 1000)
   ##estimate of c-hat:  1.89

   ##display results after overdispersion adjustment
   summaryOD(fm, c.hat = 1.89)
   summaryOD(fm, c.hat = 1.89, out.type = "nhst")

   detach(package:unmarked)
}

## End(Not run)

Gopher Tortoise Distance Sampling Data

Description

This simulated data set by Mazerolle (2015) is based on the biological parameters for the Gopher Tortoise (Gopherus polyphemus) reported by Smith et al. (2009). A half-normal distribution with a scale of 10 and without an adjustment factor was used to simulate the distance data for a study area of 120 km2km^2. An effort of 500 m in 300 line transects was deployed. A density of 72 individuals per km2km^2 was used in the simulation using the approach outlined in Buckland et al. (2001).

Usage

data(tortoise)

Format

A data frame with 410 observations on the following 5 variables.

Region.Label

a numeric identifier for the study area.

Area

a numeric variable for the surface area of the study area in square meters.

Sample.Label

a numeric identifier for each line transect relating each observation to its corresponding transect.

Effort

Effort in meters expended in each line transect.

distance

a numeric variable for the perpendicular distances in meters relative to the transect line for each of the individuals detected during the survey. Note that transects without detections have a value of NA for this variable.

Details

This data set is used to illustrate classic distance sampling (Buckland et al. 2001, Mazerolle 2015).

Source

Buckland, S. T., Anderson, D. R., Burnham, K. P., Laake, J. L., Borchers, D. L., Thomas, L. (2001) Introduction to distance sampling: estimating abundance of biological populations. Oxford University Press: Oxford.

Mazerolle, M. J. (2015) Estimating detectability and biological parameters of interest with the use of the R environment. Journal of Herpetology 49, 541–559.

Smith, L. L., Linehan, J. M., Stober, J. M., Elliott, M. J., Jensen, J. B. (2009) An evaluation of distance sampling for large-scale gopher tortoise surveys in Georgia, USA. Applied Herpetology 6, 355–368.

Examples

data(tortoise)
str(tortoise)

##plot distance data to determine if truncation is required
##(Buckland et al. 2001, pp. 15--17)
hist(tortoise$distance)

Turkey Weight Gain

Description

This one-way ANOVA data set presents turkey weight gain in pounds across five diets.

Usage

data(turkey)

Format

A data frame with 30 rows and 2 variables.

Diet

diet factor with 5 levels.

Weight.gain

weight gain in pounds.

Details

Heiberger and Holland (2004) and Ott (1993) analyze this data set to illustrate one-way ANOVA.

Source

Heiberger, R. M., Holland, B. (2004) Statistical Analysis and Data Display: an intermediate course with examples in S-Plus, R, and SAS. Springer: New York.

Ott, R. L. (1993) An Introduction to Statistical Methods and Data Analysis. Fourth edition. Duxbury: Pacific Grove, CA.

Examples

data(turkey)
str(turkey)

Computing BIC or QBIC

Description

Functions to compute the Bayesian information criterion (BIC) or a quasi-likelihood analogue (QBIC).

Usage

useBIC(mod, return.K = FALSE, nobs = NULL, ...) 

## S3 method for class 'aov'
useBIC(mod, return.K = FALSE, nobs = NULL, ...) 

## S3 method for class 'betareg'
useBIC(mod, return.K = FALSE, nobs = NULL, ...) 

## S3 method for class 'clm'
useBIC(mod, return.K = FALSE, nobs = NULL, ...) 

## S3 method for class 'clmm'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)  

## S3 method for class 'coxme'
useBIC(mod, return.K = FALSE, nobs = NULL, ...) 

## S3 method for class 'coxph'
useBIC(mod, return.K = FALSE, nobs = NULL, ...) 

## S3 method for class 'fitdist'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'fitdistr'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'glm'
useBIC(mod, return.K = FALSE, nobs = NULL, c.hat = 1,
...)

