% NOTE -- ONLY EDIT THE .Rnw FILE!!! The .tex file is
% likely to be overwritten.
\usepackage{SweaveSlides}
\title[lme4]{Mixed models in R using the lme4 package\\Part 6: Interactions}
\subject{Interactions}
\AtBeginSection[]
{
\begin{frame}
\frametitle{Outline}
\tableofcontents[currentsection]
\end{frame}
}
\begin{document}
\frame{\titlepage}
\begin{frame}
\frametitle{Outline}
\tableofcontents[pausesections,hideallsubsections]
\end{frame}
\SweaveOpts{engine=R,eps=FALSE,pdf=TRUE,width=10,height=6.5,strip.white=all}
\SweaveOpts{prefix=TRUE,prefix.string=figs/interactions,include=TRUE}
\setkeys{Gin}{width=\textwidth}
<>=
options(width=65,show.signif.stars=FALSE)
library(lattice)
library(Matrix)
library(lme4)
data(Machines, package = "MEMSS")
Machines <- Machines[with(Machines,order(Worker, Machine)),]
lattice.options(default.theme = function() standard.theme())
lattice.options(default.theme = function() standard.theme(color=FALSE))
if (file.exists("ratbrain.rda")) {
load("ratbrain.rda")
} else {
ratbrain <- within(read.delim("http://www-personal.umich.edu/~bwest/rat_brain.dat"),
{
treatment <- factor(treatment,
labels = c("Basal", "Carbachol"))
region <- factor(region, levels = c(3, 1, 2),
labels = c("VDB", "BST", "LS"))
})
save(ratbrain, file = "ratbrain.rda")
}
@
\section[Interactions]{Interactions with grouping factors}
\begin{frame}
\frametitle{Interactions of covariates and grouping factors}
\begin{itemize}
\item For longitudinal data, having a random effect for the slope
w.r.t. time by subject is reasonably easy to understand.
\item Although not generally presented in this way, these random
effects are an interaction term between the grouping factor for the
random effect (\code{Subject}) and the time covariate.
\item We can also define interactions between a categorical
covariate and a random-effects grouping factor.
\item Different ways of expressing such interactions lead to
different numbers of random effects. These different definitions
have different levels of complexity, affecting both their
expressive power and the ability to estimate all the parameters in
the model.
\end{itemize}
\end{frame}
\section[Machines data]{The Machines data}
\begin{frame}
\frametitle{Machines data}
\begin{itemize}
\item Milliken and Johnson (1989) provide (probably artificial) data
on an experiment to measure productivity according to the machine
being used for a particular operation.
\item In the experiment, a sample of six different operators used
each of the three machines on three occasions --- a total of nine
runs per operator.
\item These three machines were the specific machines of interest
and we model their effect as a fixed-effect term.
\item The operators represented a sample from the population of
potential operators. We model this factor, (\code{Worker}), as a
random effect.
\item This is a replicated ``subject/stimulus'' design with a fixed
set of stimuli that are themselves of interest. (In other
situations the stimuli may be a sample from a population of
stimuli.)
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Machines data plot}
<>=
print(dotplot(reorder(Worker, score) ~ score, Machines,
groups = Machine,
xlab = "Quality and productivity score",
ylab = "Worker", type = c("p","a"),
auto.key = list(columns = 3, lines = TRUE),
jitter.y = TRUE))
@
\end{frame}
\begin{frame}
\frametitle{Comments on the data plot}
\begin{itemize}
\item There are obvious differences between the scores on different machines.
\item It seems likely that \code{Worker} will be a significant
random effect, especially when considering the low variation
within replicates.
\item There also appears to be a significant \code{Worker:Machine}
interaction. \code{Worker} 6 has a very different pattern
w.r.t. machines than do the others.
\item We can approach the interaction in one of two ways: define
simple, scalar random effects for \code{Worker} and for the
\code{Worker:Machine} interaction or define vector-valued random
effects for \code{Worker}
\end{itemize}
\end{frame}
\section[Scalar or vector]{Scalar interactions or vector-valued random effects?}
\begin{frame}[fragile]
\frametitle{Random effects for subject and subject:stimulus}
<>=
print(fm1 <- lmer(score ~ Machine + (1|Worker) + (1|Worker:Machine),
Machines), corr = FALSE)
@
\end{frame}
\begin{frame}
\frametitle{Characteristics of the scalar interaction model}
\begin{itemize}
\item The model incorporates simple, scalar random effects
for \code{Worker} and for the \code{Worker:Machine} interaction.
