\documentclass[12pt]{article}
\usepackage{Sweave}
\usepackage{myVignette}
\usepackage[authoryear,round]{natbib}
\bibliographystyle{plainnat}
\DefineVerbatimEnvironment{Sinput}{Verbatim}
{formatcom={\vspace{-1.5ex}},fontshape=sl,
  fontfamily=courier,fontseries=b, fontsize=\scriptsize}
\DefineVerbatimEnvironment{Soutput}{Verbatim}
{formatcom={\vspace{-2.5ex}},fontfamily=courier,fontseries=b,%
  fontsize=\scriptsize}
%%\VignetteIndexEntry{Examples from Multilevel Software Reviews}
%%\VignetteDepends{lme4}
%%\VignetteDepends{lattice}
\begin{document}
\SweaveOpts{engine=R,eps=FALSE,pdf=TRUE,width=5,height=3,strip.white=true,keep.source=TRUE}
\SweaveOpts{prefix=TRUE,prefix.string=SoftRev,include=TRUE}%             ^^^^^^^^^^^^^^^^
\setkeys{Gin}{width=\textwidth}
\title{Examples from Multilevel Software Comparative Reviews}
\author{Douglas Bates\\R Development Core Team\\\email{Douglas.Bates@R-project.org}}
\date{February 2005, {\small with updates up to \today}}
\maketitle
\begin{abstract}
   The Center for Multilevel Modelling at the Institute of Education,
   London maintains a web site of ``Software reviews of multilevel
   modeling packages''.  The data sets discussed in the reviews are
   available at this web site.  We have incorporated these data sets
   in the \code{mlmRev} package for \RR{} and, in this vignette, provide
   the results of fitting several models to these data sets.
\end{abstract}
<<preliminaries,echo=FALSE,print=FALSE,results=hide>>=
options(width=80, show.signif.stars = FALSE,
        lattice.theme = function() canonical.theme("pdf", color = FALSE))
library(mlmRev)
library(lme4)
library(lattice)
set.seed(1234321)
@

\section{Introduction}
\label{sec:Intro}

\RR{} is an Open Source implementation of John Chambers' \Slang{}
language for data analysis and graphics.  \RR{} was initially developed
by Ross Ihaka and Robert Gentleman of the University of Auckland and
now is developed and maintained by an international group of
statistical computing experts.

In addition to being Open Source software, which means that anyone can
examine the source code to see exactly how the computations are being
carried out, \RR{} is freely available from a network of archive sites
on the Internet.  There are precompiled versions for installation on
the Windows operating system, Mac OS X and several distributions of
the Linux operating system.  Because the source code is available
those interested in doing so can compile their own version if they wish.

\RR{} provides an environment for interactive computing with data and for
graphical display of data.  Users and developers can extend the
capabilities of \RR{} by writing their own functions in the language
and by creating packages of functions and data sets.  Many such
packages are available on the archive network called CRAN
(Comprehensive R Archive Network) for which the parent site is
\url{https://cran.r-project.org}. One such package called \code{lme4}
(along with a companion package called \code{Matrix}) provides
functions to fit and display linear mixed models and generalized
linear mixed models, which are the statisticians' names for the models
called multilevel models or hierarchical linear models in other
disciplines.  The \code{lattice} package provides functions to
generate several high level graphics plots that help with the
visualization of the types of data to which such models are fit.
Finally, the \code{mlmRev} package provides the data sets used in the
``Software Reviews of Multilevel Modeling Packages'' from the
Multilevel Modeling Group at the Institute of Education in the UK.
This package also contains several other data sets from the multilevel
modeling literature.

The software reviews mentioned above were intended to provide
comparison of the speed and accuracy of many different packages for
fitting multilevel models.  As such, there is a standard set of
models that fit to each of the data sets in each of the packages that
were capable of doing the fit.  We will fit these models for
comparative purposes but we will also do some graphical exploration of
the data and, in some cases, discuss alternative models.

We follow the general outline of the previous reviews, beginning with
simpler structures and moving to the more complex structures.  Because
the previous reviews were performed on older and considerably slower
computers than the ones on which this vignette will be compiled, the
timings produced by the \code{system.time} function and shown in the
text should not be compared with previous timings given on the web
site.  They are an indication of the times required to fit such models
to these data sets on recent computers with processors running at
around 2 GHz or faster.

