Package 'gllm'

Summarize multiple results from gllm

Description

Compare likelihood ratio test statistics from multiple calls to gllm.

Usage

## S3 method for class 'gllm'
anova(object, ..., test=c("Chisq","none"))

Arguments

object	is an object output from gllm.
...	other objects from gllm.
test	evaluate LRTS for model, or nothing.

Value

A list with components:

Model	name of each object being compared
Resid.df	residual degrees of freedom for each model
Deviance	likelihood ratio test statistic for model versus saturated model
Pr.Fit	chi-square based P-value for model
Test	models compared in stepwise testing
Df	degrees of freedom of model comparson
LRtest	likelihood ratio test statistic comparing models
Prob	chi-square based P-value for LRTS

.

Author(s)

David L Duffy

---

Bootstrap for generalized log-linear modelling

Description

Fits log-linear models for incomplete contingency tables, including some latent class models, via EM and Fisher scoring approaches. Performs a bootstrap for the sampling distribution of the full unobserved table.

Usage

boot.gllm(y,s,X,method="hybrid",em.maxit=1,tol=0.00001,strata=NULL,R=200)

Arguments

y	is the observed contingency table.
s	is a vector of indices, one for each cell of the full (unobserved) contingency table, representing the appropriate cell of y
X	is the design matrix, or a formula.
method	chooses the EM, Fisher scoring or a hybrid (EM then scoring) method for fitting the model.
em.maxit	is the number of EM iterations.
tol	is the convergence criterion for the LR criterion.
strata	is a vector identifying the sampling strata.
R	is the number of bootstrap replicates.

Details

The generalized log-linear model allows for modelling of incomplete contingency tables, that is tables where one or more dimensions have been collapsed over. See gllm for details.

Often, functions of the full unobserved table are the main focus of the analysis. For example, in a double sampling design where there is a gold standard measure for one part of the data set and only an unreliable measure for another part, the expected value of the gold standard in the entire dataset is the outcome of interest. The standard error of this statistic may be a complex function of the observed counts and model parameters.

Bootstrapping is one way to estimate such standard errors from a complex sampling design. The bootstrap sampling may be stratified if the design implies this, e.g. product-multinomial.

Value

A matrix $R+1$ by ncol(X) containing the initial estimate of the full (unobserved) contingency table, and the $R$ bootstrap replicates of the full table.

References

Hochberg Y (1977). On the use of double sampling schemes in analyzing categorical data with misclassification errors. J Am Statist Ass 72:914-921.

Examples

#
# Fit Hochberg 1977 double sampling data
# 2x2 table of imprecise measures and 2x2x2x2 reliability data
#
# 2x2 table of imprecise measures
#
y1 <-c(1196, 13562,
       7151, 58175)
a2<- 2-as.integer(gl(2,1,4))
b2<- 2-as.integer(gl(2,2,4))
set1<-data.frame(y1,a2,b2)
#
# 2x2x2x2 reliability data
#
y2<-c(17, 3,   10, 258,
       3, 4,    4,  25,
      16, 3,   25, 197,
     100, 13, 107, 1014)

a <- 2-as.integer(gl(2,1,16))
a2<- 2-as.integer(gl(2,2,16))
b <- 2-as.integer(gl(2,4,16))
b2<- 2-as.integer(gl(2,8,16))

set2<-data.frame(y2,a,a2,b,b2)
#
# Combined analysis
#
y<-c(y1,y2)
#
# Map observed table onto underlying 2x2x2x2x2 table
#
s <-c(1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 4, 4, 3, 3, 4, 4,
      5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
#
# Model combining the tables is A*A2*B*B2 + L (dummy study variable)
#
a <- 2-as.integer(gl(2,1,32))
a2<- 2-as.integer(gl(2,2,32))
b <- 2-as.integer(gl(2,4,32))
b2<- 2-as.integer(gl(2,8,32))
l <- 2-as.integer(gl(2,16,32))

X <- model.matrix( ~ a*a2*b*b2+l)

