% NOTE -- ONLY EDIT THE .Rnw FILE!!!  The .tex file is
% likely to be overwritten.

\SweaveOpts{engine=R,eps=FALSE,pdf=TRUE,width=10,height=6.5,strip.white=all}
\SweaveOpts{prefix=TRUE,prefix.string=figs/long,include=TRUE}
\setkeys{Gin}{width=\textwidth}
<<preliminaries,echo=FALSE,results=hide>>=
options(width=65,show.signif.stars=FALSE)
library(lattice)
library(Matrix)
library(lme4)
#lattice.options(default.theme = standard.theme(color = FALSE))
@ 

\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}
<<sleepxy,echo=FALSE,fig=TRUE>>=
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}
<<Sl1,fig=TRUE,echo=FALSE,height=4.5>>=
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,echo=FALSE>>=
fm1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy)
@ 

\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}
<<sleepZ,fig=TRUE,echo=FALSE,results=hide,height=3.5>>=
print(image(fm1@Zt,xlab=NULL,ylab=NULL,sub=NULL))
@ 
  \end{center}
\end{frame}

\begin{frame}[fragile]\frametitle{Fitting the model}
<<sm1>>= 
(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}
<<fm1Images,fig=TRUE,echo=FALSE,height=3>>=
efm1 <- expand(fm1)
print(image(efm1[["T"]], xlab = NULL, ylab = NULL, sub = "T"),
      position = c(0,0,0.27,1), more = TRUE)
print(image(tcrossprod(fm1@A), xlab = NULL, ylab = NULL, sub = "AA'"),
      position = c(0.24,0,0.51,1), more = TRUE)
print(image(efm1$P, xlab = NULL, ylab = NULL, sub = "P"),
      position = c(0.49,0,0.76,1), more = TRUE)
print(image(fm1@L, xlab = NULL, ylab = NULL, sub = "L", colorkey = FALSE),
      position = c(0.73,0,1,1))
@ 
\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,echo=FALSE>>=
(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}
<<anovafm1fm2>>=
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 The \code{lme4} package has a function called \code{mcmcsamp}
    to evaluate such samples from a fitted model.  At present,
    however, there seem to be a few ``infelicities'', as Bill Venables
    calls them, in this function.
  \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}
<<fm3anova>>=
fm3 <- lmer(Reaction ~ Days + (1|Subject), sleepstudy)
anova(fm3, fm2)
@ 
\end{frame}


\begin{frame}[fragile]\frametitle{Conditional modes of the random effects}
<<rr1>>= 
(rr2 <- ranef(fm2))
@ 
\end{frame}

\begin{frame}[fragile]\frametitle{Scatterplot of the conditional modes}
  \begin{center}
<<rr2plot,echo=FALSE,fig=TRUE,results=hide>>=
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 modes of the
    random effects and the estimates of the fixed effects to get
    conditional modes of the within-subject coefficients.
  \item These conditional modess 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}
<<shrinkage,echo=FALSE,fig=TRUE>>=
df <- coef(lmList(Reaction ~ Days | Subject, sleepstudy))
cc1 <- as.data.frame(coef(fm2)[["Subject"]])
names(cc1) <- c("A", "B")
df <- cbind(df, cc1)
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]
                panel.arrows(x, y, x1, y1, type = "closed", length = 0.1,
                             angle = 15, ...)
                panel.points(x, y,
                             col = trellis.par.get("superpose.symbol")$col[2])
                panel.points(x1, y1)
            },
                   key = list(space = "top", columns = 2,
                   text = list(c("Mixed model", "Within-group")),
                   points = list(col = trellis.par.get("superpose.symbol")$col[1:2],
                   pch = trellis.par.get("superpose.symbol")$pch[1:2]))
               )))
@
  \end{center}
\end{frame}

\begin{frame}[fragile]\frametitle{Observed and fitted}
  \begin{center}
<<shrinkfit,echo=FALSE,fig=TRUE>>=
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]
                              )
             },
             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 = 2,
             text = list(c("Mixed model", "Within-group")),
             lines = list(col = trellis.par.get("superpose.line")$col[2:1],
             lty = trellis.par.get("superpose.line")$lty[2:1]))))
@ 
  \end{center}
\end{frame}

\begin{frame}[fragile]
  \frametitle{Plot of prediction intervals for the random effects}
  \begin{center}
<<caterpillar,fig=TRUE,echo=FALSE,results=hide,height=4.5>>=
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}

\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 modes) 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}