Package 'MuMIn'

Multi-model inference

Description

The package MuMIn contains functions to streamline information-theoretic model selection and carry out model averaging based on information criteria.

Details

The suite of functions includes:

dredge

performs automated model selection by generating subsets of the supplied ‘global’ model and optional choices of other model properties (such as different link functions). The set of models can be generated with ‘all possible’ combinations or tailored according to specified conditions.

model.sel

creates a model selection table from selected models.

model.avg

calculates model-averaged parameters, along with standard errors and confidence intervals. The predict method produces model-averaged predictions.

AICc

calculates the second-order Akaike information criterion. Some other criteria are provided, see below.

stdize, stdizeFit, std.coef, partial.sd

can be used to standardise data and model coefficients by standard deviation or partial standard deviation.

For a complete list of functions, use library(help = "MuMIn").

By default, AIC$_{c}$ is used to rank models and obtain model weights, although any information criterion can be used. At least the following are currently implemented in R: AIC and BIC in package stats, and QAIC, QAICc, ICOMP, CAICF, and Mallows' Cp in MuMIn. There is also a DIC extractor for MCMC models and a QIC for GEE.

Many common modelling functions in R are supported. For a complete list, see the list of supported models.

In addition to “regular” information criteria, model averaging can be performed using various types of model weighting algorithms: Bates-Granger, bootstrapped, cos-squared, jackknife, stacking, or ARM. These weighting functions are mainly applicable to glms.

Author(s)

Kamil Bartoń

References

Burnham, K. P. and Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed. New York, Springer-Verlag.

See Also

AIC, step or stepAIC for stepwise model selection by AIC.

Examples

options(na.action = "na.fail") #  change the default "na.omit" to prevent models 
                               #  from being fitted to different datasets in 
                               #  case of missing values.

fm1 <- lm(y ~ ., data = Cement)
ms1 <- dredge(fm1)

# Visualize the model selection table:

par(mar = c(3,5,6,4))
plot(ms1, labAsExpr = TRUE)

model.avg(ms1, subset = delta < 4)

confset.95p <- get.models(ms1, cumsum(weight) <= .95)
avgmod.95p <- model.avg(confset.95p)
summary(avgmod.95p)
confint(avgmod.95p)

---

Second-order Akaike Information Criterion

Description

Calculate Second-order Akaike Information Criterion for one or several fitted model objects (AIC$_{c}$, AIC for small samples).

Usage

AICc(object, ..., k = 2, REML = NULL)

Arguments

object	a fitted model object for which there exists a logLik method, or a "logLik" object.
...	optionally more fitted model objects.
k	the ‘penalty’ per parameter to be used; the default k = 2 is the classical AIC.
REML	optional logical value, passed to the logLik method indicating whether the restricted log-likelihood or log-likelihood should be used. The default is to use the method used for model estimation.

Value

If just one object is provided, returns a numeric value with the corresponding AIC$_{c}$; if more than one object are provided, returns a data.frame with rows corresponding to the objects and columns representing the number of parameters in the model (df) and AIC$_{c}$.

Note

AIC$_{c}$ should be used instead AIC when sample size is small in comparison to the number of estimated parameters (Burnham & Anderson 2002 recommend its use when $n / K < 40$).

Author(s)

Kamil Bartoń

References

Burnham, K. P. and Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed. New York, Springer-Verlag.

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

See Also

Akaike's An Information Criterion: AIC

Some other implementations:

AICc in package AICcmodavg, AICc in package bbmle, aicc in package glmulti

Examples

#Model-averaging mixed models

options(na.action = "na.fail")

data(Orthodont, package = "nlme")

# Fit model by REML
fm2 <- lme(distance ~ Sex*age, data = Orthodont,
    random = ~ 1|Subject / Sex, method = "REML")

# Model selection: ranking by AICc using ML
ms2 <- dredge(fm2, trace = TRUE, rank = "AICc", REML = FALSE)

(attr(ms2, "rank.call"))

# Get the models (fitted by REML, as in the global model)
fmList <- get.models(ms2, 1:4)

# Because the models originate from 'dredge(..., rank = AICc, REML = FALSE)',
# the default weights in 'model.avg' are ML based:
summary(model.avg(fmList))

## Not run: 
# the same result:
model.avg(fmList, rank = "AICc", rank.args = list(REML = FALSE))

## End(Not run)

---

Adaptive Regression by Mixing

Description

Combine all-subsets GLMs using the ARM algorithm. Calculate ARM weights for a set of models.

Usage

arm.glm(object, R = 250, weight.by = c("aic", "loglik"), trace = FALSE)

armWeights(object, ..., data, weight.by = c("aic", "loglik"), R = 1000)

Arguments

object	for arm.glm, a fitted “global” glm object. For armWeights, a fitted glm object, or a list of such, or an "averaging" object.
...	more fitted model objects.
R	number of permutations.
weight.by	indicates whether model weights should be calculated with AIC or log-likelihood.
trace	if TRUE, information is printed during the running of arm.glm.
data	a data frame in which to look for variables for use with prediction. If omitted, the fitted linear predictors are used.

Details

For each of all-subsets of the “global” model, parameters are estimated using randomly sampled half of the data. Log-likelihood given the remaining half of the data is used to calculate AIC weights. This is repeated R times and mean of the weights is used to average all-subsets parameters estimated using complete data.

Value

arm.glm returns an object of class "averaging" contaning only “full” averaged coefficients. See model.avg for object description.

armWeights returns a numeric vector of model weights.

Note

Number of parameters is limited to floor(nobs(object) / 2) - 1. All-subsets respect marginality constraints.

Author(s)

Kamil Bartoń

References

Yang, Y. 2001 Adaptive Regression by Mixing. Journal of the American Statistical Association 96, 574–588.

Yang, Y. 2003 Regression with multiple candidate models: selecting or mixing? Statistica Sinica 13, 783–810.

See Also

model.avg, par.avg

Weights for assigning new model weights to an "averaging" object.

Other implementation of ARM algorithm: arms in (archived) package MMIX.

Other kinds of model weights: BGWeights, bootWeights, cos2Weights, jackknifeWeights, stackingWeights.

Examples

fm <- glm(y ~ X1 + X2 + X3 + X4, data = Cement)

summary(am1 <- arm.glm(fm, R = 15))

mst <- dredge(fm)

am2 <- model.avg(mst, fit = TRUE)

Weights(am2) <- armWeights(am2, data = Cement, R = 15)

# differences are due to small R:
coef(am1, full = TRUE)
coef(am2, full = TRUE)

---

Flour beetle mortality data

Description

Mortality of flour beetles (Tribolium confusum) due to exposure to gaseous carbon disulfide CS$_{2}$, from Bliss (1935).

Usage

Beetle

Format

Beetle is a data frame with 5 elements.

Prop

a matrix with two columns named nkilled and nsurvived

mortality

observed mortality rate

dose

the dose of CS$_{2}$ in mg/L

n.tested

number of beetles tested

n.killed

number of beetles killed.

Source

Bliss, C. I. 1935 The calculation of the dosage-mortality curve. Annals of Applied Biology 22, 134–167.

References

Burnham, K. P. and Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed. New York, Springer-Verlag.

Examples

# "Logistic regression example"
# from Burnham & Anderson (2002) chapter 4.11
# Fit a global model with all the considered variables

globmod <- glm(Prop ~ dose + I(dose^2) + log(dose) + I(log(dose)^2),
   data = Beetle, family = binomial, na.action = na.fail)
# A logical expression defining the subset of models to use:
# * either log(dose) or dose
# * the quadratic terms can appear only together with linear terms
msubset <- expression(xor(dose, `log(dose)`) &
    dc(dose, `I(dose^2)`) &
    dc(`log(dose)`, `I(log(dose)^2)`))

# Table 4.6
# Use 'varying' argument to fit models with different link functions
# Note the use of 'alist' rather than 'list' in order to keep the
# 'family' objects unevaluated
varying.link <- list(family = alist(
    logit = binomial("logit"),
    probit = binomial("probit"),
    cloglog = binomial("cloglog")
    ))

(ms12 <- dredge(globmod, subset = msubset, varying = varying.link,
    rank = AIC))
 
# Table 4.7 "models justifiable a priori"
(ms3 <- subset(ms12, has(dose, !`I(dose^2)`))) 
# The same result, but would fit the models again:
# ms3 <- update(ms12, update(globmod, . ~ dose), subset =,
#    fixed = ~dose)
mod3 <- get.models(ms3, 1:3) 
# Table 4.8. Predicted mortality probability at dose 40.
# calculate confidence intervals on logit scale
logit.ci <- function(p, se, quantile = 2) {
    C. <- exp(quantile * se / (p * (1 - p))) 
    p /(p + (1 - p) * c(C., 1/C.)) 
}

mavg3 <- model.avg(mod3, revised.var = FALSE) 
# get predictions both from component and averaged models
pred <- lapply(c(component = mod3, list(averaged = mavg3)), predict,
	   newdata = list(dose = 40), type = "response", se.fit = TRUE) 
# reshape predicted values
pred <- t(sapply(pred, function(x) unlist(x)[1:2])) 
colnames(pred) <- c("fit", "se.fit") 

# build the table
tab <- cbind(
    c(Weights(ms3), NA),
    pred,
    matrix(logit.ci(pred[,"fit"], pred[,"se.fit"],
        quantile = c(rep(1.96, 3), 2)), ncol = 2) 
    ) 
colnames(tab) <- c("Akaike weight", "Predicted(40)", "SE", "Lower CI",
    "Upper CI") 
rownames(tab) <- c(as.character(ms3$family), "model-averaged") 
print(tab, digits = 3, na.print = "") 
# Figure 4.3
newdata <- list(dose = seq(min(Beetle$dose), max(Beetle$dose), length.out = 25)) 

# add model-averaged prediction with CI, using the same method as above
avpred <- predict(mavg3, newdata, se.fit = TRUE, type = "response")

avci <- matrix(logit.ci(avpred$fit, avpred$se.fit, quantile = 2), ncol = 2) 


matplot(newdata$dose, sapply(mod3, predict, newdata, type = "response"),
    type = "l", xlab = quote(list("Dose of" ~ CS[2],(mg/L))),
    ylab = "Mortality", col = 2:4, lty = 3, lwd = 1
) 
matplot(newdata$dose, cbind(avpred$fit, avci), type = "l", add = TRUE,
    lwd = 1, lty = c(1, 2, 2), col = 1)  

legend("topleft", NULL, c(as.character(ms3$family), expression(`averaged`
    %+-% CI)), lty = c(3, 3, 3, 1), col = c(2:4, 1))

---

Bates-Granger minimal variance model weights

Description

Compute empirical weights based on out of sample forecast variances, following Bates and Granger (1969).

Usage

BGWeights(object, ..., data, force.update = FALSE)

Arguments

object, ...	two or more fitted glm objects, or a list of such, or an "averaging" object.
data	a data frame containing the variables in the model.
force.update	if TRUE, the much less efficient method of updating glm function will be used rather than directly via glm.fit. This only applies to glms, in case of other model types update is always used.

Details

Bates-Granger model weights are calculated using prediction covariance. To get the estimate of prediction covariance, the models are fitted to randomly selected half of data and prediction is done on the remaining half. These predictions are then used to compute the variance-covariance between models, $\Sigma$. Model weights are then calculated as $w_{BG} = (1' \Sigma^{-1} 1)^{-1} 1 \Sigma^{-1}$, where $1$ a vector of 1-s.

Bates-Granger model weights may be outside of the $[0,1]$ range, which may cause the averaged variances to be negative. Apparently this method works best when data is large.

Value

A numeric vector of model weights.

