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\mode{\usetheme{Boadilla}\usecolortheme{seahorse}\usecolortheme{rose}}
\usepackage{SweaveSlides}
\title[Longitudinal data]{Mixed models in R using the lme4 package\\%
Part 2: Longitudinal data, modeling interactions}
\SweaveOpts{engine=R,eps=FALSE,pdf=TRUE,width=10,height=6.5,strip.white=all}
\SweaveOpts{prefix=TRUE,prefix.string=figs/long,include=TRUE}
\SweaveOpts{keep.source=TRUE}
\mode{\setkeys{Gin}{width=\textwidth}}
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<>=
options(width=65,show.signif.stars=FALSE,str=strOptions(strict.width="cut"))
library(lattice)
library(Matrix)
library(lme4a)
data(Multilocation, package = "SASmixed")
attr(Multilocation, "ginfo") <-NULL
data(Early, package="mlmRev")
lattice.options(default.theme = function() standard.theme())
lattice.options(default.theme = function() standard.theme(color=FALSE))
if (file.exists("fm11.rda")) {
load("fm11.rda")
} else {
fm11 <- lmer(Adj ~ Trt + (0+Trt|Location) + (1|Grp), Multilocation, REML=FALSE)
save(fm11, file="fm11.rda")
}
@
\begin{document}
\mode{\maketitle\tableofcontents}
\mode{\frame{\titlepage}}
\mode{\frame{\frametitle{Outline}\tableofcontents[pausesections,hideallsubsections]}}
\section[sleepstudy]{Longitudinal data: sleepstudy}
\begin{frame}\frametitle{Simple longitudinal data}
\begin{itemize}
\item \emph{Repeated measures} data consist of measurements of a
response (and, perhaps, some covariates) on several \emph{experimental}
(or observational) \emph{units}.
\item Frequently the experimental (observational) unit is
\code{Subject} and we will refer to these units as ``subjects''.
However, the methods described here are not restricted to
data on human subjects.
\item \emph{Longitudinal} data are repeated measures data in which
the observations are taken over time.
\item We wish to characterize the response over time within subjects and
the variation in the time trends between subjects.
\item Frequently we are not as interested in comparing the
particular subjects in the study as much as we are interested in
modeling the variability in the population from which the subjects
were chosen.
\end{itemize}
\end{frame}
\begin{frame}\frametitle{Sleep deprivation data}
\begin{itemize}
\item This laboratory experiment measured the effect of sleep deprivation
on cognitive performance.
\item There were 18 subjects, chosen from the population of interest
(long-distance truck drivers), in the 10 day trial. These subjects were
restricted to 3 hours sleep per night during the trial.
\item On each day of the trial each subject's reaction time was
measured. The reaction time shown here is the average of several
measurements.
\item These data are \emph{balanced} in that each subject is
measured the same number of times and on the same occasions.
\end{itemize}
\end{frame}
\begin{frame}[fragile]\frametitle{Reaction time versus days by subject}
\begin{center}
<>=
print(xyplot(Reaction ~ Days | Subject, sleepstudy, aspect = "xy",
layout = c(9,2), type = c("g", "p", "r"),
index.cond = function(x,y) coef(lm(y ~ x))[1],
xlab = "Days of sleep deprivation",
ylab = "Average reaction time (ms)"))
@
\end{center}
\end{frame}
\begin{frame}\frametitle{Comments on the sleep data plot}
\begin{itemize}
\item The plot is a ``trellis'' or ``lattice'' plot where the data
for each subject are presented in a separate panel. The axes are
consistent across panels so we may compare patterns across
subjects.
\item A reference line fit by simple linear regression to the
panel's data has been added to each panel.
\item The aspect ratio of the panels has been adjusted so that a
typical reference line lies about $45^\circ$ on the page. We have
the greatest sensitivity in checking for differences in slopes
when the lines are near $\pm 45^\circ$ on the page.
\item The panels have been ordered not by subject number (which is
essentially a random order) but according to increasing intercept
for the simple linear regression. If the slopes and the
intercepts are highly correlated we should see a pattern across
the panels in the slopes.
\end{itemize}
\end{frame}
\begin{frame}\frametitle{Assessing the linear fits}
\begin{itemize}
\item In most cases a simple linear regression provides an adequate
fit to the within-subject data.