## S3 method for class 'glmmTMB'
useBIC(mod, return.K = FALSE, nobs = NULL, c.hat = 1,
...)

## S3 method for class 'gls'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'gnls'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'hurdle'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'lavaan'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'lm'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'lme'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'lmekin'
useBIC(mod, return.K = FALSE, nobs = NULL, ...) 

## S3 method for class 'maxlikeFit'
useBIC(mod, return.K = FALSE, nobs = NULL, c.hat =
1, ...)

## S3 method for class 'mer'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'merMod'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)  

## S3 method for class 'lmerModLmerTest'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)  

## S3 method for class 'multinom'
useBIC(mod, return.K = FALSE, nobs = NULL, c.hat = 1,
...)

## S3 method for class 'nlme'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'nls'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'polr'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'rlm'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'survreg'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

## S3 method for class 'unmarkedFit'
useBIC(mod, return.K = FALSE, nobs = NULL, c.hat =
1, ...)

## S3 method for class 'vglm'
useBIC(mod, return.K = FALSE, nobs = NULL, c.hat = 1,
...)

## S3 method for class 'zeroinfl'
useBIC(mod, return.K = FALSE, nobs = NULL, ...)

Arguments

mod

an object of class aov, betareg, clm, clmm, clogit, coxme, coxph, fitdist, fitdistr, glm, glmmTMB, gls, gnls, hurdle, lavaan, lm, lme, lmekin, maxlikeFit, mer, merMod, lmerModLmerTest, multinom, nlme, nls, polr, rlm, survreg, vglm, zeroinfl, and various unmarkedFit classes containing the output of a model.

return.K

logical. If FALSE, the function returns the information criterion specified. If TRUE, the function returns K (number of estimated parameters) for a given model.

nobs

this argument allows to specify a numeric value other than total sample size to compute the BIC (i.e., nobs defaults to total number of observations). This is relevant only for mixed models or various models of unmarkedFit classes where sample size is not straightforward. In such cases, one might use total number of observations or number of independent clusters (e.g., sites) as the value of nobs.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season occupancy models (MacKenzie et al. 2002), dynamic occupancy models (MacKenzie et al. 2003), or N-mixture models (Royle 2004, Dail and Madsen 2011). If c.hat > 1, useBIC will return the quasi-likelihood analogue of the information criteria requested and multiply the variance-covariance matrix of the estimates by this value (i.e., SE's are multiplied by sqrt(c.hat)). This option is not supported for generalized linear mixed models of the mer or merMod classes.

...

additional arguments passed to the function.

Details

useBIC computes the Bayesian information criterion (BIC, Schwarz 1978):

BIC=−2∗log−likelihood+K∗log(n),BIC = -2 * log-likelihood + K * log(n),

where the log-likelihood is the maximum log-likelihood of the model, K corresponds to the number of estimated parameters, and n corresponds to the sample size of the data set.

In the presence of overdispersion, a quasi-likelihood analogue of the BIC (QBIC) will be computed, as

QBIC=−2∗log−likelihoodc−hat+K∗log(n),QBIC = \frac{-2 * log-likelihood}{c-hat} + K * log(n),

where c-hat is the overdispersion parameter specified by the user with the argument c.hat. Note that BIC or QBIC values are meaningful to select among gls or lme models fit by maximum likelihood. BIC or QBIC based on REML are valid to select among different models that only differ in their random effects (Pinheiro and Bates 2000).

Value

useBIC returns the BIC or the number of estimated parameters, depending on the values of the arguments.

Note

The actual (Q)BIC values are not really interesting in themselves, as they depend directly on the data, parameters estimated, and likelihood function. Furthermore, a single value does not tell much about model fit. Information criteria become relevant when compared to one another for a given data set and set of candidate models.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Pinheiro, J. C., Bates, D. M. (2000) Mixed-effect models in S and S-PLUS. Springer Verlag: New York.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461–464.