Note

For matrix inversion, MASS::ginv() is more stable near singularities than solve. It will be used as a fallback if solve fails and MASS is available.

Author(s)

Carsten Dormann, Kamil Bartoń

References

Bates, J. M. and Granger, C. W. J. 1969 The combination of forecasts. Journal of the Operational Research Society 20, 451-468.

Dormann, C. et al. (2018) Model averaging in ecology: a review of Bayesian, information-theoretic, and tactical approaches for predictive inference. Ecological Monographs 88, 485–504.

See Also

Weights, model.avg

Other model weights: bootWeights(), cos2Weights(), jackknifeWeights(), stackingWeights()

Examples

fm <- glm(Prop ~ mortality + dose, family = binomial, Beetle, na.action = na.fail)
models <- lapply(dredge(fm, evaluate = FALSE), eval)
ma <- model.avg(models)

# this produces warnings because of negative variances:
set.seed(78)
Weights(ma) <- BGWeights(ma, data = Beetle)
coefTable(ma, full = TRUE)

# SE for prediction is not reliable if some or none of coefficient's SE
# are available
predict(ma, data = test.data, se.fit = TRUE)
coefTable(ma, full = TRUE)

---

Bootstrap model weights

Description

Compute model weights using bootstrap.

Usage

bootWeights(object, ..., R, rank = c("AICc", "AIC", "BIC"))

Arguments

object, ...	two or more fitted glm objects, or a list of such, or an "averaging" object.
R	the number of replicates.
rank	a character string, specifying the information criterion to use for model ranking. Defaults to AICc.

Details

The models are fitted repeatedly to a resampled data set and ranked using AIC-type criterion. The model weights represent the proportion of replicates when a model has the lowest IC value.

Value

A numeric vector of model weights.

Author(s)

Kamil Bartoń, Carsten Dormann

References

Dormann, C. et al. 2018 Model averaging in ecology: a review of Bayesian, information-theoretic, and tactical approaches for predictive inference. Ecological Monographs 88, 485–504.

See Also

Weights, model.avg

Other model weights: BGWeights(), cos2Weights(), jackknifeWeights(), stackingWeights()

Examples

# To speed up the bootstrap, use 'x = TRUE' so that model matrix is included
#     in the returned object
fm <- glm(Prop ~ mortality + dose, family = binomial, data = Beetle, 
   na.action = na.fail, x = TRUE)

fml <- lapply(dredge(fm, eval = FALSE), eval)
am <- model.avg(fml)

Weights(am) <- bootWeights(am, data = Beetle, R = 25)

summary(am)

---

Cement hardening data

Description

Cement hardening data from Woods et al (1932).

Usage

Cement

Format

Cement is a data frame with 5 variables. x1-x4 are four predictor variables expressed as a percentage of weight.

y

calories of heat evolved per gram of cement after 180 days of hardening

X1

calcium aluminate

X2

tricalcium silicate

X3

tetracalcium alumino ferrite

X4

dicalcium silicate.

Source

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

References

Burnham, K. P. and Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed. New York, Springer-Verlag.

---

Plot model coefficients

Description

Produce dot-and-whisker plot of the model(-averaged) coefficients, with confidence intervals

Usage

coefplot(
  x, lci, uci,
  labels = NULL, width = 0.15,
  shift = 0, horizontal = TRUE,
  main = NULL, xlab = NULL, ylab = NULL,
  xlim = NULL, ylim = NULL,
  labAsExpr = TRUE, mar.adj = TRUE, lab.line = 0.5, 
  lty = par("lty"), lwd = par("lwd"), pch = 21,
  col = par("col"), bg = par("bg"),
  dotcex = par("cex"), dotcol = col,
  staplelty = lty, staplelwd = lwd, staplecol = col,
  zerolty = "dotted", zerolwd = lwd, zerocol = "gray",
  las = 2, ann = TRUE, axes = TRUE, add = FALSE,
  type = "p",
  ...
)

## S3 method for class 'averaging'
plot(
  x,
  full = TRUE, level = 0.95, intercept = TRUE,
  parm = NULL, labels = NULL, width = 0.1,
  shift = max(0.2, width * 2.1 + 0.05),
  horizontal = TRUE, 
  xlim = NULL, ylim = NULL,
  main = "Model-averaged coefficients",
  xlab = NULL, ylab = NULL, 
  add = FALSE,
  ...
)

Arguments

x	either a (possibly named) vector of coefficients (for coefplot), or an "averaging" object.
lci, uci	vectors of lower and upper confidence intervals. Alternatively a two-column matrix with columns containing confidence intervals, in which case uci is ignored.
labels	optional vector of coefficient names. By default, names of x are used for labels.
width	width of the staples (= end of whisker).
shift	the amount of perpendicular shift for the dots and whiskers. Useful when adding to an existing plot.
horizontal	logical indicating if the plots should be horizontal; defaults to TRUE.
main	an overall title for the plot: see title.
xlab, ylab	x- and y-axis annotation. Can be suppressed by ann=FALSE.
xlim, ylim	optional, the x and y limits of the plot.
labAsExpr	logical indicating whether the coefficient names should be transformed to expressions to create prettier labels (see plotmath)
mar.adj	logical indicating whether the (left or lower) margin should be expanded to fit the labels
lab.line	margin line for the labels
lty, lwd, pch, col, bg	default line type, line width, point character, foreground colour for all elements, and background colour for open symbols.
dotcex, dotcol	dots point size expansion and colour.
staplelty, staplelwd, staplecol	staple line type, width, and colour.
zerolty, zerolwd, zerocol	zero-line type, line width, colour. Setting zerolty to NA suppresses the line.
las	the style of labels for coefficient names. See par.
ann	logical indicating if axes should be annotated (by xlab and ylab).
axes	a logical value indicating whether both axes should be drawn on the plot.
add	logical, if true add to current plot.
type	if "n", the plot region is left empty, any other value causes the plot being drawn.
...	additional arguments passed to coefplot or more graphical parameters.
full	a logical value specifying whether the “full” model-averaged coefficients are plotted. If FALSE, the “subset”-averaged coefficients are plotted, and both types if NA. See model.avg.
level	the confidence level required.
intercept	logical indicating if intercept should be included in the plot
parm	a specification of which parameters are to be plotted, either a vector of numbers or a vector of names. If missing, all parameters are considered.

Details

Plot model(-averaged) coefficients with confidence intervals.

Value

An invisible matrix containing coordinates of points and whiskers, or, a two-element list of such, one for each coefficient type in plot.averaging when full is NA.

Author(s)

Kamil Bartoń

Examples

fm <- glm(Prop ~ dose + I(dose^2) + log(dose) + I(log(dose)^2),
  data = Beetle, family = binomial, na.action = na.fail)
ma <- model.avg(dredge(fm))

# default coefficient plot:
plot(ma, full = NA, intercept = FALSE)

# Add colours per coefficient type
# Replicate each colour n(=number of coefficients) times
clr <- c("black", "red2")
i <- rep(1:2, each = length(coef(ma)) - 1)
plot(ma, full = NA, intercept = FALSE,
   pch = 22, dotcex = 1.5,
   col = clr[i], bg = clr[i],
   lwd = 6, lend = 1, width = 0, horizontal = 0)

# Use `type = "n"` and `add` argument to e.g. add grid beneath the figure
plot(ma, full = NA, intercept = FALSE,
   width = 0, horizontal = FALSE, zerolty = NA, type = "n")
grid()
plot(ma, full = NA, intercept = FALSE,
   pch = 22, dotcex = 1.5,
   col = clr[i], bg = clr[i],
   lwd = 6, lend = 1, width = 0, horizontal = FALSE, add = TRUE)

---

Cos-squared model weights

Description

Calculate the cos-squared model weights, following the algorithm outlined in the appendix to Garthwaite & Mubwandarikwa (2010).

Usage

cos2Weights(object, ..., data, eps = 1e-06, maxit = 100, predict.args = list())

Arguments

object, ...	two or more fitted glm objects, or a list of such, or an "averaging" object. Currently only lm and glm objects are accepted.
data	a test data frame in which to look for variables for use with prediction. If omitted, the fitted linear predictors are used.
eps	tolerance for determining convergence.
maxit	maximum number of iterations.
predict.args	optionally, a list of additional arguments to be passed to predict.

Value

A numeric vector of model weights.

Author(s)

Carsten Dormann, adapted by Kamil Bartoń

References

Garthwaite, P. H. and Mubwandarikwa, E. 2010 Selection of weights for weighted model averaging. Australian & New Zealand Journal of Statistics 52, 363–382.

Dormann, C. et al. 2018 Model averaging in ecology: a review of Bayesian, information-theoretic, and tactical approaches for predictive inference. Ecological Monographs 88, 485–504.

See Also

Weights, model.avg

Other model weights: BGWeights(), bootWeights(), jackknifeWeights(), stackingWeights()

Examples

fm <- lm(y ~ X1 + X2 + X3 + X4, Cement, na.action = na.fail)
# most efficient way to produce a list of all-subsets models
models <- lapply(dredge(fm, evaluate = FALSE), eval)
ma <- model.avg(models)

test.data <- Cement
Weights(ma) <- cos2Weights(models, data = test.data)
predict(ma, data = test.data)

---

Automated model selection

Description

Generate a model selection table of models with combinations (subsets) of fixed effect terms in the global model, with optional model inclusion rules.

Usage

dredge(global.model, beta = c("none", "sd", "partial.sd"), evaluate = TRUE,
  rank = "AICc", fixed = NULL, m.lim = NULL, m.min, m.max, subset,
  trace = FALSE, varying, extra, ct.args = NULL, deps = attr(allTerms0, "deps"),
  cluster = NULL,
  ...)

## S3 method for class 'model.selection'
print(x, abbrev.names = TRUE, warnings = getOption("warn") != -1L, ...)

Arguments

global.model	a fitted ‘global’ model object. See ‘Details’ for a list of supported types.
beta	indicates whether and how the coefficients are standardized, and must be one of "none", "sd" or "partial.sd". You can specify just the initial letter. "none" corresponds to unstandardized coefficients, "sd" and "partial.sd" to coefficients standardized by SD and Partial SD, respectively. For backwards compatibility, logical value is also accepted, TRUE is equivalent to "sd" and FALSE to "none". See std.coef.
evaluate	whether to evaluate and rank the models. If FALSE, a list of unevaluated calls is returned.
rank	optionally, the rank function returning a sort of an information criterion, to be used instead AICc, e.g. AIC, QAIC or BIC. See ‘Details’.
fixed	optional, either a single-sided formula or a character vector giving names of terms to be included in all models. Not to be confused with fixed effects. See ‘Subsetting’.
m.lim, m.max, m.min	optionally, the limits c(lower, upper) for the number of terms in a single model (excluding the intercept). An NA means no limit. See ‘Subsetting’. Specifying limits as m.min and m.max is allowed for backward compatibility.
subset	logical expression or a matrix describing models to be kept in the resulting set. NULL or TRUE disables subsetting. For details, see ‘Subsetting’.
trace	if TRUE or 1, all calls to the fitting function are printed before actual fitting takes place. If trace > 1, a progress bar is displayed.
varying	optionally, a named list describing the additional arguments to vary between the generated models. Item names correspond to the arguments, and each item provides a list of choices (i.e. list(arg1 = list(choice1, choice2, ...), ...)). Complex elements in the choice list (such as family objects) should be either named (uniquely) or quoted (unevaluated, e.g. using alist, see quote), otherwise the result may be visually unpleasant. See example in Beetle.
extra	optional additional statistics to be included in the result, provided as functions, function names or a list of such (preferably named or quoted). As with the rank argument, each function must accept as an argument a fitted model object and return (a value coercible to) a numeric vector. This could be, for instance, additional information criteria or goodness-of-fit statistics. The character strings "R^2" and "adjR^2" are treated in a special way and add a likelihood-ratio based $R^{2}$ and modified-$R^{2}$ to the result, respectively (this is more efficient than using r.squaredLR directly).
x	a model.selection object, returned by dredge.
abbrev.names	Should term names in the table header be abbreviated when printed? This is the default. If full names are required, use print() explicitly with this argument set to FALSE.
warnings	if TRUE, errors and warnings issued during the model fitting are printed below the table (only with pdredge). To permanently remove the warnings, set the object's attribute "warnings" to NULL.
ct.args	optional list of arguments to be passed to coefTable (e.g. dispersion parameter for glm affecting standard errors used in subsequent model averaging).
deps	a “dependency matrix” as returned by getAllTerms, attribute "deps". Can be used to fine-tune marginality exceptions.
cluster	if a valid "cluster" object is given, it is used for parallel execution. If NULL or omitted, execution is single-threaded. With parallel calculation, an extra argument check is accepted. See pdredge for details and examples.
...	optional arguments for the rank function. Any can be an unevaluated expression, in which case any x within it will be substituted with the current model.