\item These two scalar random-effects terms have $q_1=q_2=1$ so they
contribute $n_1=6$ and $n_2=18$ random effects for a total of
$q=24$. There are $2$ variance-component parameters.
\item The random effects allow for an overall shift in level for
each worker and a separate shift for each combination of worker
and machine. The unconditional distributions of these random
effects are independent. The unconditional variances of the
interaction random effects are constant.
\item The main restriction in this model is the assumption of
constant variance and independence of the interaction random
effects.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Model matrix $\bm Z\trans$ for the scalar interaction model}
\begin{center}
<>=
print(image(fm1@Zt, sub = NULL))
@
\end{center}
\begin{itemize}
\item Because we know these are scalar random effects we can
recognize the pattern of a balanced, nested, two-factor design,
similar to that of the model for the \code{Pastes} data.
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Vector-valued random effects by subject}
<>=
print(fm2 <- lmer(score ~ Machine + (0+Machine|Worker), Machines),
corr = FALSE)
@
\end{frame}
\begin{frame}
\frametitle{Characteristics of the vector-valued r.e. model}
\begin{center}
<>=
print(image(fm2@Zt, sub = NULL, xlab = NULL, ylab = NULL))
@
\end{center}
\begin{itemize}
\item We use the specification \code{(0 + Machine|Worker)} to force
an ``indicator'' parameterization of the random effects.
\item In this image the 1's are black. The gray positions are
non-systematic zeros (initially zero but can become
nonzero).
\item Here $k=1$, $q_1=3$ and $n_1=6$ so we have $q=18$
random effects but $q_1(q_1+1)/2=6$ variance-component parameters
to estimate.
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Comparing the model fits}
\begin{itemize}
\item Although not obvious from the specifications, these model fits
are nested. If the variance-covariance matrix for the
vector-valued random effects has a special form, called
\Emph{compound symmetry}, the model reduces to model \code{fm1}.
\item The p-value of 6.5\% may or may not be significant.
\end{itemize}
<>=
fm2M <- update(fm2, REML = FALSE)
fm1M <- update(fm1, REML = FALSE)
anova(fm2M, fm1M)
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Model comparisons eliminating the unusual combination}
\begin{itemize}
\item In a case like this we may want to check if a single, unusual
combination (\code{Worker} 6 using \code{Machine} ``B'') causes
the more complex model to appear necessary. We eliminate that
unusual combination.
\end{itemize}
<>=
Machines1 <- subset(Machines, !(Worker == "6" & Machine == 'B'))
xtabs(~ Machine + Worker, Machines1)
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Machines data after eliminating the unusual combination}
<>=
print(dotplot(reorder(Worker, score) ~ score, Machines1,
groups = Machine,
xlab = "Quality and productivity score",
ylab = "Worker", type = c("p","a"),
auto.key = list(columns = 3, lines = TRUE),
jitter.y = TRUE))
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Model comparisons without the unusual combination}
<>=
fm1aM <- lmer(score ~ Machine + (1|Worker) + (1|Worker:Machine), Machines1, REML = FALSE)
fm2aM <- lmer(score ~ Machine + (0 + Machine|Worker), Machines1, REML = FALSE)
anova(fm2aM, fm1aM)
@
\end{frame}
\begin{frame}
\frametitle{Trade-offs when defining interactions}
\begin{itemize}
\item It is important to realize that estimating scale parameters (i.e.
variances and covariances) is considerably more difficult than
estimating location parameters (i.e. means or fixed-effects coefficients).
\item A vector-valued random effect term having $q_i$ random effects
per level of the grouping factor requires $q_i(q_i+1)/2$
variance-covariance parameters to be estimated. A simple, scalar
random effect for the interaction of a ``random-effects'' factor
and a ``fixed-effects'' factor requires only 1 additional
variance-covariance parameter.