See Also

AICc, bictab, bictabCustom, useBICCustom

Examples

##cement data from Burnham and Anderson (2002, p. 101)
data(cement)
##run multiple regression - the global model in Table 3.2
glob.mod <- lm(y ~ x1 + x2 + x3 + x4, data = cement)

##compute BIC with full likelihood
useBIC(glob.mod, return.K = FALSE)

##compute BIC for mixed model on Orthodont data set in Pinheiro and
##Bates (2000)
## Not run: 
require(nlme)
m1 <- lme(distance ~ age, random = ~1 | Subject, data = Orthodont,
          method= "ML")
useBIC(m1, return.K = FALSE)

## End(Not run)

Custom Computation of BIC and QBIC from User-supplied Input

Description

This function computes the Bayesian information criterion (BIC) or a quasi-likelihood counterpart (QBIC) from user-supplied input instead of extracting the values automatically from a model object. This function is particularly useful for output imported from other software or for model classes that are not currently supported by useBIC.

Usage

useBICCustom(logL, K, return.K = FALSE, nobs = NULL, c.hat = 1)

Arguments

logL

the value of the model log-likelihood.

K

the number of estimated parameters in the model.

return.K

logical. If FALSE, the function returns the information criterion specified. If TRUE, the function returns K (number of estimated parameters) for a given model.

nobs

the sample size required to compute the BIC or QBIC.

c.hat

value of overdispersion parameter (i.e., variance inflation factor) such as that obtained from c_hat. Note that values of c.hat different from 1 are only appropriate for binomial GLM's with trials > 1 (i.e., success/trial or cbind(success, failure) syntax), with Poisson GLM's, single-season or dynamic occupancy models (MacKenzie et al. 2002, 2003), N-mixture models (Royle 2004, Dail and Madsen 2011), or capture-mark-recapture models (e.g., Lebreton et al. 1992). If c.hat > 1, useBICCustom will return the quasi-likelihood analogue of the information criterion requested.

Details

useBICCustom computes one of the following two information criteria:

the Bayesian information criterion (BIC, Schwarz 1978) or the quasi-likelihood BIC (QBIC).

Value

useBICCustom returns the BIC or QBIC depending on the values of the c.hat argument.

Note

The actual (Q)BIC values are not really interesting in themselves, as they depend directly on the data, parameters estimated, and likelihood function. Furthermore, a single value does not tell much about model fit. Information criteria become relevant when compared to one another for a given data set and set of candidate models.

Author(s)

Marc J. Mazerolle

References

Burnham, K. P., Anderson, D. R. (2002) Model Selection and Multimodel Inference: a practical information-theoretic approach. Second edition. Springer: New York.

Dail, D., Madsen, L. (2011) Models for estimating abundance from repeated counts of an open population. Biometrics 67, 577–587.

Lebreton, J.-D., Burnham, K. P., Clobert, J., Anderson, D. R. (1992) Modeling survival and testing biological hypotheses using marked animals: a unified approach with case-studies. Ecological Monographs 62, 67–118.

MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A., Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology 83, 2248–2255.

MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G., Franklin, A. B. (2003) Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84, 2200–2207.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics 60, 108–115.

Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461–464.

See Also

AICc, aictabCustom, useBIC, bictab, evidence, modavgCustom

Examples

##cement data from Burnham and Anderson (2002, p. 101)
data(cement)
##run multiple regression - the global model in Table 3.2
glob.mod <- lm(y ~ x1 + x2 + x3 + x4, data = cement)

##extract log-likelihood
LL <- logLik(glob.mod)[1]

##extract number of parameters
##including residual variance
K.mod <- length(coef(glob.mod)) + 1

##compute BIC with full likelihood
useBICCustom(LL, K.mod, nobs = nrow(cement))
##compare against useBIC
useBIC(glob.mod)

Format Objects to LaTeX or HTML

Description

Functions to format various objects following model selection and multimodel inference to LaTeX or HTML tables. These functions extend the methods from the xtable package (Dahl 2014).