Details

Models are fitted through repeated evaluation of the modified call extracted from the global.model (in a similar fashion to update). This approach, while having the advantage that it can be applied to most model types through the usual formula interface, can have a considerable computational overhead.

Note that the number of combinations grows exponentially with the number of predictors ($2^{N}$, less when interactions are present, see below).

The fitted model objects are not stored in the result. To get (a subset of) the models, use get.models on the object returned by dredge. Another way to get all the models is to run lapply(dredge(..., evaluate = FALSE), eval), which avoids fitting models twice.

For a list of model types that can be used as a global.model see the list of supported models. Modelling functions that do not store a call in their result should be evaluated via a wrapper function created by updateable.

Information criterion

rank is found by a call to match.fun and may be specified as a function, a symbol, or as a character string specifying a function to be searched for from the environment of the call to dredge. It can be also a one-element named list, where the first element is taken as the rank function. The function rank must accept a model object as its first argument and always return a scalar.

Interactions

By default, marginality constraints are respected, so “all possible combinations” include only those containing interactions with their respective main effects and all lower-order terms. However, if global.model makes an exception to this principle (e.g. due to a nested design such as a / (b + d)), this will be reflected in the subset models.

Subsetting

There are three ways to constrain the resulting set of models: setting limits to the number of terms in a model with m.lim, binding term(s) to all models using fixed, and the subset argument can be used for more complex rules. For a model to be included in the selection table, its formulation must satisfy all these conditions.

subset may be an expression or a matrix. The latter should be a lower triangular matrix with logical values, where the columns and rows correspond to global.model terms. Value subset["a", "b"] == FALSE will exclude any model containing both a and b terms.
demo(dredge.subset) has examples of using the subset matrix in conjunction with correlation matrices to exclude models containing collinear predictors.

In the form of expression, the argument subset acts in a similar fashion to that in the function subset for data.frames: model terms can be referred to by name as variables in the expression, with the difference being that are interpreted as logical values (i.e. equal to TRUE if the term exists in the model).

The expression can contain any of the global.model term names, as well as names of the varying list items. global.model term names take precedence when identical to names of varying, so to avoid ambiguity varying variables in subset expression should be enclosed in V() (e.g. V(family) == "Gamma") assuming that varying is something like list(family = c("Gamma", ...))).

If item names in varying are missing, the items themselves are coerced to names. Call and symbol elements are represented as character values (via deparse), and anything other than numeric, logical, character and NULL values is replaced by item numbers (e.g. varying = list(family = list(gaussian, Gamma) should be referred to as subset = V(family) == 2. This can quickly become confusing, so it is recommended to use named lists. Examples can be found in demo(dredge.varying).

Term names appearing in fixed and subset must be given exactly as they are returned by getAllTerms(global.model), which may differ from the original term names (e.g. the interaction term components are ordered alphabetically).

The with(x) and with(+x) notation indicates, respectively, any and all interactions including the main effect term x. This is only effective with marginality exceptions. The extended form with(x, order) allows to specify the order of interaction of terms of which x is a part. For instance, with(b, 2:3) selects models with at least one second- or third-order interaction of variable b. The second (positional) argument is coerced to an integer vector. The “dot” notation .(x) is an alias for with.

The special variable `*nvar*` (backtick-quoted), in the subset expression is equal to the number of terms in the model (not the number of estimated parameters).

To make the inclusion of a model term conditional on the presence of another one, the function dc (“dependency chain”) can be used in the subset expression. dc takes any number of term names as arguments, and allows a term to be included only if all preceding ones are also present (e.g. subset = dc(a, b, c) allows for models a, a+b and a+b+c but not b, c, b+c or a+c).

subset expression can have a form of an unevaluated call, expression object, or a one-sided formula. See ‘Examples’.

Compound model terms (such as interactions, ‘as-is’ expressions within I() or smooths in gam) should be enclosed within curly brackets (e.g. {s(x,k=2)}), or backticks (like non-syntactic names, e.g. `s(x, k = 2)` ), except when they are arguments to with or dc. Backticks-quoted names must match exactly (including whitespace) the term names as returned by getAllTerms.

subset expression syntax summary

a & b

indicates that model terms a and b must be present (see Logical Operators)

{log(x,2)} or ⁠`⁠log(x, 2)⁠`⁠

represent a complex model term log(x, 2)

V(x)

represents a varying item x

with(x)

indicates that at least one term containing the main effect term x must be present

with(+x)

indicates that all the terms containing the main effect term x must be present

with(x, n:m)

indicates that at least one term containing an n-th to m-th order interaction term of x must be present

dc(a, b, c,...)

‘dependency chain’: b is allowed only if a is present, and c only if both a and b are present, etc.

`*nvar*`

the number of terms in the model.

To simply keep certain terms in all models, use of argument fixed is much more efficient. The fixed formula is interpreted in the same manner as model formula and so the terms must not be quoted.

Missing values

Use of na.action = "na.omit" (R's default) or "na.exclude" in global.model must be avoided, as it results with sub-models fitted to different data sets if there are missing values. An error is thrown if it is detected.

It is a common mistake to give na.action as an argument in the call to dredge (typically resulting in an error from the rank function to which the argument is passed through ‘...’), while the correct way is either to pass na.action in the call to the global model or to set it as a global option.

Intercept

If present in the global.model, the intercept will be included in all sub-models.

Methods

There are subset and plot methods, the latter creates a graphical representation of model weights and per-model term sum of weights. Coefficients can be extracted with coef or coefTable.

Value

An object of class c("model.selection", "data.frame"), being a data.frame, where each row represents one model. See model.selection.object for its structure.

Note

Users should keep in mind the hazards that a “thoughtless approach” of evaluating all possible models poses. Although this procedure is in certain cases useful and justified, it may result in selecting a spurious “best” model, due to the model selection bias.

“Let the computer find out” is a poor strategy and usually reflects the fact that the researcher did not bother to think clearly about the problem of interest and its scientific setting (Burnham and Anderson, 2002).

Author(s)

Kamil Bartoń

See Also

get.models, model.avg. model.sel for manual model selection tables.

Possible alternatives: glmulti in package glmulti and bestglm (bestglm). regsubsets in package leaps also performs all-subsets regression.

Variable selection through regularization provided by various packages, e.g. glmnet, lars or glmmLasso.

Examples

# Example from Burnham and Anderson (2002), page 100:

#  prevent fitting sub-models to different datasets

options(na.action = "na.fail")

fm1 <- lm(y ~ ., data = Cement)
dd <- dredge(fm1)
subset(dd, delta < 4)

# Visualize the model selection table:

par(mar = c(3,5,6,4))
plot(dd, labAsExpr = TRUE)


# Model average models with delta AICc < 4
model.avg(dd, subset = delta < 4)

#or as a 95% confidence set:
model.avg(dd, subset = cumsum(weight) <= .95) # get averaged coefficients

#'Best' model
summary(get.models(dd, 1)[[1]])

## Not run: 
# Examples of using 'subset':
# keep only models containing X3
dredge(fm1, subset = ~ X3) # subset as a formula
dredge(fm1, subset = expression(X3)) # subset as expression object
# the same, but more effective:
dredge(fm1, fixed = "X3")
# exclude models containing both X1 and X2 at the same time
dredge(fm1, subset = !(X1 && X2))
# Fit only models containing either X3 or X4 (but not both);
# include X3 only if X2 is present, and X2 only if X1 is present.
dredge(fm1, subset = dc(X1, X2, X3) && xor(X3, X4))
# the same as above, without "dc"
dredge(fm1, subset = (X1 | !X2) && (X2 | !X3) && xor(X3, X4))

# Include only models with up to 2 terms (and intercept)
dredge(fm1, m.lim = c(0, 2))

## End(Not run)

# Add R^2 and F-statistics, use the 'extra' argument
dredge(fm1, m.lim = c(NA, 1), extra = c("R^2", F = function(x)
    summary(x)$fstatistic[[1]]))

# with summary statistics:
dredge(fm1, m.lim = c(NA, 1), extra = list(
    "R^2", "*" = function(x) {
        s <- summary(x)
        c(Rsq = s$r.squared, adjRsq = s$adj.r.squared,
            F = s$fstatistic[[1]])
    })
)

# Add other information criteria (but rank with AICc):
dredge(fm1, m.lim = c(NA, 1), extra = alist(AIC, BIC, ICOMP, Cp))

---

Apply a function to calls inside an expression

Description

Apply function FUN to each occurence of a call to what() (or a symbol what) in an unevaluated expression. It can be used for advanced manipulation of expressions. Intended primarily for internal use.

Usage

exprApply(expr, what, FUN, ..., symbols = FALSE)

Arguments

expr	an unevaluated expression.
what	character string giving the name of a function. Each call to what inside expr will be passed to FUN. what can be also a character representation of an operator or parenthesis (including curly and square brackets) as these are primitive functions in R. Set what to NA to match all names.
FUN	a function to be applied.
symbols	logical value controlling whether FUN should be applied to symbols as well as calls.
...	optional arguments to FUN.

Details

FUN is found by a call to match.fun and can be either a function or a symbol (e.g., a backquoted name) or a character string specifying a function to be searched for from the environment of the call to exprApply.

Value

A (modified) expression.

Note

If expr has a source reference information ("srcref" attribute), modifications done by exprApply will not be visible when printed unless srcref is removed. However, exprApply does remove source reference from any function expression inside expr.

Author(s)

Kamil Bartoń

See Also

Expression-related functions: substitute, expression, quote and bquote.

Similar function walkCode exists in package codetools.

Functions useful inside FUN: as.name, as.call, call, match.call etc.

