% \VignetteIndexEntry{Birth Data - Loglinear Models}
% \VignetteDepends{stats}

\documentclass[a4paper]{article}

\title{Birth Data - Loglinear Models}

\begin{document}

\maketitle


<<echo=FALSE,eval=FALSE>>=
options(width=80)
@

<<results=hide,eval=FALSE>>=
library(catdata)

data(birth)
attach(birth)
@

In the following loglinear models are fitted with the binary variables Sex, Membranes, Cesarean and Induced
from the "birth" data. As an overview a contingency table is plotted.

<<eval=FALSE>>=
table1 <- table(Sex, Membranes, Cesarean, Induced)

ftable(table1)
@

Now we start fitting the models. The goal is to find a model with good fit but sparse parametrization.

First the saturated model is fitted, then the model with all 3--factor interactions and the model with all 2--factor interactions, and
finally the independence model. To control for model fit we look at the corresponding deviances and degrees of freedom.

<<eval=FALSE>>=
m4 <- loglin(table1, margin=list(c(1,2,3,4)), fit=TRUE)
cat("deviance(m4)=", m4$lrt, "df(m4)=", m4$df, "\n")

m3 <- loglin(table1, margin=list(c(1,2,3), c(1,2,4), c(1,3,4), c(2,3,4)), fit=TRUE)
cat("deviance(m3)=", m3$lrt, "df(m3)=", m3$df, "\n")

m2 <- loglin(table1, margin=list(c(1,2), c(1,3), c(1,4), c(2,3), c(2,4), c(3,4)), 
             fit=TRUE)
cat("deviance(m2)=", m2$lrt, "df(m2)=", m2$df, "\n")

m1 <- loglin(table1, margin=list(c(1), c(2), c(3), c(4)), fit=TRUE)
cat("deviance(m1)=", m1$lrt, "df(m1)=", m1$df, "\n")
@

In order to see if a model or rather the reduction of a model is appropriate we use  chi--square tests.

<<eval=FALSE>>=
(df34 <- m3$df - m4$df)
(dev34 <- m3$lrt - m4$lrt)
1-pchisq(dev34, df34)


(df23 <- m2$df - m3$df)
(dev23 <- m2$lrt - m3$lrt)
1-pchisq(dev23, df23)

(df12 <- m1$df - m2$df)
(dev12 <- m1$lrt - m2$lrt)
1-pchisq(dev12, df12)
@

Since model "m2" fits the data well but model "m1" is definitely rejected  we fit submodels of "m2" by leaving out one of the  2--factor interactions.

<<eval=FALSE>>=
m2.GM <- loglin(table1, margin=list(c(1,3), c(1,4), c(2,3), c(2,4), c(3,4)), 
                fit=TRUE)
cat("deviance(m2.GM)=", m2.GM$lrt, "df(m2.GM)=", m2.GM$df, "\n")

m2.MC <- loglin(table1, margin=list(c(1,2), c(1,3), c(1,4), c(2,4), c(3,4)), 
                fit=TRUE)
cat("deviance(m2.MC)=", m2.MC$lrt, "df(m2.MC)=", m2.MC$df, "\n")

m2.CI <- loglin(table1, margin=list(c(1,2), c(1,3), c(1,4), c(2,3), c(2,4)), 
                fit=TRUE)
cat("deviance(m2.CI)=", m2.CI$lrt, "df(m2.CI)=", m2.CI$df, "\n")

m2.GI <- loglin(table1, margin=list(c(1,2), c(1,3), c(2,3), c(2,4), c(3,4)), 
                fit=TRUE)
cat("deviance(m2.GI)=", m2.GI$lrt, "df(m2.GI)=", m2.GI$df, "\n")

m2.GC <- loglin(table1, margin=list(c(1,2), c(1,4), c(2,3), c(2,4), c(3,4)), 
                fit=TRUE)
cat("deviance(m2.GC)=", m2.GC$lrt, "df(m2.GC)=", m2.GC$df, "\n")

m2.MI <- loglin(table1, margin=list(c(1,2), c(1,3), c(1,4), c(2,3), c(3,4)), 
                fit=TRUE)
cat("deviance(m2.MI)=", m2.MI$lrt, "df(m2.MI)=", m2.MI$df, "\n")
@

These six models all have 6 degrees of freedom so that the difference of degrees of freedom corresponding to
model "m2" is 1 in each case.

<<eval=FALSE>>=
1 - pchisq(m2.GM$lrt - m2$lrt, 1)

1 - pchisq(m2.MC$lrt - m2$lrt, 1)

1 - pchisq(m2.CI$lrt - m2$lrt, 1)

1 - pchisq(m2.GI$lrt - m2$lrt, 1)

1 - pchisq(m2.GC$lrt - m2$lrt, 1)

1 - pchisq(m2.MI$lrt - m2$lrt, 1)
@

Testing of the 2--factor interactions shows that the interactions "MC", "CI" and "MI"  should be kept in the model.
In the next step the model that contains these interactions,  G|MC|MI|CI, is fitted.

<<eval=FALSE>>=
m2.GM.GI.GC<- loglin(table1, margin=list(c(1), c(2,3), c(2,4), c(3,4)), fit=TRUE)
cat("deviance(m2.GM.GI.GC)=", m2.GM.GI.GC$lrt, "df(m2.GM.GI.GC)=", m2.GM.GI.GC$df, 
    "\n")

1 - pchisq(m2.GM.GI.GC$lrt - m2$lrt, m2.GM.GI.GC$df - m2$df)
@

Comparison with model "m2" shows that reduction is possible. However, reduction to a  model in which the main effect
"G" is omitted is rejected.

<<eval=FALSE>>=
m2.G<- loglin(table1, margin=list(c(2,3), c(2,4), c(3,4)), fit=TRUE)
cat("deviance(m2.G)=", m2.G$lrt, "df(m2.G)=", m2.G$df, "\n")

1 - pchisq(m2.G$lrt - m2$lrt, m2.G$df - m2$df)
@
\end{document}