Usage

## S3 method for class 'aictab'
xtable(x, caption = NULL, label = NULL, align = NULL, 
        digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, include.AICc = TRUE,
        include.LL = TRUE, include.Cum.Wt = FALSE, ...)

## S3 method for class 'anovaOD'
xtable(x, caption = NULL, label = NULL, align = NULL,
       digits = NULL, display = NULL, auto = FALSE, 
       nice.names = TRUE, ...)

## S3 method for class 'bictab'
xtable(x, caption = NULL, label = NULL, align = NULL, 
        digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, include.BIC = TRUE, include.LL = TRUE,
        include.Cum.Wt = FALSE, ...)

## S3 method for class 'boot.wt'
xtable(x, caption = NULL, label = NULL, align = NULL,
        digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, include.AICc = TRUE, include.AICcWt = FALSE, ...)

## S3 method for class 'countDist'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, table.countDist = "distance", ...)

## S3 method for class 'checkParms'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, include.variable = TRUE, include.max.se =
        TRUE, include.n.high.se = TRUE, ...)

## S3 method for class 'countHist'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, table.countHist = "count", ...)

## S3 method for class 'detHist'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, table.detHist = "freq", ...)

## S3 method for class 'detTime'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, table.detTime = "freq", ...)

## S3 method for class 'dictab'
xtable(x, caption = NULL, label = NULL, align = NULL, 
        digits = NULL, display = NULL, auto = FALSE, 
        nice.names = TRUE, include.DIC = TRUE, include.Cum.Wt = FALSE, ...)

## S3 method for class 'ictab'
xtable(x, caption = NULL, label = NULL, align = NULL,
        digits = NULL, display = NULL, auto = FALSE, nice.names = TRUE,
        include.IC = TRUE, include.Cum.Wt = FALSE, ...)

## S3 method for class 'mb.chisq'
xtable(x, caption = NULL, label = NULL, align = NULL,
        digits = NULL, display = NULL, auto = FALSE, nice.names = TRUE,
        include.detection.histories = TRUE, ...)

## S3 method for class 'modavg'
xtable(x, caption = NULL, label = NULL, align = NULL,
        digits = NULL, display = NULL, auto = FALSE, nice.names = TRUE,
        print.table = FALSE, ...)

## S3 method for class 'modavgCustom'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, print.table = FALSE, ...)

## S3 method for class 'modavgEffect'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, print.table = FALSE, ...)

## S3 method for class 'modavgIC'
xtable(x, caption = NULL, label = NULL, align = NULL,
        digits = NULL, display = NULL, auto = FALSE, nice.names = TRUE,
        print.table = FALSE, ...)

## S3 method for class 'modavgPred'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, ...)

## S3 method for class 'modavgShrink'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, print.table = FALSE, ...)

## S3 method for class 'multComp'
xtable(x, caption = NULL, label = NULL,
        align = NULL, digits = NULL, display = NULL, auto = FALSE,
        nice.names = TRUE, print.table = FALSE, ...)

## S3 method for class 'summaryOD'
xtable(x, caption = NULL, label = NULL, align = NULL,
                           digits = NULL, display = NULL, auto = FALSE, 
                           nice.names = TRUE, ...)

Arguments

x

an object of class aictab, anovaOD, bictab, boot.wt, checkParms, countDist, countHist, detHist, detTime, dictab, ictab, mb.chisq, modavg, modavgEffect, modavgCustom, modavgIC, modavgPred, modavgShrink, multComp, or summaryOD.

caption

a character vector of length 1 or 2 storing the caption or title of the table. If the vector is of length 2, the second item is the short caption used when LaTeX generates a list of tables. The default value is NULL and suppresses the caption.

label

a character vector storing the LaTeX label or HTML anchor. The default value is NULL and suppresses the label.