\section{Two-level normal models}
\label{sec:TwoLevelNormal}

In the multilevel modeling literature a two-level model is one with
two levels of random variation; the per-observation noise term and
random effects which are grouped according to the levels of a factor.
We call this factor a \textit{grouping factor}. If the response is
measured on a continuous scale (more or less) our initial models are
based on a normal distribution for the per-observation noise and for
the random effects.  Thus such a model is called a ``two-level normal
model'' even though it has only one grouping factor for the random effects.


\subsection{The Exam data}
\label{sec:Examdata}
<<Examprep,results=hide,echo=FALSE>>=
lmer(normexam ~ standLRT + sex + schgend + (1|school), Exam)
@

The data set called \code{Exam} provides the normalized exam scores
attained by 4,059 students from 65 schools in inner London.  Some of
the covariates available with this exam score are the school the
student attended, the sex of the student, the school gender (boys,
girls, or mixed) and the student's result on the Standardised London
Reading test.

The \RR{} functions \code{str} and \code{summary} can be used to
examine the structure of a data set (or, in general, any \RR{} object)
and to provide a summary of an object.
<<ExamData>>=
str(Exam)
summary(Exam)
@


\subsection{Model fits and timings}
\label{sec:ExamFits}

The first model to fit to the Exam data incorporates fixed-effects
terms for the pretest score, the student's sex and the school gender.
The only random-effects term is an additive shift associated with the
school.

<<ExamFit>>=
(Em1 <- lmer(normexam ~ standLRT + sex + schgend + (1|school), Exam))
@

The \code{system.time} function can be used to time the execution of
an \RR{} expression.  It returns a vector of five numbers giving the
user time (time spend executing applications code), the system time
(time spent executing system functions called by the applications
code), the elapsed time, and the user and system time for any child
processes.  The first number is what is commonly viewed as the time
required to do the model fit.  (The elapsed time is unsuitable because
it can be affected by other processes running on the computer.)  These
times are in seconds.  On modern computers this fit takes only a
fraction of a second.

<<Examtime>>=
system.time(lmer(normexam ~ standLRT + sex + schgend + (1|school), Exam))
@



\subsection{Interpreting the fit}
\label{sec:ExamInterpret}

As can be seen from the output, the default method of fitting a linear
mixed model is restricted maximum likelihood (REML).  The estimates of
the variance components correspond to those reported by other packages
as given on the Multilevel Modelling Group's web site.  Note that the
estimates of the variance components are given on the scale of the
variance and on the scale of the standard deviation.  That is, the
values in the column headed \code{Std.Dev.} are simply the square
roots of the corresponding entry in the \code{Variance} column.  They
are \textbf{not} standard errors of the estimate of the variance.

The estimates of the fixed-effects are different from those quoted on
the web site because the terms for \code{sex} and \code{schgend} use a
different parameterization than in the reviews.  Here the reference
level of \code{sex} is female (\code{F}) and the coefficient labelled
\code{sexM} represents the difference for males compared to females.
Similarly the reference level of \code{schgend} is \code{mixed} and
the two coefficients represent the change from mixed to boys only and
the change from mixed to girls only.  The value of the coefficient
labelled \code{Intercept} is affected by both these changes as is the
value of the REML criterion.

To reproduce the results obtained from other packages, we must change
the reference level for each of these factors.

<<ExamRelevel>>=
Exam$sex     <- relevel(Exam$sex, "M")
Exam$schgend <- relevel(Exam$schgend, "girls")
(Em2 <- lmer(normexam ~ standLRT + sex + schgend + (1|school), Exam))
@

The coefficients now correspond to those in the tables on the web
site.  It happens that the REML criterion at the optimum in this fit
is the same as in the previous fit, but you cannot depend on this
occuring.  In general the value of the REML criterion at the optimum
depends on the parameterization used for the fixed effects.

\subsection{Further exploration}
\label{sec:ExamExplore}


\subsubsection{Checking consistency of the data}
\label{sec:consistency}

It is important to check the consistency of data before trying to fit
sophisticated models.  One should plot the data in many different ways
to see if it looks reasonableand also check relationships between
variables.

For example, each observation in these data is associated with a
particular student.  The variable \code{student} is not a unique
identifier of the student as it only has 650 unique values.  It is
intended to be a unique identifier within a school but it is not.  To
show this we create a factor that is the interaction of school and
student then drop unused levels.