\item Especially when the ``fixed-effects'' factor has a moderate to
large number of levels, the trade-off in model complexity argues
against the vector-valued approach.
\item One of the major sources of difficulty in using the
\code{lme4} package is the tendency to overspecify the number of
random effects per level of a grouping factor.
\end{itemize}
\end{frame}
\section[Brain activation]{The brain activation data}
\begin{frame}
\frametitle{Brain activation data from West, Welch and Ga\l{}ecki (2007)}
\begin{center}
<>=
print(dotplot(factor(region, levels = c("BST", "LS", "VDB")) ~ activate|treatment,
ratbrain, groups = animal, type = c("p","a"),
strip = FALSE, strip.left = TRUE,
xlab = "Activation (mean optical density)",
ylab = "Region of brain",
layout = c(1,2),
auto.key = list(columns = 5, lines = TRUE, points = FALSE)))
@
\end{center}
\begin{itemize}
\item In the experiment seven different regions of five rats' brains
were imaged in a basal condition (after injection with saline
solution) and after treatment with the drug Carbachol. The data
provided are from three regions.
\item This representation of the data is similar to the figure on
the cover of West, Welch and Ga\l{}ecki (2007).
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Brain activation data in an alternative layout}
\begin{center}
<>=
print(dotplot(region ~ activate|animal,
ratbrain, groups = treatment, type = c("p","a"),
strip = FALSE, strip.left = TRUE,
xlab = "Activation (mean optical density)",
ylab = "Region",
layout = c(1,5),
auto.key = list(columns = 2, lines = TRUE, points = FALSE)))
@
\end{center}
\begin{itemize}
\item The animals have similar patterns of changes but different magnitudes.
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Reproducing the models from West et al.}
\begin{itemize}
\item These data are analyzed in West et al. (2007) allowing for
main effects for treatment and region, a fixed-effects interaction
of these two factors and vector-valued random effects for the
intercept and the treatment by animal.
\item Note that this will require estimating three variance
component parameters from data on five animals.
\item Their final model also allowed for different residual
variances by treatment. We won't discuss that here.
\item We choose the order of the levels of \code{region} to produce
the same parameterization of the fixed effects.
\end{itemize}
<>=
str(ratbrain)
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Model 5.1 from West et al.}
<>=
print(m51 <- lmer(activate ~ region * treatment + (1|animal), ratbrain), corr = FALSE)
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Model 5.2 from West et al.}
<>=
print(m52 <- lmer(activate ~ region * treatment + (treatment|animal), ratbrain), corr = FALSE)
@
\end{frame}
\begin{frame}[fragile]
\frametitle{A variation on model 5.2 from West et al.}
<>=
print(m52a <- lmer(activate ~ region * treatment + (0+treatment|animal), ratbrain), corr = FALSE)
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Simple scalar random effects for the interaction}
<>=
print(m54 <- lmer(activate ~ region * treatment + (1|animal) +
(1|animal:treatment), ratbrain), corr = FALSE)
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Prediction intervals for the random effects}
<>=
print(dotplot(ranef(m51, post = TRUE),
strip = FALSE, strip.left = TRUE)$animal,
split = c(1,1,1,3), more = TRUE)
print(dotplot(ranef(m52, post = TRUE),
strip = FALSE, strip.left = TRUE,
scales = list(x = list(relation = "free")))$animal,
split = c(1,2,1,3), more = TRUE)
print(dotplot(ranef(m52a, post = TRUE),
strip = FALSE, strip.left = TRUE,
scales = list(x = list(relation = "free")))$animal,
split = c(1,3,1,3))
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Is this ``overmodeling'' the data?}
\begin{itemize}
\item The prediction intervals for the random effects indicate that
the vector-valued random effects are useful, as does a model comparison.
<>=
m51M <- update(m51, REML = FALSE)
m52M <- update(m52, REML = FALSE)
anova(m51M, m52M)
@
\item However, these models incorporate many fixed-effects parameters
and random effects in a model of a relatively small amount of data.
Is this too much?
\item There are several ways we can approach this:
\begin{itemize}
\item Simplify the model by considering the difference in
activation under the two conditions within the same
animal:region combination (i.e. approach it like a paired
t-test).