Examples

### simple usage:
# print all Y(...) terms in a formula (note that symbol "Y" is omitted):
# Note: if `print` is defined as S4 "standardGeneric", we need to use 
# 'print.default' rather than 'print', or put the call to 'print' inside a 
# function, i.e. as `function(x) print(x)`:
exprApply(~ X(1) + Y(2 + Y(4)) + N(Y + Y(3)), "Y", print.default)


# replace X() with log(X, base = n)
exprApply(expression(A() + B() + C()), c("A", "B", "C"), function(expr, base) {
    expr[[2]] <- expr[[1]]
    expr[[1]] <- as.name("log")
    expr$base <- base
    expr
}, base = 10)

###
# TASK: fit lm with two poly terms, varying the degree from 1 to 3 in each.
# lm(y ~ poly(X1, degree = a) + poly(X2, degree = b), data = Cement)
# for a = {1,2,3} and b = {1,2,3}

# First we create a wrapper function for lm. Within it, use "exprApply" to add
# "degree" argument to all occurences of "poly()" having "X1" or "X2" as the
# first argument. Values for "degree" are taken from arguments "d1" and "d2"

lmpolywrap <- function(formula, d1 = NA, d2 = NA, ...) { 
    cl <- origCall <- match.call()
    cl[[1]] <- as.name("lm")
    cl$formula <- exprApply(formula, "poly", function(e, degree, x) {
        i <- which(e[[2]] == x)[1]
        if(!is.na(i) && !is.na(degree[i])) e$degree <- degree[i]
        e
    }, degree = c(d1, d2), x = c("X1", "X2"))
    cl$d1 <- cl$d2 <- NULL
    fit <- eval(cl, parent.frame())
    fit$call <- origCall # replace the stored call
    fit
}

# global model:
fm <- lmpolywrap(y ~ poly(X1) + poly(X2), data = Cement)

# Use "dredge" with argument "varying" to generate calls of all combinations of
# degrees for poly(X1) and poly(X2). Use "fixed = TRUE" to keep all global model
# terms in all models.
# Since "dredge" expects that global model has all the coefficients the 
# submodels can have, which is not the case here, we first generate model calls,
# evaluate them and feed to "model.sel"

modCalls <- dredge(fm, 
    varying = list(d1 = 1:3, d2 = 1:3), 
    fixed = TRUE,
    evaluate = FALSE
)

model.sel(models <- lapply(modCalls, eval))

# Note: to fit *all* submodels replace "fixed = TRUE" with: 
# "subset = (d1==1 || {poly(X1)}) && (d2==1 || {poly(X2)})"
# This is to avoid fitting 3 identical models when the matching "poly()" term is
# absent.

---

Manipulate model formulas

Description

simplify.formula rewrites a formula into shorthand notation. Currently only the factor crossing operator * is applied, so an expanded expression such as a+b+a:b becomes a*b. expand.formula does the opposite, additionally expanding other expressions, i.e. all nesting (/), grouping and ^.

Usage

simplify.formula(x)
expand.formula(x)

Arguments

x	a formula or an object from which it can be extracted (such as a fitted model object).

Author(s)

Kamil Bartoń

See Also

formula

delete.response, drop.terms, and reformulate

Examples

simplify.formula(y ~ a + b + a:b + (c + b)^2)
simplify.formula(y ~ a + b + a:b + 0)

expand.formula(~ a * b)

---

Retrieve models from selection table

Description

Generate or extract a list of fitted model objects from a "model.selection" table or component models from the averaged model ("averaging" object), optionally using parallel computation in a cluster.

Usage

get.models(object, subset, cluster = NA, ...)

Arguments

object	object returned by dredge, model.sel or model.avg.
subset	subset of models, an expression evaluated within the model selection table (see ‘Details’).
cluster	optionally, a "cluster" object. If it is a valid cluster, models are evaluated using parallel computation.
...	additional arguments to update the models. For example, one may want to fit models with REML (e.g. argument REML = TRUE in some modelling functions) while using ML for model selection.

Details

The argument subset must be explicitely provided. This is to assure that a potentially long list of models is not fitted unintentionally. To evaluate all models, set subset to NA or TRUE.

If subset is a character vector, it is interpreted as names of rows to be selected.

Value

list of fitted model objects.

Note

"model.selection" tables created by model.sel or averaged models created by model.avg from a list of model objects (as opposed to those created with model selection tables) store the component models as part of the object - in these cases get.models simply extracts the items from these lists. Otherwise the models have to be fitted. Therefore, using get.models following dredge is not efficient as the requested models are fitted twice. If the number of generated models is reasonable, consider using lapply(dredge(..., evaluate = FALSE), eval), which generates a list of all model calls and evaluates them into a list of model objects.

Alternatively, getCall and eval can be used to compute a model out of the "model.selection" table (e.g. eval(getCall(<model.selection>, i)), where i is the model index or name).

pget.models is still available, but is deprecated.

Author(s)

Kamil Bartoń

See Also

dredge and pdredge, model.avg

makeCluster in packages parallel and snow

Examples

# Mixed models:


fm2 <- lme(distance ~ age + Sex, data = Orthodont,
    random = ~ 1 | Subject, method = "ML")
ms2 <- dredge(fm2)

# Get top-most models, but fitted by REML:
(confset.d4 <- get.models(ms2, subset = delta < 4, method = "REML"))

## Not run: 
# Get the top model:
get.models(ms2, subset = 1)[[1]]

## End(Not run)

---

Grade Point Average data

Description

First-year college Grade Point Average (GPA) from Graybill and Iyer (1994).

Usage

GPA

Format

GPA is a data frame with 5 variables. y is the first-year college Grade Point Average (GPA) and x1-x4 are four predictor variables from standardized tests (SAT) administered before matriculation.

y

GPA

x1

math score on the SAT

x2

verbal score on the SAT

x3

high school math

x4

high school English

Source

Graybill, F.A. and Iyer, H.K. (1994). Regression analysis: concepts and applications. Duxbury Press, Belmont, CA.

References

Burnham, K. P. and Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed. New York, Springer-Verlag.

---

Various information criteria

Description

Calculate Mallows' Cp and Bozdogan's ICOMP and CAIFC information criteria.

Extract or calculate Deviance Information Criterion from MCMCglmm and merMod object.

Usage

Cp(object, ..., dispersion = NULL)
ICOMP(object, ..., REML = NULL)
CAICF(object, ..., REML = NULL)
DIC(object, ...)

Arguments

object	a fitted model object (in case of ICOMP and CAICF, logLik and vcov methods must exist for the object). For DIC, an object of class "MCMCglmm" or "merMod".
...	optionally more fitted model objects.
dispersion	the dispersion parameter. If NULL, it is inferred from object.
REML	optional logical value, passed to the logLik method indicating whether the restricted log-likelihood or log-likelihood should be used. The default is to use the method used for model estimation.

Details

Mallows' Cp statistic is the residual deviance plus twice the estimate of $\sigma^{2}$ times the residual degrees of freedom. It is closely related to AIC (and a multiple of it if the dispersion is known).

ICOMP (I for informational and COMP for complexity) penalizes the covariance complexity of the model, rather than the number of parameters directly.

CAICF (C is for ‘consistent’ and F denotes the use of the Fisher information matrix) includes with penalty the natural logarithm of the determinant of the estimated Fisher information matrix.

Value

If just one object is provided, the functions return a numeric value with the corresponding IC; otherwise a data.frame with rows corresponding to the objects is returned.

References

Mallows, C. L. 1973 Some comments on Cp. Technometrics 15, 661–675.

Bozdogan, H. and Haughton, D. M. A. (1998) Information complexity criteria for regression models. Comp. Stat. & Data Analysis 28, 51–76.

Anderson, D. R. and Burnham, K. P. 1999 Understanding information criteria for selection among capture-recapture or ring recovery models. Bird Study 46, 14–21.

Spiegelhalter, D. J., Best, N. G., Carlin, B. R., van der Linde, A. 2002 Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society Series B-Statistical Methodology 64, 583–616.

See Also

AIC and BIC in stats, AICc. QIC for GEE model selection. extractDIC in package arm, on which the (non-visible) method extractDIC.merMod used by DIC is based.

---

Jackknifed model weights

Description

Compute model weights optimized for jackknifed model fits.

Usage

jackknifeWeights(
  object, ..., data, type = c("loglik", "rmse"),
  family = NULL, weights = NULL,
  optim.method = "BFGS", maxit = 1000, optim.args = list(),
  start = NULL, force.update = FALSE, py.matrix = FALSE
)

Arguments

object, ...	two or more fitted glm objects, or a list of such, or an "averaging" object.
data	a data frame containing the variables in the model. It is optional if all models are glm.
type	a character string specifying the function to minimize. Either "rmse" or "loglik".
family	used only if type = "loglik", a family object to be used for likelihood calculation. Not needed if all models share the same family and link function.
weights	an optional vector of ‘prior weights’ to be used in the model fitting process. Should be NULL or a numeric vector.
optim.method	optional, optimisation method, passed to optim.
maxit	optional, the maximum number of iterations, passed to optim.
optim.args	optional list of other arguments passed to optim.
start	starting values for model weights. Numeric of length equal the number of models.
force.update	for glm, the glm.fit function is used for fitting models to the train data, which is much more efficient. Set to TRUE to use update instead.
py.matrix	either a boolean value, then if TRUE a jackknifed prediction matrix is returned and if FALSE a vector of jackknifed model weights, or a N×M matrix (number of cases × number of models) that is interpreted as a jackknifed prediction matrix and it is used for optimisation (i.e. the jackknife procedure is skipped).

Details

Model weights are chosen (using optim) to minimise RMSE or log-likelihood of the prediction for data point i, of a model fitted omitting that data point i. The jackknife procedure is therefore run for all provided models and for all data points.

Value

The function returns a numeric vector of model weights.

Note

This procedure can give variable results depending on the optimisation method and starting values. It is therefore advisable to make several replicates using different optim.methods. See optim for possible values for this argument.

Author(s)

Kamil Bartoń. Carsten Dormann

References

Hansen, B. E. and Racine, J. S. 2012 Jackknife model averaging. Journal of Econometrics 979, 38–46

Dormann, C. et al. 2018 Model averaging in ecology: a review of Bayesian, information-theoretic, and tactical approaches for predictive inference. Ecological Monographs 88, 485–504.

See Also

Weights, model.avg

Other model weights: BGWeights(), bootWeights(), cos2Weights(), stackingWeights()

Examples

fm <- glm(Prop ~ mortality * dose, binomial(), Beetle, na.action = na.fail)

fits <- lapply(dredge(fm, eval = FALSE), eval)

amJk <- amAICc <- model.avg(fits)
set.seed(666)
Weights(amJk) <- jackknifeWeights(fits, data = Beetle)

coef(amJk)
coef(amAICc)

---

Leave-one-out cross-validation

Description

Compute RMSE/log-likelihood based on leave-one-out cross-validation.

Usage

loo(object, type = c("loglik", "rmse"), ...)

Arguments

object	a fitted object model, currently only lm/glm is accepted.
type	the criterion to use, given as a character string, either "rmse" for Root-Mean-Square Error or "loglik" for log-likelihood.
...	other arguments are currently ignored.

Details

Leave-one-out cross validation is a K-fold cross validation, with K equal to the number of data points in the set N. For all i from 1 to N, the model is fitted to all the data except for i-th row and a prediction is made for that value. The average error is computed and used to evaluate the model.

Value

A single numeric value of RMSE or mean log-likelihood.

Author(s)

Kamil Bartoń, based on code by Carsten Dormann

References

Dormann, C. et al. 2018 Model averaging in ecology: a review of Bayesian, information-theoretic, and tactical approaches for predictive inference. Ecological Monographs 88, 485–504.

Examples

fm <- lm(y ~ X1 + X2 + X3 + X4, Cement)
loo(fm, type = "l")
loo(fm, type = "r")

## Compare LOO_RMSE and AIC/c
options(na.action = na.fail)
dd <- dredge(fm, rank = loo, extra = list(AIC, AICc), type = "rmse")
plot(loo ~ AIC, dd, ylab = expression(LOO[RMSE]), xlab = "AIC/c")
points(loo ~ AICc, data = dd, pch = 19)
legend("topleft", legend = c("AIC", "AICc"), pch = c(1, 19))

---

Combine model selection tables

Description

Combine two or more model selection tables.