<<ExamIds>>=
Exam <- within(Exam, ids <- factor(school:student))
str(Exam)
@

Notice that there are 4059 observations but only 4055 unique levels of
student within school.  We can check the ones that are duplicated

<<dupExamIds>>=
as.character(Exam$ids[which(duplicated(Exam$ids))])
@

One of these cases
<<OnlyBoy>>=
subset(Exam, ids == '43:86')
xtabs(~ sex + school, Exam, subset = school %in% c(43, 50, 52), drop = TRUE)
@
is particularly interesting.  Notice that one of the students
numbered 86 in school 43 is the only male student out of 61 students
from this school who took the exam.  It is quite likely that this
student's score was attributed to the wrong school and that the school
is in fact a girls-only school, not a mixed-sex school.

The causes of the other three cases of duplicate student numbers
within a school are not as clear.  It would be necessary to go back
to the original data records to check these.

The cross-tabulation of the students by sex and school for the
mixed-sex schools
<<ExamXtabs>>=
xtabs(~ sex + school, Exam, subset = type == "Mxd", drop = TRUE)
@
shows another anomaly.  School 47 is similar to school 43 in that,
although it is classified as a mixed-sex school, 81 male students
and only one female student took the exam.  It is likely that the
school was misrecorded for this one female student and the school is a
male-only school.

Another school is worth noting. There were only eight students from
school 54 who took the exam so any within-school estimates from this
school will be unreliable.

A mosaic plot (Figure~\ref{fig:ExamMosaic}) produced with
<<ExamMosaicshow, eval = FALSE>>=
ExamMxd <- within(subset(Exam, type == "Mxd"), school <- factor(school))
mosaicplot(~ school + sex, ExamMxd)
@
helps to detect mixed-sex schools with unusually large or unusually small ratios
of females to males taking the exam.
\begin{figure}[tbp]
  \centering
<<ExamMosaic,fig=TRUE,echo=FALSE,width=8,height=8>>=
ExamMxd <- within(subset(Exam, type == "Mxd"), school <- factor(school))
mosaicplot(~ school + sex, ExamMxd)
@
  \caption{A mosaic plot of the sex distribution by school.  The areas
    of the rectangles are proportional to the number of students of
    that sex from that school who took the exam.  Schools with an
    unusally large or unusually small ratio or females to males are
    highlighted.}
  \label{fig:ExamMosaic}
\end{figure}

\subsubsection{Preliminary graphical displays}
\label{sec:Graphical}

In addition to the pretest score (\code{standLRT}), the predictor
variables used in this model are the student's sex and the school
gender, which is coded as having three levels.  There is some
redundancy in these two variables in that all the students in a
boys-only school must be male.  For graphical exploration we convert
from \code{schgend} to \code{type}, an indicator of whether the school
is a mixed-sex school or a single-sex school, and plot the response
versus the pretest score for each combination of sex and school type.
\begin{figure}[tbp]
  \centering
<<Examplot1,fig=TRUE,echo=FALSE,width=8,height=8>>=
print(xyplot(normexam ~ standLRT | sex * type, Exam,
             type = c("g", "p", "smooth"), layout = c(2,2),
             xlab = "Standardized London Reading Test score",
             ylab = "Normalized exam score", aspect = 1.2))
@
  \caption{Normalized exam score versus pretest
    (Standardized London Reading Test) score for 4095 students from 65 schools
    in inner London. The panels on the left show the male students'
    scores; those on the right show the females' scores.  The top row
    of panels shows the scores of students in single-sex schools and
    the bottom row shows the scores of students in mixed-sex
    schools. A scatterplot smoother line for each panel has been added
    to help visualize the trend.}
  \label{fig:Examplot1}
\end{figure}

This plot is created with the \code{xyplot} from the \code{lattice}
package as (essentially)
<<Examplot1show, eval = FALSE>>=
xyplot(normexam ~ standLRT | sex * type, Exam, type = c("g", "p", "smooth"))
@
The formula would be read as ``plot \code{normexam} by \code{standLRT}
given \code{sex} by (school) \code{type}''.  A few other arguments were
added in the actual call to make the axis annotations more readable.

Figure~\ref{fig:Examplot1} shows the even after accounting for a
student's sex, pretest score and school type, there is considerable
variation in the response.  We may attribute some of this variation to
differences in schools but the fitted model indicates that most of the
variation is unaccounted or ``residual'' variation.