\item Model the five animals with fixed effects and use F-tests.
\item Assess the precision of the variance estimates (done later).
\end{itemize}
\end{itemize}
\end{frame}
\section[Differences]{Considering differences}
\begin{frame}[fragile]
\frametitle{Considering differences}
\begin{itemize}
\item Before we can analyze the differences at each
\code{animal:region} combination we must first calculate them.
\item We could do this by subsetting the \code{ratbrain} data frame
for the \code{"Basal"} and \code{"Carbachol"} levels of the
\code{treatment} factor and forming the difference of the two
\code{activate} columns. For this to be correct we must have the
same ordering of levels of the \code{animal} and \code{region}
factors in each half. It turns out we do but we shouldn't count
on this (remember ``Murphy's Law''?).
\item A better approach is to reshape the data frame (but this is
complicated) or to use \code{xtabs} to align the levels. First we
should check that the data are indeed balanced and unreplicated.
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Checking for balanced and unreplicated; tabling activate}
\begin{itemize}
\item We saw the balance in the data plots but we can check too
<>=
ftable(xtabs(~ treatment + region + animal, ratbrain))
@
\item In \code{xtabs} we can use a two-sided formula to tabulate a variable
<>=
ftable(atab <- xtabs(activate ~ treatment + animal + region, ratbrain))
@
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Taking differences}
\begin{itemize}
\item The \code{atab} object is an array with additional attributes
<>=
str(atab)
@
\item Use \code{apply} to take differences over dimension 1
<>=
(diffs <- as.table(apply(atab, 2:3, diff)))
@
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Taking differences (cont'd)}
\begin{itemize}
\item Finally, convert the table of differences to a data frame.
\end{itemize}
<>=
str(diffs <- as.data.frame(diffs))
names(diffs)[3] <- "actdiff"
@
\begin{center}
<>=
print(dotplot(reorder(animal, actdiff) ~ actdiff, diffs, groups = region,
xlab = "Difference in activation with Carbachol from Basal state",
ylab = "Animal", type = c("p","a"),
auto.key = list(columns = 3, lines = TRUE, points = FALSE)))
@
\end{center}
\end{frame}
\begin{frame}[fragile]
\frametitle{A model for the differences}
<>=
print(dm1 <- lmer(actdiff ~ region + (1|animal), diffs))
@
\end{frame}
%% consider comparing a randomized blocked design with random effects
%% and with fixed effects for the blocking factor.
\section[Animals fixed]{Fixed-effects for the animals}
\begin{frame}
\frametitle{Using fixed-effects for the animals}
\begin{itemize}
\item There are five experimental units (animals) in this study.
That is about the lower limit under which we could hope to
estimate variance components.
\item We should compare with a fixed-effects model.
\item If we wish to evaluate coefficients for \code{treatment} or
\code{region} we must be careful about the ``contrasts'' that are
used to create the model. However, the analysis of variance table
does not depend on the contrasts.
\item We use \code{aov} to fit the fixed-effects model so that a
summary is the analysis of variance table.
\item The fixed-effects anova table is the sequential table with
main effects first, then two-factor interactions, etc. The anova
table for an \code{lmer} model gives the contributions of the
fixed-effects after removing the contribution of the random
effects, which include the \code{animal:treatment} interaction in
model \code{m52}.
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Fixed-effects anova versus random effects}
<>=
summary(m52f <- aov(activate ~ animal * treatment + region * treatment,
ratbrain))
anova(m52)
@
\begin{itemize}
\item Except for the \code{treatment} factor, the anova tables are
nearly identical.
\end{itemize}
\end{frame}
\section{Summary}
\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item It is possible to fit complex models to balanced data sets
from carefully designed experiments but one should always be
cautious of creating a model that is too complex.
\item I prefer to proceed incrementally, taking time to examine data
plots, rather than starting with a model incorporating all possible terms.
\item Some feel that one should be able to specify the analysis
(and, in particular, the analysis of variance table) before even
beginning to collect data. I am more of a model-builder and try
to avoid dogmatic approaches.
\item For the \code{ratbrain} data I would be very tempted to take
differences and analyze it as a randomized blocked design.
\end{itemize}
\end{frame}