#
# Table 1 using unreliable measure
#
res1<-glm(y1 ~ a2*b2, family=poisson(),data=set1)
print(summary(res1))
#
# Table 2 using reliable measure
#
res2a<-glm(y2 ~ a*b, family=poisson(),data=set2)
print(summary(res2a))
#
# Table 2 demonstrating complex relationship between gold standard and
# unreliable measure
#
res2b<-glm(y2 ~ a*a2*b*b2, family=poisson(),data=set2)
print(summary(res2b))
#
# Combined analysis
#
require(gllm)
res12<-gllm(y,s,X)
print(summary.gllm(res12))
#
# Bootstrap the collapsed table to get estimated OR for reliable measures
#
# a and b are binary vectors the length of the *full* table
# and define the variables for which the odds ratio is to be
# estimated, here the reliable measure of injury and seatbelt usage.
#
boot.hochberg <- function (y,s,X,nrep,a,b) {
  z<-boot.gllm(y,s,X,R=nrep)
  boot.tab<-cbind(apply(z[,a & b],1,sum),
                  apply(z[,!a & b],1,sum),
                  apply(z[,a & !b],1,sum),
                  apply(z[,!a & !b],1,sum))
  oddsr<-boot.tab[,1]*boot.tab[,4]/boot.tab[,2]/boot.tab[,3] 
  hochberg.tab<-data.frame( c("yes","yes","no","no"),
                            c("yes","no","yes","no"),
                            boot.tab[1,],
                            apply(boot.tab[2:(1+nrep),],2,sd))
  colnames(hochberg.tab)<-c("Precise Injury","Precise seatbelt usage",
                            "Estimated Count","Bootstrap S.E.")
  print(hochberg.tab)
  cat("\nEstimated OR=",oddsr[1],"\n")
  cat("        Bias=",oddsr[1]-mean(oddsr[2:(1+nrep)]),"\n")
  cat("Bootstrap SE=",sd(oddsr[2:(1+nrep)]),"\n\nQuantiles\n\n")
  quantile(oddsr[2:(1+nrep)],c(0.025,0.50,0.975))
}
boot.hochberg(y,s,X,nrep=20,a,b)

---

Produce one bootstrap replicate of a vector of counts

Description

Given a vector of counts from a contingency table, produce a bootstrap replicate. Sampling zeroes are replaced by 0.5.

Usage

boot.table(y,strata=NULL)

Arguments

y	is the observed contingency table.
strata	is a vector defining the strata for a stratified bootstrap.

Value

A vector of counts with the same total.

Examples

boot.table(c(1,3,4,2))
## 0.5 2.0 5.0 3.0
boot.table(c(1,3,4,2),c(1,2,1,2))
## 2 1 3 4

---

Generalized log-linear modelling by EM and iterative proportional fitting

Description

Fits log-linear models for incomplete contingency tables, including some latent class models, via an EM approach.

Usage

emgllm(y,s,X,maxit=1000,tol=0.00001)

Arguments

y	is the observed contingency table.
s	is a vector of indices, one for each cell of the full (unobserved) contingency table, representing the appropriate cell of y
X	is the design matrix, or a formula.
maxit	is the number of EM iterations.
tol	is the convergence criterion for the LR criterion.

Details

The generalized log-linear model allows for modelling of incomplete contingency tables, that is tables where one or more dimensions have been collapsed over. These include situations where imprecise measures have been calibrated using a "perfect" gold standard, and the true association between imperfectly measured variables is to be estimated; where data is missing for a subsample of the population; latent variable models where latent variables are "errorless" functions of observed variables - eg ML gene frequency estimation from counts of observed phenotypes; specialised measurement models eg where observed counts are mixtures due to perfect measures and error prone measures; standard latent class analysis; symmetry and quasi-symmetry models for square tables.

The general framework underlying these models is summarised by Espeland (1986), and Espeland & Hui (1987), and is originally due to Thompson & Baker (1981). An observed contingency table $y$, which will be treated as a vector, is modelled as arising from an underlying complete table $z$, where observed count $y_j$ is the sum of a number of elements of $z$, such that each $z_i$ contributes to no more than one $y_j$. Therefore one can write $y=F'z$, where $F$ is made up of orthogonal columns of ones and zeros.