align

a character vector of length equal to the number of columns of the table specifying the alignment of the elements. Note that the rownames are considered as an additional column and require an alignment value.

digits

a numeric vector of length one or equal to the number of columns in the table (including the rownames) specifying the number of digits to display in each column.

display

a character vector of length equal to the number of columns (including the rownames) specifying the format of each column. For example, use s for strings, f for numbers in the regular format, or d for integers. See formatC for additional possible values.

auto

Logical, indicating whether to apply automatic format when no value is passed to align, digits, or display. This autoformat (based on xalign, xdigits, and xdisplay) can be useful to quickly format a typical matrix or data.frame. Default value is FALSE.

nice.names

logical. If TRUE, column labels are modified to improve their appearance in the table. If FALSE, simpler labels are used, or the ones supplied directly by the user in the object storing the output.

include.AICc

logical. If TRUE, the column containing the information criterion (AIC, AICc, QAIC, or QAICc) of each model is printed in the table. If FALSE, the column is suppressed.

include.BIC

logical. If TRUE, the column containing the information criterion (BIC or QBIC) of each model is printed in the table. If FALSE, the column is suppressed.

include.DIC

logical. If TRUE, the column containing the deviance information criterion (DIC) of each model is printed in the table. If FALSE, the column is suppressed.

include.IC

logical. If TRUE, the column containing the information criterion of each model is printed in the table. If FALSE, the column is suppressed.

include.LL

logical. If TRUE, the column containing the log-likelihood of each model is printed in the table. If FALSE, the column is suppressed.

include.Cum.Wt

logical. If TRUE, the column containing the cumulative Akaike weights is printed in the table. If FALSE, the column is suppressed.

include.AICcWt

logical. If TRUE, the column containing the Akaike weight of each model is printed in the table. If FALSE, the column is suppressed.

include.detection.histories

logical. If TRUE, the column containing detection histories is printed in the table. If FALSE, the column is suppressed.

include.variable

logical. If TRUE, the column containing the variable name is printed in the table. If FALSE, the column is suppressed.

include.max.se

logical. If TRUE, the column containing the maximum SE in the model is printed in the table. If FALSE, the column is suppressed.

include.n.high.se

logical. If TRUE, the column containing the number of SE's greater than the threshold specified by the user is printed in the table. If FALSE, the column is suppressed.

print.table

logical. If TRUE, the model selection table is printed and other sections of the output are suppressed (e.g., model-averaged estimates). If FALSE, the model selection table is suppressed and only the other portion of the output is printed in the table.

table.detHist

character string specifying, either "freq", "prop", or "hist". If table.type = "freq", the function returns a table of frequencies of sites sampled, of sites with at least one detection, and for data with multiple primary periods, the frequencies of sites with observed extinctions and colonizations. If table.type = "prop", the table returns the proportion of sites with at least one detection, and for data with multiple periods, the proportion of sites with observed extinctions and colonizations. If table.type = "hist", the function returns the frequencies of each observed detection history.

table.detTime

character string specifying, either "freq", "prop", or "dist". If table.type = "freq", the function returns a table of frequencies of sites sampled, of sites with at least one detection, and for data with multiple primary periods, the frequencies of sites with observed extinctions and colonizations. If table.type = "prop", the table returns the proportion of sites with at least one detection, and for data with multiple periods, the proportion of sites with observed extinctions and colonizations. If table.type = "dist", the function returns the quantiles of detection times.

table.countDist

character string specifying, either "distance", "count", "freq", or "prop". If table.type = "distance", the function returns a table of counts summarized for each distance class. If table.type = "count", the function returns the table of frequencies of counts observed across sites. If table.type = "freq", the function returns a table of frequencies of sites sampled, of sites with at least one detection, and for data with multiple primary periods, the frequencies of sites with observed extinctions and colonizations. If table.type = "prop", the table returns the proportion of sites with at least one detection, and for data with multiple periods, the proportion of sites with observed extinctions and colonizations.

table.countHist

character string specifying, either "count", "freq", "prop", or "hist". If table.type = "count", the function returns the table of frequencies of counts observed across sites. If table.type = "freq", the function returns a table of frequencies of sites sampled, of sites with at least one detection, and for data with multiple primary periods, the frequencies of sites with observed extinctions and colonizations. If table.type = "prop", the table returns the proportion of sites with at least one detection, and for data with multiple periods, the proportion of sites with observed extinctions and colonizations. If table.type = "hist", the function returns the frequencies of each observed count history.