Usage

## S3 method for class 'model.selection'
merge(x, y, suffixes = c(".x", ".y"), ...)

## S3 method for class 'model.selection'
rbind(..., deparse.level = 1, make.row.names = TRUE)

Arguments

x, y, ...	model.selection objects to be combined. (... ignored in merge)
suffixes	a character vector with two elements that are appended respectively to row names of the combined tables.
make.row.names	logical indicating if unique and valid row.names should be constructed from the arguments.
deparse.level	ignored.

Value

A "model.selection" object containing models (rows) from all provided tables.

Note

Both Δ_IC values and Akaike weights are recalculated in the resulting tables.

Models in the combined model selection tables must be comparable, i.e. fitted to the same data, however only very basic checking is done to verify that. The models must also be ranked by the same information criterion.

Unlike the merge method for data.frame, this method appends second table to the first (similarly to rbind).

Author(s)

Kamil Bartoń

See Also

dredge, model.sel, merge, rbind.

Examples

## Not run: 
require(mgcv)

ms1 <- dredge(glm(Prop ~ dose + I(dose^2) + log(dose) + I(log(dose)^2),
    data = Beetle, family = binomial, na.action = na.fail))
	
fm2 <- gam(Prop ~ s(dose, k = 3), data = Beetle, family = binomial)

merge(ms1, model.sel(fm2))

## End(Not run)

---

Model utility functions

Description

These functions extract or calculate various values from provided fitted model objects(s). They are mainly meant for internal use.

coeffs extracts model coefficients;

getAllTerms extracts independent variable names from a model object;

coefTable extracts a table of coefficients, standard errors and associated degrees of freedom when possible;

get.response extracts response variable from fitted model object;

model.names generates shorthand (alpha)numeric names for one or several fitted models.

Usage

coeffs(model)

getAllTerms(x, ...)
## S3 method for class 'terms'
getAllTerms(x, intercept = FALSE, offset = TRUE, ...)

coefTable(model, ...)
## S3 method for class 'averaging'
coefTable(model, full = FALSE, adjust.se = TRUE, ...)
## S3 method for class 'lme'
coefTable(model, adjustSigma, ...)
## S3 method for class 'gee'
coefTable(model, ..., type = c("naive", "robust"))

get.response(x, data = NULL, ...)

model.names(object, ..., labels = NULL, use.letters = FALSE)

Arguments

model	a fitted model object.
object	a fitted model object or a list of such objects.
x	a fitted model object or a formula.
offset	should ‘offset’ terms be included?
intercept	should terms names include the intercept?
full, adjust.se	logical, apply to "averaging" objects. If full is TRUE, the full model-averaged coefficients are returned, and subset-averaged ones otherwise. If adjust.se is TRUE, inflated standard errors are returned. See ‘Details’ in par.avg.
adjustSigma	See summary.lme.
type	for GEE models, the type of covariance estimator to calculate returned standard errors on. Either "naive" or "robust" (‘sandwich’).
labels	optionally, a character vector with names of all the terms, e.g. from a global model. model.names enumerates the model terms in order of their appearance in the list and in the models. Therefore changing the order of the models leads to different names. Providing labels prevents that.
...	in model.names, more fitted model objects. In coefTable arguments that are passed to appropriate vcov or summary method (e.g. dispersion parameter for glm may be used here). In get.response, if data is given, arguments to be passed to model.frame. In other functions may be silently ignored.
data	a data.frame, list or environment (or object coercible to a data.frame), containing the variables in x. Required only if x is a formula, otherwise it can be used to get the response variable for a different data set.
use.letters	logical, whether letters should be used instead of numeric codes.

Details

The functions coeffs, getAllTerms and coefTable provide interface between the model object and model.avg (and dredge). Custom methods can be written to provide support for additional classes of models.

Note

coeffs's value is in most cases identical to that returned by coef, the only difference being it returns fixed effects' coefficients for mixed models, and the value is always a named numeric vector.

Use of tTable is deprecated in favour of coefTable.

Author(s)

Kamil Bartoń

---

Model averaging

Description

Model averaging based on an information criterion.

Usage

model.avg(object, ..., revised.var = TRUE)

## Default S3 method:
model.avg(object, ..., beta = c("none", "sd", "partial.sd"),
  rank = NULL, rank.args = NULL, revised.var = TRUE,
  dispersion = NULL, ct.args = NULL)

## S3 method for class 'model.selection'
model.avg(object, subset, fit = FALSE, ..., revised.var = TRUE)

Arguments

object	a fitted model object or a list of such objects, or a "model.selection" object. See ‘Details’.
...	for default method, more fitted model objects. Otherwise, arguments that are passed to the default method.
beta	indicates whether and how the component models' coefficients should be standardized. See the argument's description in dredge.
rank	optionally, a rank function (returning an information criterion) to use instead of AICc, e.g. BIC or QAIC, may be omitted if object is a model list returned by get.models or a "model.selection" object. See ‘Details’.
rank.args	optional list of arguments for the rank function. If one is an expression, an x within it is substituted with a current model.
revised.var	logical, indicating whether to use the revised formula for standard errors. See par.avg.
dispersion	the dispersion parameter for the family used. See summary.glm. This is used currently only with glm, is silently ignored otherwise.
ct.args	optional list of arguments to be passed to coefTable (besides dispersion).
subset	see subset method for "model.selection" object.
fit	if TRUE, the component models are fitted using get.models. See ‘Details’.

Details

model.avg may be used either with a list of models or directly with a model.selection object (e.g. returned by dredge). In the latter case, the models from the model selection table are not evaluated unless the argument fit is set to TRUE or some additional arguments are present (such as rank or dispersion). This results in a much faster calculation, but has certain drawbacks, because the fitted component model objects are not stored, and some methods (e.g. predict, fitted, model.matrix or vcov) would not be available with the returned object. Otherwise, get.models is called prior to averaging, and ... are passed to it.

For a list of model types that are accepted see list of supported models.

rank is found by a call to match.fun and typically is specified as a function or a symbol or a character string specifying a function to be searched for from the environment of the call to lapply. rank must be a function able to accept model as a first argument and must always return a numeric scalar.

Several standard methods for fitted model objects exist for class averaging, including summary, predict, coef, confint, formula, and vcov.

coef, vcov, confint and coefTable accept argument full that if set to TRUE, the full model-averaged coefficients are returned, rather than subset-averaged ones (when full = FALSE, being the default).

logLik returns a list of logLik objects for the component models.

Value

An object of class "averaging" is a list with components:

msTable	a data.frame with log-likelihood, IC, Δ_IC and ‘Akaike weights’ for the component models. Its attribute "term.codes" is a named vector with numerical representation of the terms in the row names of msTable.
coefficients	a matrix of model-averaged coefficients. “full” coefficients in the first row, “subset” coefficients in the second row. See ‘Note’
coefArray	a 3-dimensional array of component models' coefficients, their standard errors and degrees of freedom.
sw	object of class sw containing per-model term sum of model weights over all of the models in which the term appears.
formula	a formula corresponding to the one that would be used in a single model. The formula contains only the averaged (fixed) coefficients.
call	the matched call.

The object has the following attributes:

rank	the rank function used.
modelList	optionally, a list of all component model objects. Only if the object was created with model objects (and not model selection table).
beta	Corresponds to the function argument.
nobs	number of observations.
revised.var	Corresponds to the function argument.

Note

The ‘subset’ (or ‘conditional’) average only averages over the models where the parameter appears. An alternative, the ‘full’ average assumes that a variable is included in every model, but in some models the corresponding coefficient (and its respective variance) is set to zero. Unlike the ‘subset average’, it does not have a tendency of biasing the value away from zero. The ‘full’ average is a type of shrinkage estimator, and for variables with a weak relationship to the response it is smaller than ‘subset’ estimators.

Averaging models with different contrasts for the same factor would yield nonsense results. Currently, no checking for contrast consistency is done.

print method provides a concise output (similarly as for lm). To print more details use summary function, and confint to get confidence intervals.

Author(s)

Kamil Bartoń

References

Burnham, K. P. and Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed. New York, Springer-Verlag.

Lukacs, P. M., Burnham K. P. and Anderson, D. R. 2009 Model selection bias and Freedman’s paradox. Annals of the Institute of Statistical Mathematics 62, 117–125.

See Also

See par.avg for more details of model-averaged parameter calculation.

dredge, get.models
AICc has examples of averaging models fitted by REML.

modavg in package AICcmodavg, and coef.glmulti in package glmulti also perform model averaging.

Examples

# Example from Burnham and Anderson (2002), page 100:
fm1 <- lm(y ~ ., data = Cement, na.action = na.fail)
(ms1 <- dredge(fm1))

#models with delta.aicc < 4
summary(model.avg(ms1, subset = delta < 4))

#or as a 95% confidence set:
avgmod.95p <- model.avg(ms1, cumsum(weight) <= .95)
confint(avgmod.95p)

## Not run: 
# The same result, but re-fitting the models via 'get.models'
confset.95p <- get.models(ms1, cumsum(weight) <= .95)
model.avg(confset.95p)

# Force re-fitting the component models
model.avg(ms1, cumsum(weight) <= .95, fit = TRUE)
# Models are also fitted if additional arguments are given
model.avg(ms1, cumsum(weight) <= .95, rank = "AIC")

## End(Not run)

## Not run: 
# using BIC (Schwarz's Bayesian criterion) to rank the models
BIC <- function(x) AIC(x, k = log(length(residuals(x))))
model.avg(confset.95p, rank = BIC)
# the same result, using AIC directly, with argument k
# 'x' in a quoted 'rank' argument is substituted with a model object
# (in this case it does not make much sense as the number of observations is
# common to all models)
model.avg(confset.95p, rank = AIC, rank.args = alist(k = log(length(residuals(x)))))

## End(Not run)

---

model selection table

Description

Build a model selection table.

Usage

model.sel(object, ...)

## Default S3 method:
model.sel(object, ..., rank = NULL, rank.args = NULL,
  beta = c("none", "sd", "partial.sd"), extra)
## S3 method for class 'model.selection'
model.sel(object, rank = NULL, rank.args = NULL, fit = NA,
  ..., beta = c("none", "sd", "partial.sd"), extra)

model.sel(x) <- value

Arguments

object, value	a fitted model object, a list of such objects, or a "model.selection" object.
...	more fitted model objects.
rank	optional, custom rank function (returning an information criterion) to use instead of the default AICc, e.g. QAIC or BIC, may be omitted if object is a model list returned by get.models.
rank.args	optional list of arguments for the rank function. If one is an expression, an x within it is substituted with a current model.
fit	logical, stating whether the model objects should be re-fitted if they are not stored in the "model.selection" object. Set to NA to re-fit the models only if this is needed. See ‘Details’.
beta	indicates whether and how the component models' coefficients should be standardized. See the argument's description in dredge.
extra	optional additional statistics to include in the result, provided as functions, function names or a list of such (best if named or quoted). See dredge for details.
x	a "model.selection" object.

Details

model.sel used with "model.selection" object will re-fit model objects, unless they are stored in object (in attribute "modelList"), if argument extra is provided, or the requested beta is different than object's "beta" attribute, or the new rank function cannot be applied directly to logLik objects, or new rank.args are given (unless argument fit = FALSE).

The replacement function appends new models to the existing "model.selection" object.

Value

An object of class c("model.selection", "data.frame"), being a data.frame, where each row represents one model and columns contain useful information about each model: the coefficients, df, log-likelihood, the value of the information criterion used, Δ_IC and ‘Akaike weight’. If any arguments differ between the modelling function calls, the result will include additional columns showing them (except for formulas and some other arguments).

See model.selection.object for its structure.

