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\hypersetup{%
  pdftitle = {A Handbook of Statistical Analyses Using R},
  pdfsubject = {Book},
  pdfauthor = {Brian S. Everitt and Torsten Hothorn},
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\begin{document}

%% Title page

\title{A Handbook of Statistical Analyses Using \R{} --- 2nd Edition}

\author{Brian S. Everitt and Torsten Hothorn}

\maketitle
%%\VignetteIndexEntry{Chapter Analysing Longitudinal Data II}
%%\VignetteDepends{gee}
\setcounter{chapter}{12}


\SweaveOpts{prefix.string=figures/HSAUR,eps=FALSE,keep.source=TRUE} 

<<setup, echo = FALSE, results = hide>>=
rm(list = ls())
s <- search()[-1]
s <- s[-match(c("package:base", "package:stats", "package:graphics", "package:grDevices",
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HSAURpkg <- require("HSAUR2")
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rm(HSAURpkg)
 ### </FIXME> hm, R-2.4.0 --vanilla seems to need this
a <- Sys.setlocale("LC_ALL", "C")
 ### </FIXME>
book <- TRUE
refs <- cbind(c("AItR", "DAGD", "SI", "CI", "ANOVA", "MLR", "GLM", 
                "DE", "RP", "GAM", "SA", "ALDI", "ALDII", "SIMC", "MA", "PCA", 
                "MDS", "CA"), 1:18)
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@

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print.summary.gee <- function (x, digits = NULL, quote = FALSE, prefix = "", ...) 
{
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    if (!is.null(x$model$M)) 
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            x$model$M, "\n")
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            sep = "")
    else cat("\n\nCoefficients:\n")
    print(x$coefficients, digits = digits)
    cat("\nEstimated Scale Parameter: ", format(round(x$scale, 
        digits)))
    cat("\n...\n")

    invisible(x)
}
}

@

\chapter[Analysing Longitudinal Data II]{
         Analysing Longitudinal Data II -- Generalised Estimation Equations and Linear Mixed Effect Models:
         Treating Respiratory Illness and Epileptic Seizures
         \label{ALDII}}

\section{Introduction}


\section{Methods for Non-normal Distributions}



\section{Analysis Using \R{}: GEE}

\subsection{Beat the Blues Revisited}


To use the \Rcmd{gee} function, package \Rpackage{gee} \citep{PKG:gee}
has to be installed and attached:
<<ALDII-gee, echo = TRUE>>=
library("gee")
@
The \Rcmd{gee} function is used in a similar way to the \Rcmd{lme} function 
met in \Sexpr{ch("ALDI")} with the addition of the features of the
\Rcmd{glm} function that specify the appropriate error distribution 
for the response and the implied link function, and an argument 
to specify the structure of the working correlation matrix. Here 
we will fit an independence structure and then an exchangeable 
structure. The \R{} code for fitting generalised estimation equations to the
\Robject{BtheB\_long} data (as constructed in \Sexpr{ch("ALDI")}) with
identity working correlation matrix is as follows (note that the \Rcmd{gee} function
assumes the rows of the \Rclass{data.frame} \Robject{BtheB\_long} to be
ordered with respect to subjects):
<<ALDII-BtheB-data, echo = FALSE, results = hide>>=
data("BtheB", package = "HSAUR2")
BtheB$subject <- factor(rownames(BtheB))
nobs <- nrow(BtheB)
BtheB_long <- reshape(BtheB, idvar = "subject",
    varying = c("bdi.2m", "bdi.3m", "bdi.5m", "bdi.8m"),
    direction = "long")
BtheB_long$time <- rep(c(2, 3, 5, 8), rep(nobs, 4))
names(BtheB_long)[names(BtheB_long) == "treatment"] <- "trt"
@
<<ALDII-BtheB-geefit-indep, echo = TRUE, results = hide>>=
osub <- order(as.integer(BtheB_long$subject))
BtheB_long <- BtheB_long[osub,]
btb_gee <- gee(bdi ~ bdi.pre + trt + length + drug, 
    data = BtheB_long, id = subject, family = gaussian,
    corstr = "independence")
@
and with exchangeable correlation matrix:
<<ALDII-BtheB-geefit-ex, echo = TRUE, results = hide>>=
btb_gee1 <- gee(bdi ~ bdi.pre + trt + length + drug, 
    data = BtheB_long, id = subject, family = gaussian,
    corstr = "exchangeable")
@
The \Rcmd{summary} method can be used to inspect the fitted models; the
results are shown in Figures~\ref{ALDII-gee-summary} and
\ref{ALDII-gee1-summary}.