...

additional arguments passed to the function.

Details

xtable creates an object of the xtable class inheriting from the data.frame class. This object can then be used with print.xtable for added flexibility such as suppressing row names, modifying caption placement, and format tables in LaTeX or HTML format.

Author(s)

Marc J. Mazerolle

References

Dahl, D. B. (2014) xtable: Export tables to LaTeX or HTML. R package version 1.7-3. https://cran.r-project.org/package=xtable.

See Also

aictab, boot.wt, dictab, formatC, ictab, mb.chisq, modavg, modavgCustom, modavgIC, modavgEffect, modavgPred, modavgShrink, multComp, summaryOD, anovaOD, xtable, print.xtable

Examples

if(require(xtable)) {
##model selection example
data(dry.frog)
##setup candidate models
Cand.models <- list( )
Cand.models[[1]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2,
                       data = dry.frog)
Cand.models[[2]] <- lm(log_Mass_lost ~ Shade + Substrate +
                       cent_Initial_mass + Initial_mass2 +
                       Shade:Substrate, data = dry.frog)
Cand.models[[3]] <- lm(log_Mass_lost ~ cent_Initial_mass +
                       Initial_mass2, data = dry.frog)
Model.names <- c("additive", "interaction", "no shade")

##model selection table - AICc
out <- aictab(cand.set = Cand.models, modnames = Model.names)

xtable(out)
##exclude AICc and LL
xtable(out, include.AICc = FALSE, include.LL = FALSE)
##remove row names and add caption
print(xtable(out, caption = "Model selection based on AICc"),
      include.rownames = FALSE, caption.placement = "top")


##model selection table - BIC
out2 <- bictab(cand.set = Cand.models, modnames = Model.names)

xtable(out2)
##exclude AICc and LL
xtable(out2, include.BIC = FALSE, include.LL = FALSE)
##remove row names and add caption
print(xtable(out2, caption = "Model selection based on BIC"),
      include.rownames = FALSE, caption.placement = "top")


##model-averaged estimate of Initial_mass2
mavg.mass <- modavg(cand.set = Cand.models, parm = "Initial_mass2",
                    modnames = Model.names)
#model-averaged estimate
xtable(mavg.mass, print.table = FALSE)
#table with contribution of each model
xtable(mavg.mass, print.table = TRUE)  


##model-averaged predictions for first 10 observations
preds <- modavgPred(cand.set = Cand.models, modnames = Model.names,
                    newdata = dry.frog[1:10, ])
xtable(preds)
}


##example of diagnostics
## Not run: 
if(require(unmarked)){
##distance sampling example from ?distsamp
data(linetran)
ltUMF <- with(linetran, {
              unmarkedFrameDS(y = cbind(dc1, dc2, dc3, dc4),
              siteCovs = data.frame(Length, area, habitat),
              dist.breaks = c(0, 5, 10, 15, 20),
              tlength = linetran$Length * 1000, survey = "line",
              unitsIn = "m")
              })

##summarize counts across distance classes
xtable(countDist(ltUMF), table.countDist = "distance")
##summarize counts across all sites
xtable(countDist(ltUMF), table.countDist = "count")

##Half-normal detection function
fm1 <- distsamp(~ 1 ~ 1, ltUMF)
##determine parameters with highest SE's
xtable(checkParms(fm1))
}

## End(Not run)