In some ways the high level of residual variation obscures the pattern
in the data.  By removing the data points and overlaying the
scatterplot smoothers we can concentrate on the relationships between
the covariates.
\begin{figure}[tbp]
  \centering
<<Examplot2,fig=TRUE,echo=FALSE,width=8,height=4>>=
print(xyplot(normexam ~ standLRT, Exam, groups = sex:type,
             type = c("g", "smooth"), xlim = c(-3,3), ylim = c(-2,2),
             xlab = "Standardized London Reading Test score",
             ylab = "Normalized exam score",
             auto.key = list(space = 'right', points = FALSE, lines = TRUE),
             aspect = 1))
@
  \caption{Overlaid scatterplot smoother lines of the normalized test
    scores versus the pretest (Standardized London Reading Test) score
    for female (F) and male (M) students in single-sex (Sngl) and
    mixed-sex (Mxd) schools.}
  \label{fig:Examplot2}
\end{figure}
The call to \code{xyplot} is essentially
<<Examplot2show,eval=FALSE>>=
xyplot(normexam ~ standLRT, Exam, groups = sex:type, type = c("g", "smooth"))
@

Figure~\ref{fig:Examplot2} is a remarkable plot in that it shows
nearly a perfect ``main effects'' relationship of the response with
the three covariates and almost no interaction.  It is rare to see
real data follow a simple theoretical relationship so closely.

To elaborate, we can see that for each of the four groups the smoothed
relationship between the exam score and the pretest score is close to
a straight line and that the lines are close to being parallel.  The
only substantial deviation is in the smoothed relationship for the
males in single-sex schools and this is the group with the fewest
observations and hence the least precision in the estimated
relationship.  The lack of parallelism for this group is most apparent
in the region of extremely low pretest scores where there are few
observations and a single student who had a low pretest score and a
moderate post-test score can substantially influence the curve.  Five
or six such points can be seen in the upper left panel of
Figure~\ref{fig:Examplot1}.

We should also notice the ordering of the lines and the spacing
between the lines.  The smoothed relationships for students in
single-sex schools are consistently above those in the mixed-sex
schools and, except for the region of low pretest scores described
above, the relationship for the females in a given type of school is
consistently above that for the males.  Furthermore the distance
between the female and male lines in the single-sex schools is
approximately the same as the corresponding distance in the mixed-sex
schools.  We would summarize this by saying that there is a positive
effect for females versus males and a positive effect for single-sex
versus mixed-sex and no indication of interaction between these
factors.


\subsubsection{The effect of schools}
\label{sec:schoolEffects}

We can check for patterns within and between schools by plotting the
response versus the pretest by school.  Because there appear to be
differences in this relationship for single-sex versus mixed-sex
schools and for females versus males we consider these separately.
\begin{figure}[tbp]
  \centering
<<Examplot3,fig=TRUE,echo=FALSE,width=8,height=8>>=
print(xyplot(normexam ~ standLRT | school, Exam,
             type = c("g", "p", "r"),
             xlab = "Standardized London Reading Test score",
             ylab = "Normalized exam score",
             subset = sex == "F" & type == "Sngl"))
@
  \caption{Normalized exam scores versus pretest (Standardized London
    Reading Test) score by school for female students in single-sex
    schools.}
  \label{fig:Examplot3}
\end{figure}

In Figure~\ref{fig:Examplot3} we plot the normalized exam scores
versus the pretest score by school for female students in single-sex
schools.  The plot is produced as
<<Examplot3show>>=
xyplot(normexam ~ standLRT | school, Exam,
             type = c("g", "p", "r"),
             subset = sex == "F" & type == "Sngl")
@
The \code{"r"} in the \code{type} argument adds a simple linear
regression line to each panel.

The first thing we notice in Figure~\ref{fig:Examplot3} is that
school 48 is an anomaly because only two students in this school took
the exam.  Because within-school results based on only two students
are unreliable, we will exclude this school from further plots (but we
do include these data when fitting comprehensive models).