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method 
                            for the \Robject{btb\_gee} model (slightly abbreviated).
                            \label{ALDII-gee-summary}}
\SchunkLabel
<<ALDII-BtheB-geesummary, echo = TRUE>>=
summary(btb_gee)
@
\SchunkRaw

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method 
                            for the \Robject{btb\_gee1} model (slightly abbreviated).
                            \label{ALDII-gee1-summary}}
\SchunkLabel
<<ALDII-BtheB-gee1summary, echo = TRUE>>=
summary(btb_gee1)
@
\SchunkRaw


\subsection{Respiratory Illness \label{ALDII:resp}}


The baseline status, i.e., the status for \Robject{month == 0}, will enter
the models as an explanatory variable and thus we have to rearrange the
\Rclass{data.frame} \Robject{respiratory} 
in order to create a new variable \Robject{baseline}:
<<ALDII-respiratory-data, echo = TRUE>>=
data("respiratory", package = "HSAUR2")
resp <- subset(respiratory, month > "0")
resp$baseline <- rep(subset(respiratory, month == "0")$status,
                     rep(4, 111))
resp$nstat <- as.numeric(resp$status == "good")
resp$month <- resp$month[, drop = TRUE]
@
<<ALDII-respiratory-names, echo = FALSE, results = hide>>=
names(resp)[names(resp) == "treatment"] <- "trt"
levels(resp$trt)[2] <- "trt"
@

The new variable \Robject{nstat} is simply a dummy coding for a poor
respiratory status. Now we can use the data \Robject{resp} to fit a logistic regression model
and GEE models with an independent and an exchangeable correlation structure
as follows.
<<ALDII-respiratory-fit, echo = TRUE, results = hide>>=
resp_glm <- glm(status ~ centre + trt + gender + baseline 
    + age, data = resp, family = "binomial")
resp_gee1 <- gee(nstat ~ centre + trt + gender + baseline 
    + age, data = resp, family = "binomial", id = subject, 
    corstr = "independence", scale.fix = TRUE, 
    scale.value = 1)
resp_gee2 <- gee(nstat ~ centre + trt + gender + baseline 
    + age, data = resp, family = "binomial", id = subject, 
    corstr = "exchangeable", scale.fix = TRUE, 
    scale.value = 1)
@

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method
                            for the \Robject{resp\_glm} model.
                            \label{ALDII-resp-glm-summary}}
\SchunkLabel
<<ALDII-resp-glm-summary, echo = TRUE>>=
summary(resp_glm)
@
\SchunkRaw

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method
                            for the \Robject{resp\_gee1} model (slightly abbreviated).
                            \label{ALDII-resp-gee1-summary}}
\SchunkLabel
<<ALDII-resp-gee1summary, echo = TRUE>>=
summary(resp_gee1)
@
\SchunkRaw

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method
                            for the \Robject{resp\_gee2} model (slightly abbreviated).
                            \label{ALDII-resp-gee2-summary}}
\SchunkLabel
<<ALDII-resp-gee2-summary, echo = TRUE>>=
summary(resp_gee2)
@
\SchunkRaw



The estimated treatment effect taken from the exchangeable 
structure GEE model is \Sexpr{round(coef(resp_gee2)["trttrt"], 3)} 
which, using the robust standard 
errors, has an associated $95\%$ confidence interval 
<<ALDII-resp-confint, echo = TRUE>>=
se <- summary(resp_gee2)$coefficients["trttrt",
                                      "Robust S.E."]
coef(resp_gee2)["trttrt"] +  
    c(-1, 1) * se * qnorm(0.975)
@
These values reflect effects on the log-odds scale. Interpretation 
becomes simpler if we exponentiate the values to get the effects 
in terms of odds. This gives a treatment effect of
\Sexpr{round(exp(coef(resp_gee2)["trttrt"]), 3)}
and a  $95\%$ confidence interval of 
<<ALDII-resp-confint-exp, echo = TRUE>>=
exp(coef(resp_gee2)["trttrt"] + 
    c(-1, 1) * se * qnorm(0.975))
@
The odds of achieving a `good' respiratory status with the active treatment is between  %'
about twice and seven times the corresponding odds for the placebo. 