We then specify a loglinear model for $z$, so that $log(E(z))=X'b$, where $X$ is a design matrix, and $b$ a vector of loglinear parameters. The loglinear model for $z$ and thus $y$, can be fitted via an iterative proportional fitting algorithm for $b$ and $z$, with an EM fitting for $y$, $z$ and $b$ (Haber 1984).

The emgllm function is a wrapper for C code implementing the approach in Haber (1984).

Value

A list with components:

deviance	the final model deviance (-2 log likelihood)
observed.values	the observed counts in y
fitted.values	the expected counts under the fitted model
full.table	the expected counts for the full (unobserved) table.

References

Espeland MA (1986). A general class of models for discrete multivariate data. Commun. Statist.-Simula 15:405-424.

Espeland MA, Hui SL (1987). A general approach to analyzing epidemiologic data that contains misclassification errors. Biometrics 43:1001-1012.

Haber M (1984). AS207: Fitting a general log-linear model. Appl Statist 33:358-362.

Thompson R, Baker RJ (1981). Composite link functions in generalized linear models. Appl Stat 30: 125-131.

Examples

#
# latent class analysis: two latent classes
#
# Data matrix 2x2x2x2x2 table of responses to five binary items
#
y<-c( 3,   6,   2,   11,   1,   1,   3,    4,
      1,   8,   0,   16,   0,   3,   2,   15,
     10,  29,  14,   81,   3,  28,  15,   80,
     16,  56,  21,  173,  11,  61,  28,  298)
#
# Scatter matrix: full table is 2x2x2x2x2x2
#
s<-  c(1:32,1:32)
#
# Design matrix: x is the latent variable (2 levels), 
# a-e are the observed variables
#
i<-rep(1,64)
x<-as.integer(gl(2,32,64))-1
a<-as.integer(gl(2,16,64))-1
b<-as.integer(gl(2,8 ,64))-1
c<-as.integer(gl(2,4 ,64))-1
d<-as.integer(gl(2,2 ,64))-1
e<-as.integer(gl(2,1 ,64))-1
X<-cbind(i,x,a,b,c,d,e,x*cbind(a,b,c,d,e))
colnames(X)<-c("Int","X","A","B","C","D","E","AX","BX","CX","DX","EX")
res<-emgllm(y,s,X, tol=0.01)
res
#
# Obtain standard errors for parameter estimates
#
summary(scoregllm(y,s,X,as.array(res$full.table)))

---

Generalized log-linear modelling by EM and iterative proportional fitting

Description

Fits log-linear models for incomplete contingency tables, via an EM approach.

Usage

emgllmfitter(y,s,X,maxit,tol)

Arguments

y	is the observed contingency table.
s	is a vector of indices, one for each cell of the full (unobserved) contingency table, representing the appropriate cell of y
X	is the design matrix.
maxit	is the number of EM iterations.
tol	is the convergence criterion for the LR criterion.

Details

The call to Andreas Borg's C code that fits the model by EM/IPF. The algorithm follows the approach in Haber (1984).

Value

A list with components:

y	the observed table
ji	s, the scatter vector
c	the design matrix
istop	maximum EM iterations
conv	the convergence tolerance
e	expected counts for the full (unobserved) table
ni	nrow(X)
nj	length(y)
nk	ncol(X)-1
f	expected counts

References

Haber M (1984). AS207: Fitting a general log-linear model. Appl Statist 33:358-362.

---

Generalized log-linear modelling

Description

Fits log-linear models for incomplete contingency tables, including some latent class models, via EM and Fisher scoring approaches.

Usage

gllm(y,s,X,method="hybrid",em.maxit=1,tol=0.00001)

Arguments

y	is the observed contingency table.
s	is a vector of indices, one for each cell of the full (unobserved) contingency table, representing the appropriate cell of y
X	is the design matrix, or a formula.
method	chooses the EM, Fisher scoring or a hybrid (EM then scoring) method for fitting the model.
em.maxit	is the number of EM iterations.
tol	is the convergence criterion for the LR criterion.