Author(s)

Kamil Bartoń

See Also

dredge, AICc, list of supported models.

Possible alternatives: ICtab (in package bbmle), or aictab (AICcmodavg).

Examples

Cement$X1 <- cut(Cement$X1, 3)
Cement$X2 <- cut(Cement$X2, 2)

fm1 <- glm(formula = y ~ X1 + X2 * X3, data = Cement)
fm2 <- update(fm1, . ~ . - X1 - X2)
fm3 <- update(fm1, . ~ . - X2 - X3)

## ranked with AICc by default
(msAICc <- model.sel(fm1, fm2, fm3))

## ranked with BIC
model.sel(fm1, fm2, fm3, rank = AIC, rank.args = alist(k = log(nobs(x))))
# or
# model.sel(msAICc, rank = AIC, rank.args = alist(k = log(nobs(x))))
# or
# update(msAICc, rank = AIC, rank.args = alist(k = log(nobs(x))))

# appending new models:
model.sel(msAICc) <- update(fm1, . ~ 1)

---

Description of Model Selection Objects

Description

An object of class "model.selection" holds a table of model coefficients and ranking statistics. It is produced by dredge or model.sel.

Value

The object is a data.frame with additional attributes. Each row represents one model. The models are ordered by the information criterion value specified by rank (lowest on top).

Data frame columns:

<model terms>	For numeric covariates these columns hold coefficent value, for factors their presence in the model. If the term is not present in a model, value is NA.
<varying arguments>	Optional. If any arguments differ between the modelling function calls (except for formulas and some other arguments), these will be held in additional columns (of class "factor").
df	Number of model parameters
logLik	Log-likelihood (or quasi-likelihood for GEE)
<rank>	Information criterion value
delta	Δ_IC
weight	‘Akaike weights’.

Attributes:

model.calls	A list containing model calls (arranged in the same order as in the table). A model call can be retrieved with getCall(*, i) where i is a vector of model index or name (if given as character string).
global	The global.model object
global.call	Call to the global.model
terms	A character string holding all term names. Attribute "interceptLabel" gives the name of the intercept term.
rank	The rank function used
beta	A character string, representing the coefficient standardizing method used. Either "none", "sd" or "partial.sd"
coefTables	List of matrices of class "coefTable" containing each model's coefficients with std. errors and associated dfs
nobs	Number of observations
warnings	optional (pdredge only). A list of errors and warnings issued by the modelling function during the fitting, with a model number appended to each.

It is not recommended to directly access the attributes. Instead, use extractor functions if possible. These include getCall for retrieving model calls, coefTable and coef for coefficients, and nobs. logLik extracts list of model log-likelihoods (as "logLik" objects), and Weights extracts ‘Akaike weights’.

The object has class c("model.selection", "data.frame").

See Also

dredge, model.sel.

---

List of supported models

Description

List of model classes accepted by model.avg, model.sel, and dredge.

Details

Fitted model objects that can be used with model selection and model averaging functions include those produced by:

• lm, glm (package stats);

• rlm, glm.nb and polr (MASS);

• multinom (nnet);

• lme, gls (nlme);

• lmer, glmer (lme4);

• cpglm, cpglmm (cplm);

• gam, gamm* (mgcv);

• gamm4* (gamm4);

• gamlss (gamlss);

• glmmML (glmmML);

• glmmadmb (glmmADMB from R-Forge);

• glmmTMB (glmmTMB);

• MCMCglmm* (MCMCglmm);

• asreml (non-free commercial package asreml; allows only for REML comparisons);

• hurdle, zeroinfl (pscl);

• negbin, betabin (class "glimML"), package aod);

• aodml, aodql (aods3);

• betareg (betareg);

• brglm (brglm);

• *sarlm models, spautolm (spatialreg);

• spml* (if fitted by ML, splm);

• coxph, survreg (survival);

• coxme, lmekin (coxme);

• rq (quantreg);

• clm and clmm (ordinal);

• logistf (logistf);

• crunch*, pgls (caper);

• maxlike (maxlike);

• most "unmarkedFit" objects from package unmarked);

• mark and related functions (class mark from package RMark). Currently dredge can only manipulate formula element of the argument model.parameters, keeping its other elements intact.

Generalized Estimation Equation model implementations: geeglm from package geepack, gee from gee, geem from geeM, wgeesel from wgeesel, and yags from yags (on R-Forge) can be used with QIC as the selection criterion.

Other classes are also likely to be supported, in particular if they inherit from one of the above classes. In general, the models averaged with model.avg may belong to different types (e.g. glm and gam), provided they use the same data and response, and if it is valid to do so. This applies also to constructing model selection tables with model.sel.

Note

* In order to use gamm, gamm4, spml (> 1.0.0), crunch or MCMCglmm with dredge, an updateable wrapper for these functions should be created.

See Also

model.avg, model.sel and dredge.

---

Identify nested models

Description

Find models that are ‘nested’ within each model in the model selection table.

Usage

nested(x, indices = c("none", "numeric", "rownames"), rank = NULL)

Arguments

x	a "model.selection" object (result of dredge or model.sel).
indices	if omitted or "none" then the function checks if, for each model, there are any higher ranked models nested within it. If "numeric" or "rownames", indices or names of all nested models are returned. See “Value”.
rank	the name of the column with the ranking values (defaults to the one before “delta”). Only used if indices is "none".

Details

In model comparison, a model is said to be “nested” within another model if it contains a subset of parameters of the latter model, but does not include other parameters (e.g. model ‘A+B’ is nested within ‘A+B+C’ but not ‘A+C+D’).

This function can be useful in a model selection approach suggested by Richards (2008), in which more complex variants of any model with a lower IC value are excluded from the candidate set.

Value

A vector of length equal to the number of models (table rows).

If indices = "none" (the default), it is a vector of logical values where i-th element is TRUE if any model(s) higher up in the table are nested within it (i.e. if simpler models have lower IC pointed by rank).

For indices other than "none", the function returns a list of vectors of numeric indices or names of models nested within each i-th model.

Note

This function determines nesting based only on fixed model terms, within groups of models sharing the same ‘varying’ parameters (see dredge and example in Beetle).

Author(s)

Kamil Bartoń

References

Richards, S. A., Whittingham, M. J., Stephens, P. A. 2011 Model selection and model averaging in behavioural ecology: the utility of the IT-AIC framework. Behavioral Ecology and Sociobiology 65, 77–89.

Richards, S. A. 2008 Dealing with overdispersed count data in applied ecology. Journal of Applied Ecology 45, 218–227.

See Also

dredge, model.sel

Examples

fm <- lm(y ~ X1 + X2 + X3 + X4, data = Cement, na.action = na.fail)
ms <- dredge(fm)

# filter out overly complex models according to the 
# "nesting selection rule":
subset(ms, !nested(.)) # dot represents the ms table object

# print model "4" and all models nested within it
nst <- nested(ms, indices = "row")
ms[c("4", nst[["4"]])]

ms$nested <- sapply(nst, paste, collapse = ",")

ms

---

Parameter averaging

Description

Average a coefficient with standard errors based on provided weights. This function is intended chiefly for internal use.

Usage

par.avg(x, se, weight, df = NULL, level = 1 - alpha, alpha = 0.05,
  revised.var = TRUE, adjusted = TRUE)

Arguments

x	vector of parameters.
se	vector of standard errors.
weight	vector of weights.
df	optional vector of degrees of freedom.
alpha, level	significance level for calculating confidence intervals.
revised.var	logical, should the revised formula for standard errors be used? See ‘Details’.
adjusted	logical, should the inflated standard errors be calculated? See ‘Details’.

Details

Unconditional standard errors are square root of the variance estimator, calculated either according to the original equation in Burnham and Anderson (2002, equation 4.7), or a newer, revised formula from Burnham and Anderson (2004, equation 4) (if revised.var = TRUE, this is the default). If adjusted = TRUE (the default) and degrees of freedom are given, the adjusted standard error estimator and confidence intervals with improved coverage are returned (see Burnham and Anderson 2002, section 4.3.3).

Value

par.avg returns a vector with named elements:

Coefficient	model coefficients
SE	unconditional standard error
Adjusted SE	adjusted standard error
Lower CI, Upper CI	unconditional confidence intervals.

Author(s)

Kamil Bartoń

References

Burnham, K. P. and Anderson, D. R. 2002 Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed.

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

See Also

model.avg for model averaging.

---

Automated model selection using parallel computation

Description

Parallelized version of dredge.

Usage

pdredge(global.model, cluster = NULL, 
  beta = c("none", "sd", "partial.sd"), evaluate = TRUE, rank = "AICc", 
  fixed = NULL, m.lim = NULL, m.min, m.max, subset, trace = FALSE, 
  varying, extra, ct.args = NULL, deps = attr(allTerms0, "deps"),
  check = FALSE, ...)

Arguments

global.model, beta, rank, fixed, m.lim, m.max, m.min, subset, varying, extra, ct.args, deps, ...	see dredge.
evaluate	whether to evaluate and rank the models. If FALSE, a list of unevaluated calls is returned and cluster is not used.
trace	displays the generated calls, but may not work as expected since the models are evaluated in batches rather than one by one.
cluster	either a valid "cluster" object, or NULL for a single threaded execution.
check	either integer or logical value controlling how much checking for existence and correctness of dependencies is done on the cluster nodes. See ‘Details’.

Details

All the dependencies for fitting the global.model, including the data and any objects that the modelling function will use must be exported to the cluster worker nodes (e.g. via clusterExport). The required packages must be also loaded thereinto (e.g. via clusterEvalQ(..., library(package)), before the cluster is used by pdredge.

If check is TRUE or positive, pdredge tries to check whether all the variables and functions used in the call to global.model are present in the cluster nodes' .GlobalEnv before proceeding further. This will cause false errors if some arguments of the model call (other than subset) would be evaluated in the data environment. In that case is desirable to use check = FALSE (the default).

If check is TRUE or greater than one, pdredge will compare the global.model updated on the cluster nodes with the one given as an argument.

Value

See dredge.

Note

As of version 1.45.0, using pdredge directly is deprecated. Use dredge instead and provide cluster argument.

Author(s)

Kamil Bartoń

See Also

makeCluster and other cluster related functions in packages parallel or snow.

Examples

# One of these packages is required:
## Not run: require(parallel) || require(snow)

# From example(Beetle)

Beetle100 <- Beetle[sample(nrow(Beetle), 100, replace = TRUE),]

fm1 <- glm(Prop ~ dose + I(dose^2) + log(dose) + I(log(dose)^2),
    data = Beetle100, family = binomial, na.action = na.fail)

msubset <- expression(xor(dose, `log(dose)`) & (dose | !`I(dose^2)`)
    & (`log(dose)` | !`I(log(dose)^2)`))
varying.link <- list(family = alist(logit = binomial("logit"),
    probit = binomial("probit"), cloglog = binomial("cloglog") ))

# Set up the cluster
clusterType <- if(length(find.package("snow", quiet = TRUE))) "SOCK" else "PSOCK"
clust <- try(makeCluster(getOption("cl.cores", 2), type = clusterType))

clusterExport(clust, "Beetle100")

# noticeable gain only when data has about 3000 rows (Windows 2-core machine)
print(system.time(dredge(fm1, subset = msubset, varying = varying.link)))
print(system.time(dredge(fm1, cluster = FALSE, subset = msubset,
    varying = varying.link)))
print(system.time(pdd <- dredge(fm1, cluster = clust, subset = msubset,
    varying = varying.link)))

print(pdd)

## Not run: 
# Time consuming example with 'unmarked' model, based on example(pcount).
# Having enough patience you can run this with 'demo(pdredge.pcount)'.
library(unmarked)
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
    obsCovs = mallard.obs)
(ufm.mallard <- pcount(~ ivel + date + I(date^2) ~ length + elev + forest,
    mallardUMF, K = 30))
clusterEvalQ(clust, library(unmarked))
clusterExport(clust, "mallardUMF")

# 'stats4' is needed for AIC to work with unmarkedFit objects but is not
# loaded automatically with 'unmarked'.
require(stats4)
invisible(clusterCall(clust, "library", "stats4", character.only = TRUE))

#system.time(print(pdd1 <- dredge(ufm.mallard,
#   subset = `p(date)` | !`p(I(date^2))`, rank = AIC)))

system.time(print(pdd2 <- dredge(ufm.mallard, cluster = clust,
    subset = `p(date)` | !`p(I(date^2))`, rank = AIC, extra = "adjR^2")))


# best models and null model
subset(pdd2, delta < 2 | df == min(df))

# Compare with the model selection table from unmarked
# the statistics should be identical:
models <- get.models(pdd2, delta < 2 | df == min(df), cluster = clust)

modSel(fitList(fits = structure(models, names = model.names(models,
    labels = getAllTerms(ufm.mallard)))), nullmod = "(Null)")

## End(Not run)

stopCluster(clust)

---

Visualize model selection table

Description

Produces a graphical representation of model weights and terms.

