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\begin{document}
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\title{\textbf{\textsf{lmer} for \textsf{SAS PROC MIXED} Users}}
\author{Douglas Bates\\Department of Statistics\\University of
  Wisconsin -- Madison\\\email{Bates@wisc.edu}}
\date{}
\maketitle
<<preliminaries,echo=FALSE,results=hide>>=
options(width=75, contrasts=c(unordered="contr.SAS",ordered="contr.poly"))
library(SASmixed)
library(lme4)
library(lattice)
@
\section{Introduction}
\label{sec:intro}

The \code{lmer} function from the \code{lme4} package for \textsf{R} is used
to fit linear mixed-effects models.  It is similar in scope to the
\textsf{SAS} procedure \code{PROC MIXED} described in
\citet{litt:mill:stro:wolf:1996}.

A file on the SAS Institute web site (\textsf{https://www.sas.com})
contains all the data sets in the book and all the SAS programs used
in \citet{litt:mill:stro:wolf:1996}.  We have converted the data
sets from the tabular representation used for SAS to the
\code{data.frame} objects used by \code{lmer}.  To help users familiar
with \code{SAS PROC MIXED} get up to speed with \code{lmer} more quickly,
we provide transcripts of some \code{lmer} analyses paralleling the
\code{SAS PROC MIXED} analyses in \citet{litt:mill:stro:wolf:1996}.

In this paper we highlight some of the similarities and differences of
\code{lmer} analysis and \code{SAS PROC MIXED} analysis.

\section{Similarities between lmer and SAS PROC MIXED}
\label{sec:similarities}

Both \code{SAS PROC MIXED} and \code{lmer} can fit linear mixed-effects
models expressed in the Laird-Ware formulation.  For a single level of
grouping \citet{lair:ware:1982} write the $n_i\/$-dimensional
response vector $\by_i$ for the $i\/$th experimental unit as
\begin{gather}
  \label{eqn:oneLevel}
  \by_i = \bX_i \bbeta + \bZ_i \bb_i + \beps_i,\quad i=1,\dots,M\\
  \bb_i\sim\mathcal{N}(\bzer,\bSigma),
  \quad\beps_i\sim\mathcal{N}(\bzer,\sigma^2 \bI)\notag
\end{gather}
where $\bbeta$ is the $p$-dimensional vector of \emph{fixed effects},
$\bb_i$ is the $q$-dimensional vector of \emph{random effects},
$\bX_i$ (of size $n_i\times p$) and $\bZ_i$ (of size $n_i\times q$)
are known fixed-effects and random-effects regressor matrices, and
$\beps_i$ is the $n_i\/$-dimensional \emph{within-group error} vector
with a spherical Gaussian distribution.  The assumption
$\mathrm{Var}(\beps_i)=\sigma^2\bI$ can be relaxed using additional
arguments in the model fitting.

The basic specification of the model requires a linear model
expression for the fixed effects and a linear model expression for the
random effects.  In \code{SAS PROC MIXED} the fixed-effects part is
specified in the \code{model} statement and the random-effects
part in the \code{random} statement.  In \code{lmer} the fixed effects
and the random effects are both specified as terms in the
\code{formula} argument to \code{lmer}.

Both \code{SAS PROC MIXED} and \code{lmer} allow a mixed-effects model
to be fit by maximum likelihood (\code{method = ml} in SAS) or by
maximum residual likelihood, sometimes also called restricted maximum
likelihood or \textsf{REML}.  This is the default criterion in
\code{SAS PROC MIXED} and in \code{lmer}.  To get \textsf{ML}
estimates use the optional argument \code{REML=FALSE} in the call to
\code{lmer}.

\section{Important differences}
\label{sec:differences}

The output from \code{PROC MIXED} typically includes values of the
Akaike Information Criterion (\textsf{AIC}) and Schwartz's Bayesian
Criterion (\textsf{SBC}).  These are used to compare different models
fit to the same data.  The output of the \code{summary} function
applied to the object created by \code{lmer} also produces values of
\textsf{AIC} and \textsf{BIC} but the definitions used in older
versions of \code{PROC MIXED} are different from those used in more
recent versions of \code{PROC MIXED} and in \code{lmer}.  In
\code{lmer} the definitions are such that ``smaller is better''.  In
some older versions of \code{PROC MIXED} the definitions are such that
``bigger is better''.