Details

The generalized log-linear model allows for modelling of incomplete contingency tables, that is tables where one or more dimensions have been collapsed over. These include situations where imprecise measures have been calibrated using a "perfect" gold standard, and the true association between imperfectly measured variables is to be estimated; where data is missing for a subsample of the population; latent variable models where latent variables are "errorless" functions of observed variables - eg ML gene frequency estimation from counts of observed phenotypes; specialised measurement models eg where observed counts are mixtures due to perfect measures and error prone measures; standard latent class analysis; symmetry and quasi-symmetry models for square tables.

The general framework underlying these models is summarised by Espeland (1986), and Espeland & Hui (1987), and is originally due to Thompson & Baker (1981). An observed contingency table $y$, which will be treated as a vector, is modelled as arising from an underlying complete table $z$, where observed count $y_j$ is the sum of a number of elements of $z$, such that each $z_i$ contributes to no more than one $y_j$. Therefore one can write $y=F'z$, where $F$ is made up of orthogonal columns of ones and zeros.

We then specify a loglinear model for $z$, so that $log(E(z))=X'b$, where $X$ is a design matrix, and $b$ a vector of loglinear parameters. The loglinear model for $z$ and thus $y$, can be fitted using two methods, both of which are available in gllm. The first was presented as AS207 by Michael Haber (1984) and combines an iterative proportional fitting algorithm for $b$ and $z$, with an EM fitting for $y$, $z$ and $b$. The second is a Fisher scoring approach, presented in Espeland (1986).

The gllm function is actually a simple wrapper for scoregllm().

Value

A list with components:

iter	the number of scoring iterations until convergence
deviance	the final model deviance (-2 log likelihood)
df	the model degrees of freedom
coefficients	the model parameter estimates
se	the standard errors for the model parameter estimates
V	the variance-covariance matrix for the model parameter estimates
observed.values	the observed counts in y
fitted.values	the expected counts under the fitted model
residuals	Pearsonian residuals under the fitted model
full.table	the expected counts for the full (unobserved) table.

References

Espeland MA (1986). A general class of models for discrete multivariate data. Commun. Statist.-Simula 15:405-424.

Espeland MA, Hui SL (1987). A general approach to analyzing epidemiologic data that contains misclassification errors. Biometrics 43:1001-1012.

Haber M (1984). AS207: Fitting a general log-linear model. Appl Statist 33:358-362.

Thompson R, Baker RJ (1981). Composite link functions in generalized linear models. Appl Statist 30: 125-131.

Examples

#
# latent class analysis: two latent classes
#
# Data matrix 2x2x2x2x2 table of responses to five binary items
# (items 11-15 of sections 6-7 of the Law School Admission Test)
#
y<-c( 3,   6,   2,   11,   1,   1,   3,    4,
      1,   8,   0,   16,   0,   3,   2,   15,
     10,  29,  14,   81,   3,  28,  15,   80,
     16,  56,  21,  173,  11,  61,  28,  298)
#
# Scatter matrix: full table is 2x2x2x2x2x2
#
s<-  c(1:32,1:32)
#
# Design matrix: x is the latent variable (2 levels), 
# a-e are the observed variables
#
x<-as.integer(gl(2,32,64))-1
a<-as.integer(gl(2,16,64))-1
b<-as.integer(gl(2,8 ,64))-1
c<-as.integer(gl(2,4 ,64))-1
d<-as.integer(gl(2,2 ,64))-1
e<-as.integer(gl(2,1 ,64))-1

res1<-gllm(y,s,~x*(a+b+c+d+e),method="em",tol=0.01)
res1
#
# An example of model fitting: gametic association between two diallelic loci
#
# Data matrix
#
y<-c( 187,386,156,
      352,310,20,
      136,0  ,0)
#
# Scatter matrix
#
s<-  c( 1, 2, 2, 3,
        4, 5, 5, 6,
        4, 5, 5, 6,
        7, 8, 8, 9)
#
# Design matrix
#
X<-  matrix(c( 1,0,0,0,0,0,1,
               1,0,1,0,0,0,0,
               1,0,1,0,0,0,0,
               1,0,2,0,1,0,0,
               1,1,0,0,0,0,0,
               1,1,1,0,0,1,0,
               1,1,1,0,0,0,1,
               1,1,2,0,1,1,1,
               1,1,0,0,0,0,0,
               1,1,1,0,0,0,1,
               1,1,1,0,0,1,0,
               1,1,2,0,1,1,1,
               1,2,0,1,0,0,0,
               1,2,1,1,0,1,1,
               1,2,1,1,0,1,1,
               1,2,2,1,1,2,2), byrow=TRUE, ncol=7) 