\subsection{Epilepsy} 

Moving on to the count data in \Robject{epilepsy} from 
Table~\ref{ALDII-epilepsy-tab}, we begin by calculating 
the means and variances of the number of seizures for all 
interactions between treatment and period:
<<ALDII-epilepsy, echo = TRUE>>=
data("epilepsy", package = "HSAUR2")
itp <- interaction(epilepsy$treatment, epilepsy$period)
tapply(epilepsy$seizure.rate, itp, mean)
tapply(epilepsy$seizure.rate, itp, var)
@
Some of the variances are considerably larger than the corresponding means, which for 
a Poisson variable may suggest that overdispersion may be a problem, see
\Sexpr{ch("GLM")}.

\begin{figure}
\begin{center}
<<ALDII-plot1, echo = TRUE, fig = TRUE, height = 4>>=
layout(matrix(1:2, nrow = 1))
ylim <- range(epilepsy$seizure.rate)
placebo <- subset(epilepsy, treatment == "placebo")
progabide <- subset(epilepsy, treatment == "Progabide")
boxplot(seizure.rate ~ period, data = placebo,
        ylab = "Number of seizures", 
        xlab = "Period", ylim = ylim, main = "Placebo")
boxplot(seizure.rate ~ period, data = progabide,
        main  = "Progabide", ylab = "Number of seizures", 
        xlab = "Period", ylim = ylim)
@
\caption{Boxplots of numbers of seizures in each two-week
         period post randomisation for placebo and active treatments.
         \label{ALDII-plot1}}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
<<ALDII-plot2, echo = TRUE, fig = TRUE, height = 4>>=
layout(matrix(1:2, nrow = 1))
ylim <- range(log(epilepsy$seizure.rate + 1))
boxplot(log(seizure.rate + 1) ~ period, data = placebo,
        main = "Placebo", ylab = "Log number of seizures", 
        xlab = "Period", ylim = ylim)
boxplot(log(seizure.rate + 1) ~ period, data = progabide,
        main = "Progabide", ylab = "Log number of seizures", 
        xlab = "Period", ylim = ylim)
@
\caption{Boxplots of log of numbers of seizures in
         each two-week period post randomisation for placebo and active 
         treatments. \label{ALDII-plot2}}
\end{center}
\end{figure}


We can now fit a Poisson regression model to the data 
assuming independence using the \Rcmd{glm} function. 
We also use the GEE approach to fit an 
independence structure, followed by an exchangeable structure
using the following \R{} code:
<<ALDII-epilepsy-gee, echo = TRUE, results = hide>>=
per <- rep(log(2),nrow(epilepsy))
epilepsy$period <- as.numeric(epilepsy$period)
names(epilepsy)[names(epilepsy) == "treatment"] <- "trt"
fm <- seizure.rate ~ base + age + trt + offset(per)
epilepsy_glm <- glm(fm, data = epilepsy, family = "poisson")
epilepsy_gee1 <- gee(fm, data = epilepsy, family = "poisson", 
    id = subject, corstr = "independence", scale.fix = TRUE, 
    scale.value = 1)
epilepsy_gee2 <- gee(fm, data = epilepsy, family = "poisson", 
    id = subject, corstr = "exchangeable", scale.fix = TRUE, 
    scale.value = 1)
epilepsy_gee3 <- gee(fm, data = epilepsy, family = "poisson", 
    id = subject, corstr = "exchangeable", scale.fix = FALSE,
    scale.value = 1)
@
As usual we inspect the fitted models using the \Rcmd{summary} method, the
results are given in Figures~\ref{ALDII-epilepsy-glm-summary},
\ref{ALDII-epilepsy-gee1-summary}, \ref{ALDII-epilepsy-gee2-summary}, and 
\ref{ALDII-epilepsy-gee3-summary}.