When models are fit by \textsf{REML}, the values of \textsf{AIC},
\textsf{SBC} (or \textsf{BIC}) and the log-likelihood can only be
compared between models with exactly the same fixed-effects structure.
When models are fit by maximum likelihood these criteria can be
compared between any models fit to the same data.  That is, these
quality-of-fit criteria can be used to evaluate different
fixed-effects specifications or different random-effects
specifications or different specifications of both fixed effects and
random effects.

We encourage developing and testing the model using likelihood ratio
tests or the \textsf{AIC} and \textsf{BIC} criteria.  Once a form
for both the random effects and the fixed effects has been determined,
the model can be refit with \code{REML = TRUE} if the restricted
estimates of the variance components are desired.  Note that the
\code{update} function provides a convenient way of refitting a model
with changes to one or more arguments.

\section{Data manipulation}
\label{sec:data}

Both \code{PROC MIXED} and \code{lmer} work with data in a tabular form
with one row per observation.  There are, however, important
differences in the internal representations of variables in the data.

In \textsf{SAS} a qualitative factor can be stored either as numerical
values or alphanumeric labels.  When a factor stored as numerical
values is used in \code{PROC MIXED} it is listed in the \code{class}
statement to indicate that it is a factor.  In \s{} this information
is stored with the data itself by converting the variable to a factor
when it is first stored.  If the factor represents an ordered set of
levels, it should be converted to an \code{ordered} factor.

For example the SAS code
\begin{Example}
data animal;
 input trait animal y;
 datalines;
1 1 6
1 2 8
1 3 7
2 1 9
2 2 5
2 3 .
;
\end{Example}
would require that the \code{trait} and \code{animal} variables be
specified in a class statement in any model that is fit.

In \R{} these data could be read from a file, say \texttt{animal.dat},
and converted to factors by
\begin{Schunk}
\begin{Sinput}
animal <- within(read.table("animal.dat", header = TRUE),
             {
                 trait <- factor(trait)
                 animal <- factor(animal)
             })
\end{Sinput}
\end{Schunk}
In general it is a good idea to check the types of variables in a data
frame before working with it.  One way of doing this is to apply
the function \textsf{data.class} to each variable in turn using the
\code{sapply} function.
<<applyClass>>=
sapply(Animal, data.class)
str(Animal)
@

\subsection{Unique levels of factors}
\label{sec:nested}

Designs with nested grouping factors are indicated differently in the
two languages.  An example of such an experimental design is the
semiconductor experiment described in section 2.2 of
\citet{litt:mill:stro:wolf:1996} where twelve wafers are
assigned to four experimental treatments with three wafers per
treatment.  The levels for the wafer factor are \code{1}, \code{2}, and
\code{3} but the wafer factor is only meaningful within the same level
of the treatment factor, \code{et}.  There is nothing associating wafer
\code{1} of the third treatment group with wafer \code{1} of the first
treatment group.

In \code{SAS} this nesting of factors is denoted by \code{wafer(et)}.  In
\s{} the nesting is written with \code{~ ET/Wafer} and read ``wafer
within ET''.  If both levels of nested factors are to be associated
with random effects then this is all you need to know.  You would use
an expression with a \code{"/"} in the grouping factor part of the
formula in the call to \code{lmer} object.  The random effects term
would be either
\begin{Example}
  (1 | ET/Wafer)
\end{Example}
or, equivalently
\begin{Example}
  (1 | ET:Wafer) + (1 | ET)
\end{Example}