colnames(X)<-c("Intercept", "A", "B", "P1", "P2", "Delta", "Epsilon")
res2<-gllm(y,s,X[,c(1:6)],method="hybrid",em.maxit=1,tol=0.00001)
res2
#

---

Invasive Cervical Cancer v exposure to Herpes Simplex Virus

Description

The case-control study of Hildesheim et al (1991) has been reanalysed by several authors (Carroll et al 1993; Spiegelhalter et al 1999; Prescott et al 2002). Exposure to Herpes Simplex Virus in cases suffering from invasive cervical cancer and in unaffected controls was assessed by Western Blot in all cases and controls and by a gold-standard refined Western blotting in a subset of 115 subjects.

Usage

data(hildesheim)

Format

A data frame table.

Source

Hildesheim et al (1991) Herpes simplex virus type 2: A possible interaction with human papillomavirus types 16/18 in the development of invasive cervical cancer. Int J Cancer 49, 335-340.

References

Carroll NJ, Gail MH, Lubin JH (1993) Case-control studies with errors in covariates. J Am Statist Assoc 88, 185-199.

Prescott GJ, Garthwaite PH (2002) A simple bayesian analysis of misclassified binary data with a validation substudy. Biometrics 58, 454-458.

Spiegelhalter DJ, Thomas A, Best NG (1999) Win-Bugs, Version 1.2. Technical Report. Cambridge: UK.

Examples

data(hildesheim)
ftable(xtabs(Freq ~ case+HSV.inac+HSV.gold, hildesheim))
fisher.test(xtabs(Freq ~ case+HSV.inac, hildesheim))
fisher.test(xtabs(Freq ~ case+HSV.gold, hildesheim, subset=HSV.gold!="?"))

#
# Combined analysis (ordered as incomplete then complete data)
#
y<-hildesheim$Freq[c(3,9,6,12,1,2,7,8,4,5,10,11)]
#
# Map observed table onto underlying 2x2x2x2 table
#
s <-c(1, 1, 2, 2, 3, 3, 4, 4,
      5, 6, 7, 8, 9, 10, 11, 12)
#
substudy  <- 2-as.integer(gl(2,8,16))
hsv.inac  <- 2-as.integer(gl(2,4,16))
hsv.gold  <- 2-as.integer(gl(2,2,16))
cancer    <- 2-as.integer(gl(2,1,16))

require(gllm)
res<-gllm(y,s, ~substudy+hsv.inac*hsv.gold*cancer)
print(summary.gllm(res))
#
# Bootstrap the collapsed table to get estimated OR for reliable measures
#
# a and b are binary vectors the length of the *full* table
# and define the variables for which the odds ratio is to be
# estimated, here the reliable measure of HSV exposure and Ca Cx
#
boot.hildesheim <- function (y,s,X,nrep,a,b) {
  z<-boot.gllm(y,s,X,R=nrep)
  boot.tab<-cbind(apply(z[,a & b],1,sum),
                  apply(z[,!a & b],1,sum),
                  apply(z[,a & !b],1,sum),
                  apply(z[,!a & !b],1,sum))
  oddsr<-boot.tab[,1]*boot.tab[,4]/boot.tab[,2]/boot.tab[,3] 
  hildesheim.tab<-data.frame( c("yes","yes","no","no"),
                              c("yes","no","yes","no"),
                              boot.tab[1,],
                              apply(boot.tab[2:(1+nrep),],2,sd))
  colnames(hildesheim.tab)<-c("Precise HSV","Cervical Cancer",
                            "Estimated Count","Bootstrap S.E.")
  print(hildesheim.tab)
  cat("\nEstimated OR=",oddsr[1],"\n")
  cat("        Bias=",oddsr[1]-mean(oddsr[2:(1+nrep)]),"\n")
  cat("Bootstrap SE=",sd(oddsr[2:(1+nrep)]),"\n\nQuantiles\n\n")
  print(quantile(oddsr[2:(1+nrep)],c(0.025,0.50,0.975)))