\item Patterns for some subjects (e.g.{} 350, 352 and 371) deviate
from linearity but the deviations are neither widespread nor
consistent in form.
\item There is considerable variation in the intercept (estimated
reaction time without sleep deprivation) across subjects -- 200
ms.{} up to 300 ms.{} -- and in the slope (increase in reaction time
per day of sleep deprivation) -- 0 ms./day up to 20 ms./day.
\item We can examine this variation further by plotting
confidence intervals for these intercepts and slopes. Because we use a
pooled variance estimate and have balanced data, the intervals
have identical widths.
\item We again order the subjects by increasing intercept so we can
check for relationships between slopes and intercepts.
\end{itemize}
\end{frame}
\begin{frame}[fragile]\frametitle{95\% conf int on within-subject intercept and slope}
\begin{center}
<>=
print(plot(confint(lmList(Reaction ~ Days | Subject, sleepstudy),
pooled = TRUE), order = 1))
@
\end{center}
These intervals reinforce our earlier impressions of considerable
variability between subjects in both intercept and slope but little
evidence of a relationship between intercept and slope.
\end{frame}
<>=
fm1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy)
@
\section[Random slopes]{A model with random effects for intercept and slope}
\begin{frame}\frametitle{A preliminary mixed-effects model}
\begin{itemize}
\item We begin with a linear mixed model in which the fixed effects
$[\beta_1,\beta_2]\trans$ are the representative intercept and slope
for the population and the random effects $\bm b_i=[b_{i1},b_{i2}]\trans,
i=1,\dots,18$ are the deviations in intercept and slope associated
with subject $i$.
\item The random effects vector, $\bm b$, consists of the $18$
intercept effects followed by the $18$ slope effects.
\end{itemize}
\begin{center}
<>=
print(image(fm1@re@Zt,xlab=NULL,ylab=NULL,sub=NULL))
@
\end{center}
\end{frame}
\begin{frame}[fragile]\frametitle{Fitting the model}
<>=
(fm1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy))
@
\end{frame}
\begin{frame}[fragile]\frametitle{Terms and matrices}
\begin{itemize}
\item The term \code{Days} in the formula generates a model matrix
$\bm X$ with two columns, the intercept column and the numeric
\code{Days} column. (The intercept is included unless
suppressed.)
\item The term \code{(Days|Subject)} generates a vector-valued
random effect (intercept and slope) for each of the $18$ levels of
the \code{Subject} factor.
\end{itemize}
\end{frame}
\begin{frame}\frametitle{A model with uncorrelated random effects}
\begin{itemize}
\item The data plots gave little indication of a systematic
relationship between a subject's random effect for slope and
his/her random effect for the intercept. Also, the estimated
correlation is quite small.
\item We should consider a model with uncorrelated random effects.
To express this we use two random-effects terms with the same
grouping factor and different left-hand sides. In the formula for
an \code{lmer} model, distinct random effects terms are modeled as
being independent. Thus we specify the model with two distinct
random effects terms, each of which has \code{Subject} as the
grouping factor. The model matrix for one term is intercept only
(\code{1}) and for the other term is the column for \code{Days}
only, which can be written \code{0+Days}. (The expression
\code{Days} generates a column for \code{Days} and an intercept.
To suppress the intercept we add \code{0+} to the expression;
\code{-1} also works.)
\end{itemize}
\end{frame}
\begin{frame}[fragile]\frametitle{A mixed-effects model with
independent random effects}
<>=
(fm2 <- lmer(Reaction ~ Days + (1|Subject) + (0+Days|Subject), sleepstudy))
@
\end{frame}
\begin{frame}[fragile]\frametitle{Comparing the models}
\begin{itemize}
\item Model \code{fm1} contains model \code{fm2} in the sense that
if the parameter values for model \code{fm1} were constrained so
as to force the correlation, and hence the covariance, to be zero,
and the model were re-fit, we would get model \code{fm2}.
\item The value \code{0}, to which the correlation is constrained, is
not on the boundary of the allowable parameter values.
\item In these circumstances a likelihood ratio test and a reference
distribution of a $\chi^2$ on 1 degree of freedom is suitable.
