% \VignetteIndexEntry{Knee Injuries - Marginal Models}
% \VignetteDepends{gee, geepack}

\documentclass[a4paper]{article}

\title{Knee Injuries - Marginal Models}

\begin{document}

\maketitle


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

First the dataset knee is loaded:

<<results=hide,eval=FALSE>>=
library(catdata)
data(knee)
attach(knee)
@


To obtain  a simple binary model the response variables are dichotomized. The groups are constructed by
pain level up to level 2 und pain level higher than level 2.

<<eval=FALSE>>=
R2D <- rep(0, length(R2))
R3D <- rep(0, length(R3))
R4D <- rep(0, length(R3))

R2D[R2>2] <- 1
R3D[R3>2] <- 1
R4D[R4>2] <- 1
@

Now the covariates have to be transformed  so that they can be used for the functions "gee" from the
"gee"--library and "geeglm" from the "geepack"--library, which will be employed for fitting the models.

<<eval=FALSE>>=
N <- rep(knee$N, each=3)
Th <- rep(knee$Th, each=3)
Age <- rep(knee$Age, each=3)
Sex <- rep(knee$Sex, each=3)
@

Now the response vector is built and the quadratic age--effect "Age2" is computed.

<<eval=FALSE>>=
Response <- c(rbind(R2D,R3D,R4D))
Age2 <- Age^2
@

The covariates therapy and sex are treated as factors:

<<eval=FALSE>>=
Th <- as.factor(Th)
Sex <- as.factor(Sex)
@

First the GEEs are fitted with the funtion "gee" from library "gee".

<<eval=FALSE>>=
library(gee)
@

The first model is a GEE with independent correlation structure:

<<results=hide,eval=FALSE>>=
gee1a <- gee(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit))
@

<<eval=FALSE>>=
summary(gee1a)
@

The second model is a GEE with exchangeable correlation structure:

<<results=hide,eval=FALSE>>=
gee2a <- gee(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit), corstr="exchangeable")
@

<<eval=FALSE>>=
summary(gee2a)
@

Finally a GEE with exponential correlation structure is fitted:

<<results=hide,eval=FALSE>>=
gee3a <- gee(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit), corstr="AR-M", Mv=1)
@

<<eval=FALSE>>=
summary(gee3a)
@


In the following the corresponding marginal models are fitted with the function "geeglm" from the library "geepack".

<<eval=FALSE>>=
library(geepack)
@

Model with independent correlation structure:

<<results=hide,eval=FALSE>>=
gee1b <- geeglm(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit))
@

<<eval=FALSE>>=
summary(gee1b)
@

Model with exchangeable correlation structure:

<<results=hide,eval=FALSE>>=
gee2b <- geeglm(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit), corstr="exchangeable")
@

<<eval=FALSE>>=
summary(gee2b)
@

Model with exponential correlation structure:

<<results=hide,eval=FALSE>>=
gee3b <- geeglm(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit), corstr="ar1")
@

<<eval=FALSE>>=
summary(gee3b)
@

\bigskip

For comparison a simple GLM with logit--link is fitted with the same covariates as in the marginal models above:

<<eval=FALSE>>=
glm1 <- glm(Response ~ Th + Sex + Age + Age2,
family=binomial(link=logit))
summary(glm1)
@

It is often advatageous to center the  variables like age around a value in the middle of its range. So 
now the marginal models from above are replicated with  age  centered around 30 years. 

<<eval=FALSE>>=
Age <- Age-30
Age2 <- Age^2
@

Again we use the function "gee" from the "gee"--library for fitting those models.

\bigskip

Model with independent correlation structure and centered age:

<<results=hide,eval=FALSE>>=
gee1c <- gee(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit))
@

<<eval=FALSE>>=
summary(gee1c)
@

Model with exchangeable correlation structure and centered age:

<<results=hide,eval=FALSE>>=
gee2c <- gee(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit), corstr="exchangeable")
@

<<eval=FALSE>>=
summary(gee2c)
@

Model with exponential correlation structure and centered age:

<<results=hide,eval=FALSE>>=
gee3c <- gee(Response ~ Th + Sex + Age + Age2, id=N,
family=binomial(link=logit), corstr="AR-M", Mv=1)
@

<<eval=FALSE>>=
summary(gee3c)
@



\end{document}