  b<-mean(log(oddsr[2:(1+nrep)]))
  se<-sd(log(oddsr[2:(1+nrep)]))
  ztest<-b/se
  cat("\n      Estimated log(OR)=",log(oddsr[1]),"\n",
      "Bootstrap mean log(OR)=",b,"\n",
      "          Bootstrap SE=",se,"\n",
      "                Wald Z=",ztest," (P=",2*pnorm(ztest,lower=FALSE),")\n")
}
boot.hildesheim(y,s,~substudy+hsv.inac*hsv.gold*cancer,nrep=50,cancer,hsv.gold)

---

Estimate linkage disequilibrium between two codominant autosomal loci

Description

Fits a log-linear model for allelic association between two codominant autosomal loci. Measures of LD are odds ratios.

Usage

ld2(locus1, locus2)

Arguments

locus1	is a character vector containing the genotypes at the first locus, or a RxC contingency table of genotype counts.
locus2	is a character vector containing the genotypes at the second locus.

Value

m0	base model
m1	estimating LD coefficient(s) assuming HWE
m2	testing HWE at locus 1
m3	testing HWE at locus 2
m4	estimating LD and HWD coefficient(s)

Examples

MNS<-matrix(c(91,32,5,147,78,17,85,75,7), nr=3)
colnames(MNS)<-c("S/S","S/s","s/s")
rownames(MNS)<-c("M/M","M/N","N/N")
class(MNS)<-"table"
print(MNS)
res<-ld2(MNS)
print(res)

---

Write design and filter matrices for log-linear model of linkage disequilibrium between two codominant autosomal loci

Description

Write design and filter matrices for log-linear model of linkage disequilibrium between two codominant autosomal loci.

Usage

ld2.model(nall1, nall2, formula="~a1+a2+p1+p2+d")

Arguments

nall1	is number of alleles at first codominant locus.
nall2	is number of alleles at first codominant locus.
formula	is character string listing terms to be included in model, where a1 denotes allele frequencies for locus 1, p1 the deviation from Hardy-Weinberg expectations for locus1, and d the intragametic allelic association parameters.

Value

A list with components:

Geno	is a dummy contingency table showing the expected order.
s	is a vector of indices, one for each cell of the full (unobserved) contingency table, representing the appropriate cell of y
X	is the design matrix.

---

Five dichotomous items from the Law School Admission Test (LSAT)

Description

Small dataset (items 11-15 from sections 6 and 7 of the Law School Admission Test) used by Bock and Lieberman (1970), Christoffersson (1975) and Joreskog and Sorbom (1986) to test methods for factor analysis of binary data.

Source

Bock RD, Lieberman M (1970). Fitting a response model for n dichotomously scored items. Psychometrika 35, 179-197.

---

Create a filter matrix from a summary array of indices

Description

Create a filter matrix that multiplying the vector of counts from a complete contingency table, gives a collapsed contingency table.

Usage

scatter(y,s)

Arguments

y	is the observed contingency table. Provides a target length only.
s	is a vector of indices, one for each cell of the full (unobserved) contingency table, representing the appropriate cell of y

Value

S	A matrix of orthogonal columns of 1s and 0s

Author(s)

David L Duffy

Examples

y<-double(3)
z<-1:5
z %*% scatter(y,c(1,1,2,3,3))
## 1+2, 3, 4+5

---

Generalized log-linear modelling via Fisher scoring

Description

Fits log-linear models for incomplete contingency tables, including some latent class models, via Fisher scoring approaches.

Usage

scoregllm(y,s,X,m,tol=1e-5)

Arguments

y	is the observed contingency table.
s	is a vector of indices, one for each cell of the full (unobserved) contingency table, representing the appropriate cell of y
X	is the design matrix or a formula.
m	is a vector of starting values for the full (unobserved) contingency table.
tol	is the convergence criterion for the LR criterion.