\end{itemize}
<>=
anova(fm2, fm1)
@
\end{frame}
\begin{frame}\frametitle{Conclusions from the likelihood ratio test}
\begin{itemize}
\item Because the large p-value indicates that we would not reject
\code{fm2} in favor of \code{fm1}, we prefer the more parsimonious
\code{fm2}.
\item This conclusion is consistent with the AIC (Akaike's
Information Criterion) and the BIC (Bayesian Information
Criterion) values for which ``smaller is better''.
\item We can also use a Bayesian approach, where we regard the
parameters as themselves being random variables, is assessing the
values of such parameters. A currently popular Bayesian method is
to use sequential sampling from the conditional distribution of
subsets of the parameters, given the data and the values of the
other parameters. The general technique is called \Emph{Markov
chain Monte Carlo} sampling.
\item We will expand on the use of likelihood-ratio tests in the next section.
\end{itemize}
\end{frame}
% \begin{frame}[fragile]\frametitle{Likelihood ratio tests on variance components}
% \begin{itemize}
% \item As for the case of a covariance, we can fit the model with and
% without the variance component and compare the fit quality.
% \item As mentioned previously, the likelihood ratio is a reasonable
% test statistic for the comparison but the ``asymptotic'' reference
% distribution of a $\chi^2$ does not apply because the parameter
% value being tested is on the boundary.
% \item The p-value computed using the $\chi^2$ reference distribution
% should be conservative (i.e. greater than the p-value that would
% be obtained through simulation).
% \end{itemize}
% <>=
% fm3 <- lmer(Reaction ~ Days + (1|Subject), sleepstudy)
% anova(fm3, fm2)
% @
% \end{frame}
\section{Conditional means}
\begin{frame}[fragile]\frametitle{Conditional means of the random effects}
<>=
(rr2 <- ranef(fm2))
@
\end{frame}
\begin{frame}[fragile]\frametitle{Scatterplot of the conditional means}
\begin{center}
<>=
print(plot(rr2, aspect = 1, type = c("g", "p"))[[1]])
@
\end{center}
\end{frame}
\begin{frame}\frametitle{Comparing within-subject coefficients}
\begin{itemize}
\item For this model we can combine the conditional means of the
random effects and the estimates of the fixed effects to get
conditional means of the within-subject coefficients.
\item These conditional means will be ``shrunken'' towards the
fixed-effects estimates relative to the estimated coefficients
from each subject's data. John Tukey called this ``borrowing
strength'' between subjects.
\item Plotting the shrinkage of the within-subject coefficients
shows that some of the coefficients are considerably shrunken
toward the fixed-effects estimates.
\item However, comparing the within-group and mixed model fitted
lines shows that large changes in coefficients occur in the noisy
data. Precisely estimated within-group coefficients are not
changed substantially.
\end{itemize}
\end{frame}
\begin{frame}[fragile]\frametitle{Estimated within-group coefficients and BLUPs}
\begin{center}
<>=
df <- coef(lmList(Reaction ~ Days | Subject, sleepstudy))
fclow <- subset(df, `(Intercept)` < 251)
fchigh <- subset(df, `(Intercept)` > 251)
cc1 <- as.data.frame(coef(fm2)$Subject)
names(cc1) <- c("A", "B")
df <- cbind(df, cc1)
ff <- fixef(fm2)
with(df,
print(xyplot(`(Intercept)` ~ Days, aspect = 1,
x1 = B, y1 = A,
panel = function(x, y, x1, y1, subscripts, ...) {
panel.grid(h = -1, v = -1)
x1 <- x1[subscripts]
y1 <- y1[subscripts]
larrows(x, y, x1, y1, type = "closed", length = 0.1,
angle = 15, ...)