Although the regression lines on the panels can help us to look for
variation in the schools, the ordering of the panels is, for our
purposes, random.  We recreate this plot in Figure~\ref{fig:Examplot4} using
<<Examplot4show,eval=FALSE>>=
xyplot(normexam ~ standLRT | school, Exam, type = c("g", "p", "r"),
             subset = sex == "F" & type == "Sngl" & school != 48,
             index.cond = function(x, y) coef(lm(y ~ x))[1])
@
\begin{figure}[tbp]
  \centering
<<Examplot4,fig=TRUE,echo=FALSE,width=8,height=8>>=
print(xyplot(normexam ~ standLRT | school, Exam,
             type = c("g", "p", "r"),
             xlab = "Standardized London Reading Test score",
             ylab = "Normalized exam score",
             subset = sex == "F" & type == "Sngl" & school != 48,
             index.cond = function(x, y) coef(lm(y ~ x))[1]))
@
  \caption{Normalized exam scores versus pretest (Standardized London
    Reading Test) score by school for female students in single-sex
    schools. School 48 where only two students took the exam has been
    eliminated and the panels have been ordered by increasing
    intercept (predicted normalized score for a pretest score of 0) of
    the regression line.}
  \label{fig:Examplot4}
\end{figure}
so that the panels are ordered (from left to right starting at the
bottom row) by increasing intercept for the regression line (i.e.{} by
increasing predicted exam score for a student with a pretest score of 0).

Alternatively, we could order the panels by increasing slope of the
within-school regression lines, as in Figure~\ref{fig:Examplot4a}.
\begin{figure}[tbp]
  \centering
<<Examplot4a,fig=TRUE,echo=FALSE,width=8,height=8>>=
print(xyplot(normexam ~ standLRT | school, Exam,
             type = c("g", "p", "r"),
             xlab = "Standardized London Reading Test score",
             ylab = "Normalized exam score",
             subset = sex == "F" & type == "Sngl" & school != 48,
             index.cond = function(x, y) coef(lm(y ~ x))[2]))
@
  \caption{Normalized exam scores versus pretest (Standardized London
    Reading Test) score by school for female students in single-sex
    schools. School 48 has been eliminated and the panels have been
    ordered by increasing slope of the within-school regression
    lines.}
  \label{fig:Examplot4a}
\end{figure}

Although it is informative to plot the within-school regression lines
we need to assess the variability in the estimates of the coefficients
before concluding if there is ``significant'' variability between
schools.  We can obtain the individual regression fits with the
\code{lmList} function
<<ExamLmListFS>>=
show(ExamFS <- lmList(normexam ~ standLRT | school, Exam,
                      subset = sex == "F" & type == "Sngl" & school != 48))
@
and compare the confidence intervals on these coefficients.
<<Examplot4cshow>>=
plot(confint(ExamFS, pool = TRUE), order = 1)
@
\begin{figure}[tbp]
  \centering
<<Examplot4c,fig=TRUE,echo=FALSE,width=8,height=3>>=
print(plot(confint(ExamFS, pool = TRUE), order = 1))
@
  \caption{Confidence intervals on the coefficients of the
    within-school regression lines for female students in single-sex
    schools. School 48 has been eliminated and the schools have been
    ordered by increasing estimated intercept.}
  \label{fig:Examplot4c}
\end{figure}


\begin{figure}[tbp]
  \centering
<<Examplot5,fig=TRUE,echo=FALSE,width=8,height=4>>=
print(xyplot(normexam ~ standLRT | school, Exam,
             type = c("g", "p", "r"),
             xlab = "Standardized London Reading Test score",
             ylab = "Normalized exam score",
             subset = sex == "M" & type == "Sngl", layout = c(5,2),
             index.cond = function(x, y) coef(lm(y ~ x))[1]))
@
  \caption{Normalized exam scores versus pretest (Standardized London
    Reading Test) score by school for male students in single-sex
    schools.}
  \label{fig:Examplot5}
\end{figure}

<<ExamLmListMS>>=
show(ExamMS <- lmList(normexam ~ standLRT | school, Exam,
                      subset = sex == "M" & type == "Sngl"))
@
The corresponding plot of the confidence intervals is shown in
Figure~\ref{fig:Examplot5b}.
\begin{figure}[tbp]
  \centering
<<Examplot5b,fig=TRUE,echo=FALSE,width=8,height=2.5>>=
print(plot(confint(ExamMS, pool = TRUE), order = 1))
@
  \caption{Confidence intervals on the coefficients of the
    within-school regression lines for female students in single-sex
    schools. School 48 has been eliminated and the schools have been
    ordered by increasing estimated intercept.}
  \label{fig:Examplot5b}
\end{figure}