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method   
                            for the \Robject{epilepsy\_glm} model.
                            \label{ALDII-epilepsy-glm-summary}}
\SchunkLabel
<<ALDII-espilepsy-glm-summary, echo = TRUE>>=
summary(epilepsy_glm)
@
\SchunkRaw

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method   
                            for the \Robject{epilepsy\_gee1} model (slightly abbreviated).
                            \label{ALDII-epilepsy-gee1-summary}}
\SchunkLabel
<<ALDII-espilepsy-gee1-summary, echo = TRUE>>=
summary(epilepsy_gee1)
@
\SchunkRaw

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method   
                            for the \Robject{epilepsy\_gee2} model (slightly abbreviated).
                            \label{ALDII-epilepsy-gee2-summary}}
\SchunkLabel
<<ALDII-espilepsy-gee2-summary, echo = TRUE>>=
summary(epilepsy_gee2)
@
\SchunkRaw

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method   
                            for the \Robject{epilepsy\_gee3} model (slightly abbreviated).
                            \label{ALDII-epilepsy-gee3-summary}}
\SchunkLabel
<<ALDII-espilepsy-gee3-summary, echo = TRUE>>=
summary(epilepsy_gee3)
@
\SchunkRaw


\section{Analysis Using \R{}: Random Effects}

As an example of using generalised mixed models for the analysis 
of longitudinal data with a non-normal response, the following 
logistic model will be fitted to the respiratory illness data
\begin{eqnarray*}
\text{logit}(\P(\text{status} = \text{good})) & = &
\beta_0 + \beta_1 \text{treatment} + \beta_2 \text{time} + \beta_3 \text{gender} \\%
& & + \beta_4 \text{age} + \beta_5 \text{centre} + 
      \beta_6 \text{baseline} + u
\end{eqnarray*}
where $u$ is a subject-specific random effect.

The necessary \R{} code for fitting the model using the \Rcmd{glmer}
function from package \Rpackage{lme4} \citep{PKG:lme4,HSAUR:Bates2005}
is:
<<ALDII-respiratory-lmer, echo = TRUE>>=
library("lme4")
resp_lmer <- glmer(status ~ baseline + month + 
    trt + gender + age + centre + (1 | subject), 
    family = binomial(), data = resp)
exp(fixef(resp_lmer))
@


The significance of the effects as estimated by this random effects 
model and by the GEE model described in Section~\ref{ALDII:resp} 
is generally similar. 
But as expected from our previous discussion the estimated coefficients 
are substantially larger. While the estimated effect of treatment 
on a randomly sampled individual, given the set of observed covariates, 
is estimated by the marginal model using GEE to increase the log-odds 
of being disease free by $\Sexpr{round(coef(resp_gee2)["trttrt"], 3)}$, 
the corresponding estimate from 
the random effects model is 
$\Sexpr{round(fixef(resp_lmer)["trttrt"], 3)}$. 
These are not inconsistent 
results but reflect the fact that the models are estimating 
different parameters. The random effects estimate is conditional 
upon the patient’s random effect, a quantity that is rarely known 
in practise. Were we to examine the log-odds of the average predicted 
probabilities with and without treatment (averaged over the random 
effects) this would give an estimate comparable to that estimated 
within the marginal model.

<<ALDII-resp-lmer-dirty, echo = FALSE>>=
su <- summary(resp_lmer)
if (!interactive()) {
    summary <- function(x) {
        cat("\n...\n")
        cat("Fixed effects:\n")
        lme4V <- packageDescription("lme4")$Version
        if (compareVersion("0.999999-2", lme4V) >= 0) {
            printCoefmat(su@coefs)
        } else { 
            printCoefmat(su$coefficients)
        }
        cat("\n...\n")
    }
}
@

\renewcommand{\nextcaption}{\R{} output of the \Rcmd{summary} method
                            for the \Robject{resp\_lmer} model (abbreviated).
                            \label{ALDII-resp-lmer-summary}}
\SchunkLabel
<<ALDII-resp-lmer-summary, echo = TRUE>>=
summary(resp_lmer)
@
\SchunkRaw

\clearpage


\bibliographystyle{LaTeXBibTeX/refstyle}
\bibliography{LaTeXBibTeX/HSAUR}   
\end{document}