Usage

## S3 method for class 'model.selection'
plot(
  x,
  ylab = NULL, xlab = NULL, main = "Model selection table",
  labels = NULL, terms = NULL, labAsExpr = TRUE,
  vlabels = rownames(x), mar.adj = TRUE,
  col = NULL, col.mode = 2,
  bg = "white", border = par("col"),
  par.lab = NULL, par.vlab = NULL,
  axes = TRUE, ann = TRUE,
  ...
)

Arguments

x	a "model.selection" object.
xlab, ylab, main	labels for the x and y axes, and the main title for the plot.
labels	optional, a character vector or an expression containing model term labels (to appear on top side of the plot). Its length must be equal to number of displayed model terms. Defaults to the model term names.
terms	which terms to include (default NULL means all terms).
labAsExpr	logical, indicating whether the term names should be interpreted (parsed) as R expressions for prettier labels. See also plotmath.
vlabels	alternative labels for the table rows (i.e. model names)
mar.adj	logical indicating whether the top and right margin should be enlarged if necessary to fit the labels.
col	vector or a matrix of colours for the non-empty grid cells. See 'Details'. If col is given as a matrix, the colours are applied to rows and columns. How it is done is governed by the argument col.mode.
col.mode	either numeric or "value", specifies cell colouring mode. See 'Details'.
bg	background colour for the empty cells.
border	border colour for cells and axes.
par.lab, par.vlab	optional lists of arguments and graphical parameters for drawing term labels (top axis) and model names (right axis), respectively. Items of par.lab are passed as arguments to mtext, and those of par.vlab are passed to axis.
axes, ann	logical values indicating whether the axis and annotation should appear on the plot.
...	further graphical parameters to be set for the plot.

Details

Colours

If col.mode = 0, the colours are recycled: if col is a matrix, recycling takes place both per row and per column. If col.mode > 0, the colour values in the columns are interpolated and assigned according to the model weights. Higher values shift the colours for models with lower model weights more forward. See also colorRamp. If col.mode < 0 or "value" (partially matched, case-insensitive) and col has two or more elements, colours are used to represent coefficient values: the first element in col is used for categorical predictors, the rest for continuous values. The default is grey for factors and HCL palette "Blue-Red 3" otherwise, ranging from blue for negative values to red for positive ones.

The following arguments are useful for adjusting label size and position in par.lab and par.vlab : cex, las (see par), line and hadj (see mtext and axis).

Author(s)

Kamil Bartoń

See Also

plot.default, par, MuMIn-package

Examples

dd <- dredge(lm(formula = y ~ ., data = Cement, na.action = na.fail))
plot(dd,
    # colours by coefficient value:
    col.mode = "value",
    par.lab = list(las = 2, line = 1.2, cex = 1),
    bg = "gray30",
    # change labels for the models to Akaike weights:
    vlabels = parse(text = paste("omega ==", round(Weights(dd), 2)))
    )
plot(dd, col = 2:3, col.mode = 0) # colour recycled by row
plot(dd, col = cbind(2:3, 4:5), col.mode = 0) # colour recycled by row and column
plot(dd, col = 2:3, col.mode = 1) # colour gradient by model weight

---

Predict method for averaged models

Description

Model-averaged predictions, optionally with standard errors.

Usage

## S3 method for class 'averaging'
predict(object, newdata = NULL, se.fit = FALSE,
  interval = NULL, type = NA, backtransform = FALSE, full = TRUE, ...)

Arguments

object	an object returned by model.avg.
newdata	optional data.frame in which to look for variables with which to predict. If omitted, the fitted values are used.
se.fit	logical, indicates if standard errors should be returned. This has any effect only if the predict methods for each of the component models support it.
interval	currently not used.
type	the type of predictions to return (see documentation for predict appropriate for the class of used component models). If omitted, the default type is used. See ‘Details’.
backtransform	if TRUE, the averaged predictions are back-transformed from link scale to response scale. This makes sense provided that all component models use the same family, and the prediction from each of the component models is calculated on the link scale (as specified by type. For glm, use type = "link"). See ‘Details’.
full	if TRUE, the full model-averaged coefficients are used (only if se.fit = FALSE and the component objects are a result of lm).
...	arguments to be passed to respective predict method (e.g. level for lme model).

Details

predicting is possible only with averaging objects with "modelList" attribute, i.e. those created via model.avg from a model list, or from model.selection object with argument fit = TRUE (which will recreate the model objects, see model.avg).

If all the component models are ordinary linear models, the prediction can be made either with the full averaged coefficients (the argument full = TRUE this is the default) or subset-averaged coefficients. Otherwise the prediction is obtained by calling predict on each component model and weighted averaging the results, which corresponds to the assumption that all predictors are present in all models, but those not estimated are equal zero (see ‘Note’ in model.avg). Predictions from component models with standard errors are passed to par.avg and averaged in the same way as the coefficients are.

Predictions on the response scale from generalized models can be calculated by averaging predictions of each model on the link scale, followed by inverse transformation (this is achieved with type = "link" and backtransform = TRUE). This is only possible if all component models use the same family and link function. Alternatively, predictions from each model on response scale may be averaged (with type = "response" and backtransform = FALSE). Note that this leads to results differing from those calculated with the former method. See also predict.glm.

Value

If se.fit = FALSE, a vector of predictions, otherwise a list with components: fit containing the predictions, and se.fit with the estimated standard errors.

Note

This method relies on availability of the predict methods for the component model classes (except when all component models are of class lm).

The package MuMIn includes predict methods for lme, and gls that calculate standard errors of the predictions (with se.fit = TRUE). They enhance the original predict methods from package nlme, and with se.fit = FALSE they return identical result. MuMIn's versions are always used in averaged model predictions (so it is possible to predict with standard errors), but from within global environment they will be found only if MuMIn is before nlme on the search list (or directly extracted from namespace as MuMIn:::predict.lme).

Author(s)

Kamil Bartoń

See Also

model.avg, and par.avg for details of model-averaged parameter calculation.

predict.lme, predict.gls

Examples

# Example from Burnham and Anderson (2002), page 100:
fm1 <- lm(y ~ X1 + X2 + X3 + X4, data = Cement)

ms1 <- dredge(fm1)
confset.95p <- get.models(ms1, subset = cumsum(weight) <= .95)
avgm <- model.avg(confset.95p)

nseq <- function(x, len = length(x)) seq(min(x, na.rm = TRUE),
    max(x, na.rm=TRUE), length = len)

# New predictors: X1 along the range of original data, other
# variables held constant at their means
newdata <- as.data.frame(lapply(lapply(Cement[, -1], mean), rep, 25))
newdata$X1 <- nseq(Cement$X1, nrow(newdata))

n <- length(confset.95p)

# Predictions from each of the models in a set, and with averaged coefficients
pred <- data.frame(
	model = sapply(confset.95p, predict, newdata = newdata),
	averaged.subset = predict(avgm, newdata, full = FALSE),
    averaged.full = predict(avgm, newdata, full = TRUE)
	)

opal <- palette(c(topo.colors(n), "black", "red", "orange"))
matplot(newdata$X1, pred, type = "l",
	lwd = c(rep(2,n),3,3), lty = 1,
    xlab = "X1", ylab = "y", col=1:7)

# For comparison, prediction obtained by averaging predictions of the component
# models
pred.se <- predict(avgm, newdata, se.fit = TRUE)
y <- pred.se$fit
ci <- pred.se$se.fit  * 2
matplot(newdata$X1, cbind(y, y - ci, y + ci), add = TRUE, type="l",
	lty = 2, col = n + 3, lwd = 3)

legend("topleft",
    legend=c(lapply(confset.95p, formula),
        paste(c("subset", "full"), "averaged"), "averaged predictions + CI"),
    lty = 1, lwd = c(rep(2,n),3,3,3),  cex = .75, col=1:8)

palette(opal)

---

Quasi AIC or AICc

Description

Calculate a modification of Akaike's Information Criterion for overdispersed count data (or its version corrected for small sample, “quasi-AIC$_{c}$”), for one or several fitted model objects.

Usage

QAIC(object, ..., chat, k = 2, REML = NULL)
QAICc(object, ..., chat, k = 2, REML = NULL)

Arguments

object	a fitted model object.
...	optionally, more fitted model objects.
chat	$\hat{c}$, the variance inflation factor.
k	the ‘penalty’ per parameter.
REML	optional logical value, passed to the logLik method indicating whether the restricted log-likelihood or log-likelihood should be used. The default is to use the method used for model estimation.

Value

If only one object is provided, returns a numeric value with the corresponding QAIC or QAIC$_{c}$; otherwise returns a data.frame with rows corresponding to the objects.

Note

$\hat{c}$ is the dispersion parameter estimated from the global model, and can be calculated by dividing model's deviance by the number of residual degrees of freedom.

In calculation of QAIC, the number of model parameters is increased by 1 to account for estimating the overdispersion parameter. Without overdispersion, $\hat{c} = 1$ and QAIC is equal to AIC.

Note that glm does not compute maximum-likelihood estimates in models within the quasi- family. In case it is justified, it can be worked around by ‘borrowing’ the aic element from the corresponding ‘non-quasi’ family (see ‘Example’).

Consider using negative binomial family with overdispersed count data.

Author(s)

Kamil Bartoń

See Also

AICc, quasi family used for models with over-dispersion.

Tests for overdispersion in GLM[M]: check_overdispersion.

Examples

options(na.action = "na.fail")

# Based on "example(predict.glm)", with one number changed to create
# overdispersion
budworm <- data.frame(
    ldose = rep(0:5, 2), sex = factor(rep(c("M", "F"), c(6, 6))),
    numdead = c(10, 4, 9, 12, 18, 20, 0, 2, 6, 10, 12, 16))
budworm$SF = cbind(numdead = budworm$numdead,
    numalive = 20 - budworm$numdead)

budworm.lg <- glm(SF ~ sex*ldose, data = budworm, family = binomial)
(chat <- deviance(budworm.lg) / df.residual(budworm.lg))

dredge(budworm.lg, rank = "QAIC", chat = chat)
dredge(budworm.lg, rank = "AIC")


## Not run: 
# A 'hacked' constructor for quasibinomial family object that allows for
# ML estimation
hacked.quasibinomial <- function(...) {
    res <- quasibinomial(...)
    res$aic <- binomial(...)$aic
    res
}
QAIC(update(budworm.lg, family = hacked.quasibinomial), chat = chat)

## End(Not run)

---

QIC and quasi-Likelihood for GEE

Description

Calculate quasi-likelihood under the independence model criterion (QIC) for Generalized Estimating Equations.