lpoints(x, y,
pch = trellis.par.get("superpose.symbol")$pch[2],
col = trellis.par.get("superpose.symbol")$col[2])
lpoints(x1, y1,
pch = trellis.par.get("superpose.symbol")$pch[1],
col = trellis.par.get("superpose.symbol")$col[1])
lpoints(ff[2], ff[1],
pch = trellis.par.get("superpose.symbol")$pch[3],
col = trellis.par.get("superpose.symbol")$col[3])
ltext(fclow[,2], fclow[,1], row.names(fclow),
adj = c(0.5, 1.7))
ltext(fchigh[,2], fchigh[,1], row.names(fchigh),
adj = c(0.5, -0.6))
},
key = list(space = "top", columns = 3,
text = list(c("Mixed model", "Within-group", "Population")),
points = list(col = trellis.par.get("superpose.symbol")$col[1:3],
pch = trellis.par.get("superpose.symbol")$pch[1:3]))
)))
@
\end{center}
\end{frame}
\begin{frame}[fragile]\frametitle{Observed and fitted}
\begin{center}
<>=
print(xyplot(Reaction ~ Days | Subject, sleepstudy, aspect = "xy",
layout = c(9,2), type = c("g", "p", "r"),
coef.list = df[,3:4],
panel = function(..., coef.list) {
panel.xyplot(...)
panel.abline(as.numeric(coef.list[packet.number(),]),
col.line = trellis.par.get("superpose.line")$col[2],
lty = trellis.par.get("superpose.line")$lty[2]
)
panel.abline(fixef(fm2),
col.line = trellis.par.get("superpose.line")$col[4],
lty = trellis.par.get("superpose.line")$lty[4]
)
},
index.cond = function(x,y) coef(lm(y ~ x))[1],
xlab = "Days of sleep deprivation",
ylab = "Average reaction time (ms)",
key = list(space = "top", columns = 3,
text = list(c("Within-subject", "Mixed model", "Population")),
lines = list(col = trellis.par.get("superpose.line")$col[c(2:1,4)],
lty = trellis.par.get("superpose.line")$lty[c(2:1,4)]))))
@
\end{center}
\end{frame}
\begin{frame}[fragile]
\frametitle{Plot of prediction intervals for the random effects}
\begin{center}
<>=
print(dotplot(ranef(fm1,post=TRUE),
scales = list(x = list(relation = 'free')))[["Subject"]])
@
\end{center}
Each set of prediction intervals have constant width because of the
balance in the experiment.
\end{frame}
\section{Conclusions}
\begin{frame}\frametitle{Conclusions from the example}
\begin{itemize}
\item Carefully plotting the data is enormously helpful in
formulating the model.
\item It is relatively easy to fit and evaluate models to data like
these, from a balanced designed experiment.
\item We consider two models with random effects for the slope and
the intercept of the response w.r.t. time by subject. The models
differ in whether the (marginal) correlation of the vector of
random effects per subject is allowed to be nonzero.
\item The ``estimates'' (actually, the conditional means) of the
random effects can be considered as penalized estimates of these
parameters in that they are shrunk towards the origin.
\item Most of the prediction intervals for the random effects
overlap zero.
\end{itemize}
\end{frame}
\section[Other interactions]{Other forms of interactions}
\begin{frame}\frametitle{Random slopes and interactions}
\begin{itemize}
\item In the \code{sleepstudy} model fits we allowed for random
effects for \code{Days} by \code{Subject}.
\item These random effects can be considered as an interaction
between the fixed-effects covariate \code{Days} and the
random-effects factor \code{Subject}.
\item When we have both fixed-levels categorical covariates and
random-levels categorical covariates we have many different ways
in which interactions can be expressed.
\item Often the wide range of options provides ``enough rope to hang
yourself'' in the sense that it is very easy to create an
overly-complex model.
\end{itemize}
\end{frame}
\begin{frame}
[fragile]
\frametitle{The \code{Multilocation} data set}
\begin{itemize}
\item Data from a multi-location trial of several treatments are
described in section 2.8 of Littell, Milliken, Stroup and
Wolfinger (1996) \textbf{SAS System for Mixed Models} and are
available as \code{Multilocation} in package \code{SASmixed}.
\item Littell et al. don't cite the source of the data. Apparently
\code{Adj} is an adjusted response of some sort for 4 different
treatments applied at each of 3 blocks in each of 9 locations.
Because \code{Block} is implicitly nested in \code{Location}, the
\code{Grp} interaction variable was created.