For the mixed-sex schools we can consider the effect of the pretest
score and sex in the plot (Figure~\ref{fig:Examplot6}) and in the
\begin{figure}[tbp]
  \centering
<<Examplot6,fig=TRUE,echo=FALSE,width=8,height=9>>=
print(xyplot(normexam ~ standLRT | school, Exam, groups = sex,
             type = c("g", "p", "r"),
             xlab = "Standardized London Reading Test score",
             ylab = "Normalized exam score",
             subset = !school %in% c(43, 47) & type == "Mxd",
             index.cond = function(x, y) coef(lm(y ~ x))[1],
             auto.key = list(space = 'top', lines = TRUE,
             columns = 2), layout = c(7,5),
             aspect = 1.2))
@
  \caption{Normalized exam scores versus pretest
    score by school and sex for students in mixed-sex
    schools.}
  \label{fig:Examplot6}
\end{figure}
separate model fits for each school.
<<ExamLmListM>>=
show(ExamM <- lmList(normexam ~ standLRT + sex| school, Exam,
                     subset = type == "Mxd" & !school %in% c(43,47,54)))
@
The confidence intervals for these separately fitted models,
\begin{figure}[tbp]
  \centering
<<Examplot6b,fig=TRUE,echo=FALSE,width=8,height=4>>=
print(plot(confint(ExamM, pool = TRUE), order = 1))
@
  \caption{Confidence intervals on the coefficients of the
    within-school regression lines for female students in single-sex
    schools. School 48 has been eliminated and the schools have been
    ordered by increasing estimated intercept.}
  \label{fig:Examplot6b}
\end{figure}
shown in Figure~\ref{fig:Examplot6b} indicate differences in the
intercepts and possibly differences in the slopes with respect to the
pretest scores.  However, there is not a strong indication of
variation by school in the effect of sex.


\subsection{Multilevel models for the exam data}
\label{sec:ExamModels}

We begin with a model that has a random effects for the intercept by
school plus additive fixed effects for the pretest score, the
student's sex and the school type.
<<Em3>>=
(Em3 <- lmer(normexam ~ standLRT + sex + type + (1|school), Exam))
@
Our data exploration indicated that the slope with respect to the
pretest score may vary by school.  We can fit a model with random
effects by school for both the slope and the intercept as
<<Em4>>=
(Em4 <- lmer(normexam ~ standLRT + sex + type + (standLRT|school), Exam))
@
and compare this fit to the previous fit with
<<EmAnova>>=
anova(Em3, Em4)
@
There is a strong evidence of a significant random effect for the
slope by school, whether judged by AIC, BIC or the p-value for the
likelihood ratio test.

The p-value for the likelihood ratio test is based on a $\chi^2$ distribution
with degrees of freedom calculated as the difference in the number of
parameters in the two models.  Because one of the parameters
eliminated from the full model in the submodel is at its boundary the
usual asymptotics for the likelihood ratio test do not apply.
However, it can be shown that the p-value quoted for the test is
conservative in the sense that it is an upper bound on the
p-value that would be calculated say from a parametric bootstrap.

Having an upper bound of $1.9\times 10^{-10}$ on the p-value can be
regarded as ``highly significant'' evidence of the utility of the
random effect for the slope by school.

We could also add a random effect for the student's sex by school
<<Em5>>=
(Em5 <- lmer(normexam ~ standLRT + sex + type + (standLRT + sex|school), Exam))
@
Notice that the estimate of the variance of the \code{sexM} term is
essentially zero so there is no need to test the significance of this
variance component.  We proceed with the analysis of \code{Em4}.

\section{Growth curve model for repeated measures data}
\label{sec:GrowthCurve}

<<Oxboys>>=
str(Oxboys)
system.time(mX1 <- lmer(height ~ age + I(age^2) + I(age^3) + I(age^4) + (age + I(age^2)|Subject),
                       Oxboys))
summary(mX1)
system.time(mX2 <- lmer(height ~ poly(age,4) + (age + I(age^2)|Subject), Oxboys))
summary(mX2)
@

\section{Cross-classification model}
\label{sec:CrossClassified}

<<ScotsSec>>=
str(ScotsSec)
system.time(mS1 <- lmer(attain ~ sex + (1|primary) + (1|second), ScotsSec))
summary(mS1)
@

\section{Session Info}

<<sessionInfo, results=tex>>=
toLatex(sessionInfo())
@

%\bibliography{MlmSoftRev}
\end{document}