Usage

QIC(object, ..., typeR = FALSE)
QICu(object, ..., typeR = FALSE)
quasiLik(object, ...)

Arguments

object	a fitted model object of class "gee", "geepack", "geem", "wgee", or "yags".
...	for QIC and QIC$_{u}$, optionally more fitted model objects.
typeR	logical, whether to calculate QIC(R). QIC(R) is based on quasi-likelihood of a working correlation $R$ model. Defaults to FALSE, and QIC(I) based on independence model is returned.

Value

If just one object is provided, returns a numeric value with the corresponding QIC; if more than one object are provided, returns a data.frame with rows corresponding to the objects and one column representing QIC or QIC$_{u}$.

Note

This implementation is based partly on (revised) code from packages yags (R-Forge) and ape.

Author(s)

Kamil Bartoń

References

Pan, W. 2001 Akaike's Information Criterion in Generalized Estimating Equations. Biometrics 57, 120–125

Hardin J. W., Hilbe, J. M. 2003 Generalized Estimating Equations. Chapman & Hall/CRC

See Also

Methods exist for gee (package gee), geeglm (geepack), geem (geeM), wgee (wgeesel, the package's QIC.gee function is used), and yags (yags on R-Forge). There is also a QIC function in packages MESS and geepack, returning some extra information (such as CIC and QICc). yags and compar.gee from package ape both provide QIC values.

Examples

data(ohio)

fm1 <- geeglm(resp ~ age * smoke, id = id, data = ohio,
    family = binomial, corstr = "exchangeable", scale.fix = TRUE)
fm2 <- update(fm1, corstr = "ar1")
fm3 <- update(fm1, corstr = "unstructured")

# QIC function is also defined in 'geepack' but is returns a vector[6], so
# cannot be used as 'rank'. Either use `MuMIn::QIC` syntax or make a wrapper
# around `geepack::QIC`

QIC <- MuMIn::QIC
## Not run: 
QIC <- function(x) geepack::QIC(x)[1]

## End(Not run)

model.sel(fm1, fm2, fm3, rank = QIC)

#####
library(geepack)
library(MuMIn)

## Not run: 
# same result:
    dredge(fm1, m.lim = c(3, NA), rank = QIC, varying = list(
    corstr = list("exchangeable", "unstructured", "ar1")
    ))      

## End(Not run)

---

Pseudo-R-squared for Generalized Mixed-Effect models

Description

Calculate conditional and marginal coefficient of determination for Generalized mixed-effect models ($R_{GLMM}^{2}$).

Usage

r.squaredGLMM(object, null, ...)
## S3 method for class 'merMod'
r.squaredGLMM(object, null, envir = parent.frame(), pj2014 = FALSE, ...)

Arguments

object	a fitted linear model object.
null	optionally, a null model, including only random effects. See ‘Details’.
envir	optionally, the environment in which the null model is to be evaluated. Defaults to the current frame. See eval.
pj2014	logical, if TRUE and object is of poisson family, the result will include $R_{GLMM}^{2}$ using original formulation of Johnson (2014). This requires fitting object with an observation-level random effect term added.
...	additional arguments, ignored.

Details

There are two types of $R_{GLMM}^{2}$: marginal and conditional.

Marginal $R_{GLMM}^{2}$ represents the variance explained by the fixed effects, and is defined as:

$R_{GLMM(m)}^{2}= \frac{\sigma_f^2}{\sigma_f^2 + \sigma_{\alpha}^2 + \sigma_{\varepsilon }^2}$

Conditional $R_{GLMM}^{2}$ represents the variance explained by the entire model, including both fixed and random effects. It is calculated by the equation:

$R_{GLMM(c)}^{2}= \frac{\sigma_f^2 + \sigma_{\alpha}^2}{\sigma_f^2 + \sigma_{\alpha}^2 + \sigma_{\varepsilon }^2}$

where $\sigma_f^2$ is the variance of the fixed effect components, $\sigma_{\alpha}$ is the variance of the random effects, and $\sigma_\epsilon^2$ is the “observation-level” variance.

Three methods are available for deriving the observation-level variance $\sigma_\varepsilon$: the delta method, lognormal approximation and using the trigamma function.

The delta method can be used with for all distributions and link functions, while lognormal approximation and trigamma function are limited to distributions with logarithmic link. Trigamma-estimate is recommended whenever available. Additionally, for binomial distributions, theoretical variances exist specific for each link function distribution.

Null model. Calculation of the observation-level variance involves in some cases fitting a null model containing no fixed effects other than intercept, otherwise identical to the original model (including all the random effects). When using r.squaredGLMM for several models differing only in their fixed effects, in order to avoid redundant calculations, the null model object can be passed as the argument null. Otherwise, a null model will be fitted via updating the original model. This assumes that all the variables used in the original model call have the same values as when the model was fitted. The function warns about this when fitting the null model is required. This warnings can be disabled by setting options(MuMIn.noUpdateWarning = TRUE).

Value

r.squaredGLMM returns a two-column numeric matrix, each (possibly named) row holding values for marginal and conditional $R_{GLMM}^{2}$ calculated with different methods, such as “delta”, “log-normal”, “trigamma”, or “theoretical” for models of binomial family.

Note

Important: as of MuMIn version 1.41.0, r.squaredGLMM returns a revised statistics based on Nakagawa et al. (2017) paper. The returned value's format also has changed (it is a matrix rather than a numeric vector as before). Pre-1.41.0 version of the function calculated the “theoretical” $R_{GLMM}^{2}$ for binomial models.

$R_{GLMM}^{2}$ can be calculated also for fixed-effect models. In the simpliest case of OLS it reduces to var(fitted) / (var(fitted) + deviance / 2). Unlike likelihood-ratio based $R^{2}$ for OLS, value of this statistic differs from that of the classical $R^{2}$.

Currently methods exist for classes: merMod, lme, glmmTMB, glmmADMB, glmmPQL, cpglm(m) and (g)lm.

For families other than gaussian, Gamma, poisson, binomial and negative binomial, the residual variance is obtained using get_variance from package insight.

See note in r.squaredLR help page for comment on using $R^{2}$ in model selection.

Author(s)

Kamil Bartoń. This implementation is based on R code from ‘Supporting Information’ for Nakagawa et al. (2014), (the extension for random-slopes) Johnson (2014), and includes developments from Nakagawa et al. (2017).

References

Nakagawa, S., Schielzeth, H. 2013 A general and simple method for obtaining $R^{2}$ from Generalized Linear Mixed-effects Models. Methods in Ecology and Evolution 4, 133–142.

Johnson, P. C. D. 2014 Extension of Nakagawa & Schielzeth’s $R_{GLMM}^{2}$ to random slopes models. Methods in Ecology and Evolution 5, 44–946.

Nakagawa, S., Johnson, P. C. D., Schielzeth, H. 2017 The coefficient of determination $R^{2}$ and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. J. R. Soc. Interface 14, 20170213.

See Also

summary.lm, r.squaredLR

r2 from package performance calculates $R_{GLMM}^{2}$ also for variance at different levels, with optional confidence intervals. r2glmm has functions for $R^{2}$ and partial $R^{2}$.

Examples

data(Orthodont, package = "nlme")

fm1 <- lme(distance ~ Sex * age, ~ 1 | Subject, data = Orthodont)

fmnull <- lme(distance ~ 1, ~ 1 | Subject, data = Orthodont)

r.squaredGLMM(fm1)
r.squaredGLMM(fm1, fmnull)
r.squaredGLMM(update(fm1, . ~ Sex), fmnull)

r.squaredLR(fm1)
r.squaredLR(fm1, null.RE = TRUE)
r.squaredLR(fm1, fmnull) # same result

## Not run: 
if(require(MASS)) {
    fm <- glmmPQL(y ~ trt + I(week > 2), random = ~ 1 | ID, 
        family = binomial, data = bacteria, verbose = FALSE)
    fmnull <- update(fm, . ~ 1)
    r.squaredGLMM(fm)

    # Include R2GLMM (delta method estimates) in a model selection table:
    # Note the use of a common null model
    dredge(fm, extra = list(R2 = function(x) r.squaredGLMM(x, fmnull)["delta", ]))
    
}

## End(Not run)

---

Likelihood-ratio based pseudo-R-squared

Description

Calculate a coefficient of determination based on the likelihood-ratio test ($R_{LR}^{2}$).

Usage

r.squaredLR(object, null = NULL, null.RE = FALSE, ...)

null.fit(object, evaluate = FALSE, RE.keep = FALSE, envir = NULL, ...)

Arguments

object	a fitted model object.
null	a fitted null model. If not provided, null.fit will be used to construct it. null.fit's capabilities are limited to only a few model classes, for others the null model has to be specified manually.
null.RE	logical, should the null model contain random factors? Only used if no null model is given, otherwise omitted, with a warning.
evaluate	if TRUE evaluate the fitted model object else return the call.
RE.keep	if TRUE, the random effects of the original model are included.
envir	the environment in which the null model is to be evaluated, defaults to the environment of the original model's formula.
...	further arguments, of which only x would be used, to maintain compatibility with older versions (x has been replaced with object).

Details

This statistic is is one of the several proposed pseudo-$R^{2}$'s for nonlinear regression models. It is based on an improvement from null (intercept only) model to the fitted model, and calculated as

$R_{LR}^{2}=1-\exp(-\frac{2}{n}(\log\mathcal{L}(x)-\log\mathcal{L}(0)))$

where $\log\mathcal{L}(x)$ and $\log\mathcal{L}(0)$ are the log-likelihoods of the fitted and the null model respectively. ML estimates are used if models have been fitted by REstricted ML (by calling logLik with argument REML = FALSE). Note that the null model can include the random factors of the original model, in which case the statistic represents the ‘variance explained’ by fixed effects.

For OLS models the value is consistent with classical $R^{2}$. In some cases (e.g. in logistic regression), the maximum $R_{LR}^{2}$ is less than one. The modification proposed by Nagelkerke (1991) adjusts the $R_{LR}^{2}$ to achieve 1 at its maximum: $\bar{R}^{2} = R_{LR}^{2} / \max(R_{LR}^{2})$ where $\max(R_{LR}^{2}) = 1 - \exp(\frac{2}{n}\log\mathcal{L}(\textrm{0}))$.

null.fit tries to guess the null model call, given the provided fitted model object. This would be usually a glm. The function will give an error for an unrecognised class.

Value

r.squaredLR returns a value of $R_{LR}^{2}$, and the attribute "adj.r.squared" gives the Nagelkerke's modified statistic. Note that this is not the same as nor equivalent to the classical ‘adjusted R squared’.

null.fit returns the fitted null model object (if evaluate = TRUE) or an unevaluated call to fit a null model.

Note

$R^{2}$ is a useful goodness-of-fit measure as it has the interpretation of the proportion of the variance ‘explained’, but it performs poorly in model selection, and is not suitable for use in the same way as the information criteria.

References

Cox, D. R. and Snell, E. J. 1989 The analysis of binary data, 2nd ed. London, Chapman and Hall.

Magee, L. 1990 $R^{2}$ measures based on Wald and likelihood ratio joint significance tests. Amer. Stat. 44, 250–253.

Nagelkerke, N. J. D. 1991 A note on a general definition of the coefficient of determination. Biometrika 78, 691–692.

See Also

summary.lm, r.squaredGLMM

r2 from package performance calculates many different types of $R^{2}$.

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