\end{itemize}
<>=
str(Multilocation)
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Response by \code{Grp} and \code{Trt}}
<>=
print(dotplot(reorder(Grp, Adj) ~ Adj, Multilocation,
groups=Trt, type=c("p","a"),
auto.key=list(columns=4,lines=TRUE)))
@
\begin{itemize}
\item From this one plot (Littell et al. do not provide any plots but
instead immediately jump into fitting several ``cookie-cutter''
models) we see differences between locations, not as much between
blocks within location, and treatment 2 providing a lower adjusted
response.
\end{itemize}
\end{frame}
\begin{frame}
[fragile]
\frametitle{Response by \code{Block} and \code{Trt} within \code{Location}}
<>=
ll <- with(Multilocation, reorder(Location, Adj))
print(dotplot(reorder(reorder(Grp, Adj), as.numeric(ll)) ~ Adj|ll, Multilocation,
groups=Trt, type=c("p","a"), strip=FALSE, strip.left=TRUE, layout=c(1,9),
auto.key=list(columns=4,lines=TRUE),
scales = list(y=list(relation="free"))))
@
\end{frame}
\begin{frame}
\frametitle{Fixed-levels categorical covariates and ``contrasts''}
\begin{itemize}
\item In this experiment we are interested in comparing the
effectiveness of these four levels of \code{Trt}.
\item That is, the levels of \code{Trt} are fixed levels and we
should incorporate them in the fixed-effects part of the model.
\item Unlike the situation with random effects, we cannot separately
estimate ``effects'' for each level of a categorical covariate in
the fixed-effects and an overall intercept term.
\item We could suppress the intercept term but even then we still
encounter redundancies in effects for each level when we have more
than one categorical covariate in the fixed-effects.
\item Because of this we estimate coefficients for $k-1$ ``contrasts''
associated with the $k$ levels of a factor.
\item The default contrasts (called \code{contr.treatment}) measure
changes relative to a reference level which is the first level of
the factor. Other contrasts can be used when particular
comparisons are of interest.
\end{itemize}
\end{frame}
\begin{frame}
[fragile]
\frametitle{A simple model for \code{Trt} controlling for \code{Grp}}
<>=
print(fm3 <- lmer(Adj ~ Trt + (1|Grp), Multilocation), corr=FALSE)
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Interpretation of the results}
\begin{itemize}
\item We see that the variability between the Location/Block
combinations (levels of \code{Grp}) is greater than the residual
variability, indicating the importance of controlling for it.
\item The contrast between levels 2 and 1 of \code{Trt}, labeled
\code{Trt2} is the greatest difference and apparently significant.
\item If we wish to evaluate the ``significance'' of the levels of
\code{Trt} as a group, however, we should fit the trivial
model and perform a LRT.
\end{itemize}
<>=
fm4 <- lmer(Adj ~ 1 + (1|Grp), Multilocation)
anova(fm4, fm3)
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{\code{Location} as a fixed-effect}
\begin{itemize}
\item We have seen that \code{Location} has a substantial effect on
\code{Adj}. If we are interested in these specific 9 locations
we could incorporate them as fixed-effects parameters.
\item Instead of examining 8 coefficients separately we will
consider their cumulative effect using the single-argument form of
\code{anova}.
\end{itemize}
<>=
anova(fm5 <- lmer(Adj ~ Location + Trt + (1|Grp), Multilocation))
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{An interaction between fixed-effects factors}
\begin{itemize}
\item We could ask if there is an interaction between the levels of
\code{Trt} and those of \code{Location} considered as fixed effects.
\end{itemize}
<>=
anova(fm6 <- lmer(Adj ~ Location*Trt + (1|Grp), Multilocation))
anova(fm5, fm6)
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Considering levels of \code{Location} as random effects}
<>=
print(fm7 <- lmer(Adj ~ Trt + (1|Location) + (1|Grp), Multilocation), corr = FALSE)
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Is \code{Grp} needed in addition to \code{Location}?}
\begin{itemize}
\item At this point we may want to check whether the random effect
for \code{Block} within \code{Location} is needed in addition to
the random effect for \code{Location}.
\end{itemize}
<>=
fm8 <- lmer(Adj ~ Trt + (1|Location), Multilocation)
anova(fm8, fm7)
@
\begin{itemize}
\item Apparently not, but we may want to revisit this issue after
checking for interactions.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Ways of modeling random/fixed interactions}
\begin{itemize}
\item There are two ways we can model the interaction between a
fixed-levels factor (\code{Trt}) and a random-levels factor
(\code{Location}, as we are currently viewing this factor).