In this case, however, there would not usually be any random effects
associated with the ``experimental treatment'' or \code{ET} factor.  The
only random effects are at the \code{Wafer} level.  It is necessary to
create a factor that will have unique levels for each \code{Wafer}
within each level of \code{ET}.  One way to do this is to assign
<<semiconductorGrp>>=
Semiconductor <- within(Semiconductor, Grp <- factor(ET:Wafer))
@
after which we could specify a random effects term of \code{(1 |
  Grp)}.  Alternatively, we can use the explicit term
\begin{Example}
  (1 | ET:Wafer)
\end{Example}

\subsection{General approach}
\label{sec:generalApproach}

As a general approach to importing data into \R{} for mixed-effects
analysis you should:
\begin{itemize}
\item Create a \code{data.frame} with one row per observation and one
  column per variable.
\item Use \code{factor} or \code{as.factor} to explicitly convert any
  ordered factors to class \code{ordered}.
\item Use \code{ordered} or \code{as.ordered} to explicitly convert any
  ordered factors to class \code{ordered}.
\item If necessary, use interaction terms to create a factor with unique
  levels from inner nested factors.
\item Plot the data.  Plot it several ways.  The use of lattice
  graphics is closely integrated with the \code{lme4} library.
  Lattice plots can provide invaluable insight into the structure of
  the data.  Use them.
\end{itemize}

\section{Contrasts}
\label{sec:contrasts}

When comparing estimates produced by \code{SAS PROC MIXED} and by
\code{lmer} one must be careful to consider the contrasts that are
used to define the effects of factors.  In \textsf{SAS} a model with
an intercept and a qualitative factor is defined in terms of the
intercept and the indicator variables for all but the last level of
the factor.  The default behaviour in \s{} is to use the Helmert
contrasts for the factor.  On a balanced factor these provide a set of
orthogonal contrasts.  In \R{} the default is the ``treatment''
contrasts which are almost the same as the SAS parameterization except
that they drop the indicator of the first level, not the last level.

When in doubt, check which contrasts are being used with the
\textsf{contrasts} function.

To make comparisons easier, you may find it worthwhile to declare
<<contrasts,echo=TRUE,eval=FALSE>>=
options(contrasts = c(factor = "contr.SAS", ordered = "contr.poly"))
@
at the beginning of your session.

\bibliography{Usinglmer}
\appendix

\section{AvgDailyGain}
\label{sec:AvgDailyGain}

<<adg1,fig=TRUE,echo=TRUE,width=5,height=6>>=
print(xyplot(adg ~ Treatment | Block, AvgDailyGain, type = c("g", "p", "r"),
       xlab = "Treatment (amount of feed additive)",
       ylab = "Average daily weight gain (lb.)", aspect = "xy",
       index.cond = function(x, y) coef(lm(y ~ x))[1]))
@
\begin{figure}[tbp]
  \centering
  \includegraphics{figs/f-adg1}
  \caption{Average daily weight gain}
  \label{fig:adg1}
\end{figure}
<<adg>>=
## compare with output 5.1, p. 178
(fm1Adg <- lmer(adg ~ (Treatment - 1)*InitWt + (1 | Block), AvgDailyGain))
anova(fm1Adg)   # checking significance of terms
## common slope model
(fm2Adg <- lmer(adg ~ InitWt + Treatment + (1 | Block), AvgDailyGain))
anova(fm2Adg)
(fm3Adg <- lmer(adg ~ InitWt + Treatment - 1 + (1 | Block), AvgDailyGain))
@


\section{BIB}
\label{sec:BIB}
<<bib1,fig=TRUE,echo=TRUE,width=6,height=6>>=
print(xyplot(y ~ x | Block, BIB, groups = Treatment, type = c("g", "p"),
             aspect = "xy", auto.key = list(points = TRUE, space = "right",
             lines = FALSE)))
@
\begin{figure}[tbp]
  \centering
  \includegraphics{figs/f-bib1}
  \caption{Balanced incomplete block design}
  \label{fig:bib1}
\end{figure}
<<bib>>=
## compare with Output 5.7, p. 188
(fm1BIB <- lmer(y ~ Treatment * x + (1|Block), BIB))
anova(fm1BIB)     # strong evidence of different slopes
## compare with Output 5.9, p. 193
(fm2BIB <- lmer(y ~ Treatment + x:Grp + (1|Block), BIB))
anova(fm2BIB)
@