Details

The generalized log-linear model allows for modelling of incomplete contingency tables, that is tables where one or more dimensions have been collapsed over. These include situations where imprecise measures have been calibrated using a "perfect" gold standard, and the true association between imperfectly measured variables is to be estimated; where data is missing for a subsample of the population; latent variable models where latent variables are "errorless" functions of observed variables - eg ML gene frequency estimation from counts of observed phenotypes; specialised measurement models eg where observed counts are mixtures due to perfect measures and error prone measures; standard latent class analysis; symmetry and quasi-symmetry models for square tables.

The general framework underlying these models is summarised by Espeland (1986), and Espeland & Hui (1987), and is originally due to Thompson & Baker (1981). An observed contingency table $y$, which will be treated as a vector, is modelled as arising from an underlying complete table $z$, where observed count $y_j$ is the sum of a number of elements of $z$, such that each $z_i$ contributes to no more than one $y_j$. Therefore one can write $y=F'z$, where $F$ is made up of orthogonal columns of ones and zeros.

We then specify a loglinear model for $z$, so that $log(E(z))=X'b$, where $X$ is a design matrix, and $b$ a vector of loglinear parameters. The loglinear model for $z$ and thus $y$, can be fitted by a Fisher scoring approach, presented in Espeland (1986).

The gllm function is actually a simple wrapper for scoregllm().

Value

A list with components:

iter	the number of scoring iterations until convergence
deviance	the final model deviance (-2 log likelihood)
df	the model degrees of freedom
coefficients	the model parameter estimates
se	the standard errors for the model parameter estimates
V	the variance-covariance matrix for the model parameter estimates
observed.values	the observed counts in y
fitted.values	the expected counts under the fitted model
residuals	Pearsonian residuals under the fitted model
full.table	the expected counts for the full (unobserved) table.

References

Espeland MA (1986). A general class of models for discrete multivariate data. Commun. Statist.-Simula 15:405-424.

Espeland MA, Hui SL (1987). A general approach to analyzing epidemiologic data that contains misclassification errors. Biometrics 43:1001-1012.

Thompson R, Baker RJ (1981). Composite link functions in generalized linear models. Appl Statist 30: 125-131.

Examples

#
# An example of model fitting: gametic association between two diallelic loci
# Data matrix
#
y<-c( 187,386,156,
      352,310,20,
      136,0  ,0)
#
# Scatter matrix
#
s<-  c( 1, 2, 2, 3,
        4, 5, 5, 6,
        4, 5, 5, 6,
        7, 8, 8, 9)
#
# Design matrix
#
X<-  matrix(c( 1,0,0,0,0,0,1,
               1,0,1,0,0,0,0,
               1,0,1,0,0,0,0,
               1,0,2,0,1,0,0,
               1,1,0,0,0,0,0,
               1,1,1,0,0,1,0,
               1,1,1,0,0,0,1,
               1,1,2,0,1,1,1,
               1,1,0,0,0,0,0,
               1,1,1,0,0,0,1,
               1,1,1,0,0,1,0,
               1,1,2,0,1,1,1,
               1,2,0,1,0,0,0,
               1,2,1,1,0,1,1,
               1,2,1,1,0,1,1,
               1,2,2,1,1,2,2), byrow=TRUE, ncol=7) 

colnames(X)<-c("Intercept", "A", "B", "P1", "P2", "Delta", "Epsilon")
res<-scoregllm(y,s,X[,c(1:6)], 
               c(255,176,176,121,164,37,113,25,164,113,37,25,90,20,20,5))
summary(res)
#

---

Summarize results of gllm

Description

Summarizes contents of result of call to gllm. The print method pretty prints the summary object.

Usage

## S3 method for class 'gllm'
summary(object,...)

Arguments

object	is the object output from gllm.
...	other arguments

Value

A list with components:

nobs	the number of cells in the observed table
nfull	the number of cells in the full table
mean.cell	the mean cell count in the observed table
deviance	the final model deviance (-2 log likelihood)
model.df	the model degrees of freedom
coefficients	the model parameter estimates, standard errors
residuals	Observed and fitted counts, plus Pearsonian residuals

Author(s)

David L Duffy

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