\item The first, and generally preferable, way is to incorporate a
simple scalar random-effects term with the interaction as the grouping
factor.
\item The second, more complex, way is to use vector-valued random
effects for the random-levels factor. We must be careful when
using this approach because it often produces a degenerate model,
but not always obviously degenerate.
\end{itemize}
\end{frame}
\begin{frame}
[fragile]
\frametitle{Scalar random effects for interaction}
<>=
(fm9 <- lmer(Adj ~ Trt + (1|Trt:Location) + (1|Location), Multilocation, REML=FALSE))
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Both interaction and Block-level random effects}
<>=
(fm10 <- update(fm9, . ~ . + (1|Grp)))
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Scalar interaction random effects are still not significant}
<>=
anova(fm10, fm8)
@
\begin{itemize}
\item We have switched to ML fits because we are comparing models
using \code{anova}. In a comparative \code{anova} any REML fits are
refit as ML before comparison so we start with the ML fits.
\item In model \code{fm9} the estimated variance for the scalar
interaction random effects was exactly zero in the ML fit. In
\code{fm10} the estimate is positive but still not significant.
\end{itemize}
\end{frame}
\begin{frame}
[fragile]
\frametitle{Vector-valued random effects}
\begin{itemize}
\item An alternative formulation for an interaction between
\code{Trt} and \code{Location} (viewed as a random-levels factor)
is to use vector-valued random effects.
\item We have used a similar construct in model \code{fm1} with
vector-valued random effects (intercept and slope) for each level
of \code{Subject}.
\item One way to fit such a model is
<>=
fm11 <- lmer(Adj ~ Trt + (Trt|Location) + (1|Grp), Multilocation, REML=FALSE)
@
but interpretation is easier when fit as
<>=
fm11 <- lmer(Adj ~ Trt + (0+Trt|Location) + (1|Grp), Multilocation, REML=FALSE)
@
\end{itemize}
\end{frame}
\begin{frame}
[fragile]
\frametitle{Examining correlation of random effects}
\begin{itemize}
\item The random effects summary for \code{fm11}
<>=
cat(paste(capture.output(print(fm11))[4:15], collapse="\n"), "\n")
@
shows very high correlations between the random effects for the levels
of \code{Trt} within each level of \code{Location}.
\item Such a situation may pass by unnoticed if estimates of variances and
covariances are all that is reported.
\item In this case (and many other similar cases) the
variance-covariance matrix of the vector-valued random effects is
effectively singular.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Singular variance-covariance for random effects}
\begin{itemize}
\item When we incorporate too many fixed-effects terms in a model we
usually find out because the standard errors become very large.
\item For random effects terms, especially those that are
vector-valued, overparameterization is sometimes more difficult to detect.
\item The REML and ML criteria for mixed-effects models seek to
balance the complexity of the model versus the fidelity of the
fitted values to the observed responses.
\item The way ``complexity'' is measured in this case, a model with a singular
variance-covariance matrix for the random effects is considered a
good thing - it is optimally simple.
\item When we have only scalar random-effects terms singularity
means that one of the variance components must be exactly zero
(and ``near singularity'' means very close to zero).
\end{itemize}
\end{frame}
\begin{frame}
[fragile]
\frametitle{Detecting singular random effects}
\begin{itemize}
\item The \code{Lambda} slot in a \code{merMod} object is the
triangular factor of the variance-covariance matrix.
\item We can directly assess its condition number using the
\code{kappa} (condition number) or \code{rcond} (reciprocal
condition number) functions. Large condition numbers are bad.
\item We do need to be cautious when we have a large number of
levels for the grouping factors because \code{Lambda} will be
\textbf{very} large (but also very sparse). At present the
\code{kappa} and \code{rcond} functions transform the sparse
matrix to a dense matrix, which could take a very long time.
\end{itemize}
<>=
kappa(fm11@re@Lambda)
rcond(fm11@re@Lambda)
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Using verbose model fits}
\begin{itemize}
\item An alternative, which is recommended whenever you have doubts
about a model fit, is to use \code{verbose=TRUE} (the lines don't
wrap and we miss the interesting part here).