\section{Bond}
\label{sec:Bond}

<<bond>>=
## compare with output 1.1 on p. 6
(fm1Bond <- lmer(pressure ~ Metal + (1|Ingot), Bond))
anova(fm1Bond)
@

\section{Cultivation}
\label{sec:Cultivation}

<<Cultivation>>=
str(Cultivation)
xtabs(~Block+Cult, Cultivation)
(fm1Cult <- lmer(drywt ~ Inoc * Cult + (1|Block) + (1|Cult), Cultivation))
anova(fm1Cult)
(fm2Cult <- lmer(drywt ~ Inoc + Cult + (1|Block) + (1|Cult), Cultivation))
anova(fm2Cult)
(fm3Cult <- lmer(drywt ~ Inoc + (1|Block) + (1|Cult), Cultivation))
anova(fm3Cult)
@



\section{Demand}
\label{sec:Demand}

<<Demand>>=
## compare to output 3.13, p. 132
(fm1Demand <-
 lmer(log(d) ~ log(y) + log(rd) + log(rt) + log(rs) + (1|State) + (1|Year),
      Demand))
@

\section{HR}
\label{sec:HR}
<<HR>>=
## linear trend in time
(fm1HR <- lmer(HR ~ Time * Drug + baseHR + (Time|Patient), HR))
anova(fm1HR)
## remove interaction
(fm3HR <- lmer(HR ~ Time + Drug + baseHR + (Time|Patient), HR))
anova(fm3HR)
## remove Drug term
(fm4HR <- lmer(HR ~ Time + baseHR + (Time|Patient), HR))
anova(fm4HR)
@


\section{Mississippi}
\label{sec:Mississippi}

<<Mississippi>>=
## compare with output 4.1, p. 142
(fm1Miss <- lmer(y ~ 1 + (1 | influent), Mississippi))
## compare with output 4.2, p. 143
(fm1MLMiss <- lmer(y ~ 1 + (1 | influent), Mississippi, REML=FALSE))
ranef(fm1MLMiss)          # BLUP's of random effects on p. 144
ranef(fm1Miss)            # BLUP's of random effects on p. 142
VarCorr(fm1Miss)          # compare to output 4.7, p. 148
## compare to output 4.8 and 4.9, pp. 150-152
(fm2Miss <- lmer(y ~ Type + (1 | influent), Mississippi))
anova(fm2Miss)
@

\section{Multilocation}
\label{sec:Multilocation}

%% --- keep in sync with ../man/Multilocation.Rd
%%                       ~~~~~~~~~~~~~~~~~~~~~~~
<<Multilocation>>=
str(Multilocation)
### Create a Block %in% Location factor
Multilocation$Grp <- with(Multilocation, Block:Location)
(fm1Mult <- lmer(Adj ~ Location * Trt + (1|Grp), Multilocation))
anova(fm1Mult)
(fm2Mult <- lmer(Adj ~ Location + Trt + (1|Grp), Multilocation))
(fm3Mult <- lmer(Adj ~ Location       + (1|Grp), Multilocation))
(fm4Mult <- lmer(Adj ~            Trt + (1|Grp), Multilocation))
(fm5Mult <- lmer(Adj ~ 1              + (1|Grp), Multilocation))
anova(fm2Mult)
fm2MultR <- lmer(Adj ~ Trt + (Trt - 1|Location) + (1|Block), Multilocation,
                 verbose = TRUE)
## non convergence in 10000 evaluations
fm2MultR
@


\section{PBIB}
\label{sec:PBIB}

<<PBIB>>=
str(PBIB)
## compare with output 1.7  pp. 24-25
(fm1PBIB <- lmer(response ~ Treatment + (1 | Block), PBIB))
@


\section{SIMS}
\label{sec:SIMS}

<<SIMS>>=
str(SIMS)
## compare to output 7.4, p. 262
(fm1SIMS <- lmer(Gain ~ Pretot + (Pretot | Class), SIMS))
@
\end{document}

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