\end{itemize}
<>=
fm11 <- lmer(Adj ~ Trt + (0+Trt|Location) + (1|Grp), Multilocation, REML=FALSE, verbose=TRUE)
@
<>=
fm11@re@theta
@
\end{frame}
\begin{frame}
\frametitle{What to watch for in the verbose output}
\begin{itemize}
\item In this model the criterion is being optimized with respect to
11 parameters.
\item These are the variance component parameters, $\bm\theta$. The
fixed-effects coefficients, $\bm\beta$, and the common scale
parameter, $\sigma$, are at their conditionally optimal values.
\item Generally it is more difficult to estimate a variance
parameter (either a variance or a covariance) than it is to
estimate a coefficient. Estimating 11 such parameters requires a
considerable amount of information.
\item Some of these parameters are required to be non-negative.
When they become zero or close to zero ($2.7\times10^{-7}$, in
this case) the variance-covariance matrix is degenerate.
\item The \code{@re@lower} slot contains the lower bounds.
Parameter components for which \code{@re@lower} is \code{-Inf} are
unbounded. The ones to check are those for which \code{@re@lower}
is \code{0}.
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Data on early childhood cognitive development}
<>=
print(xyplot(cog ~ age | id, Early, type = c("g",'b'), aspect = 'xy',
layout = c(29,4), between = list(y = c(0,0.5)),
# skip = rep(c(FALSE,TRUE),c(58,11)),
xlab = "Age (yr)",
ylab = "Cognitive development score",
scales = list(x = list(tick.number = 3, alternating = TRUE,
labels = c("1","","2"), at = c(1,1.5,2))),
par.strip.text = list(cex = 0.7)))
@
\end{frame}
\begin{frame}[fragile]
\frametitle{Fitting a model to the Early data}
\begin{itemize}
\item The \code{Early} data in the \code{mlmRev} package are from a
study on early childhood cognitive development as influenced by a
treatment. These data are discussed in \textbf{Applied
Longitudinal Data Analysis} (2003) by Singer and Willett.
\item A model with random effects for slope and intercept is
\end{itemize}
<>=
Early <- within(Early, tos <- age-0.5)
fm12 <- lmer(cog ~ tos+trt:tos+(tos|id), Early, verbose=TRUE)
@
\end{frame}
\begin{frame}
[fragile]
\frametitle{Fitted model for the Early data}
<>=
print(fm12, corr=FALSE)
@
Here is it obvious that there is a problem. However, Singer and
Willett did not detect this in model fits from SAS PROC MIXED or
MLWin, both of which reported a covariance estimate.
\end{frame}
\begin{frame}
\frametitle{Other practical issues}
\begin{itemize}
\item In some disciplines there is an expectation that data will be
analyzed starting with the most complex model and evaluating terms
according to their p-values.
\item This can be appropriate for carefully balanced, designed
experiments. It is rarely a good approach on observational,
imbalanced data.
\item Bear in mind that this approach was formulated when graphical
and computational capabilities were very limited.
\item A more appropriate modern approach is to explore the data
graphically and to fit models sequentially, comparing these fitted
models with tests such as the LRT.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Fixed-effects or random-effects?}
\begin{itemize}
\item Earlier we described the distinction between fixed and
random effects as dependent on the repeatability of the levels.
\item This is the basis for the distinction but the number of levels
observed must also be considered.
\item Fitting mixed-effects models requires data from several levels
of the grouping factor. Even when a factor represents a random
selection (say sample transects in an ecological study) it is not
practical to estimate a variance component from only two or three
observed levels.
\item At the other extreme, a census of a large number of levels can
be modeled with random effects even though the observed levels are
not a sample.
\end{itemize}
\end{frame}
\section{Summary}
\label{sec:summary}
\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item In models of longitudinal data on several subjects we often
incorporate random effects for the intercept and/or the slope of
the response with respect to time.
\item By default we allow for a general variance-covariance matrix
for the random effects for each subject.
\item The model can be restricted to independent random effects when
appropriate.
\item For other interactions of fixed-effects factors and
random-effects grouping factors, the general term can lead to
estimation of many variance-covariance parameters. We may want to
restrict to independent random effects for the subject and the
subject/type interaction.
\end{itemize}
\end{frame}
\end{document}