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  pdftitle = {A Handbook of Statistical Analyses Using R},
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  pdfauthor = {Brian S. Everitt and Torsten Hothorn},
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\begin{document}

%% Title page

\title{A Handbook of Statistical Analyses Using \R}

\author{Brian S. Everitt and Torsten Hothorn}

\maketitle
%%\VignetteIndexEntry{Chapter Analysis of Variance}
\setcounter{chapter}{3}


\SweaveOpts{prefix.string=figures/HSAUR,eps=FALSE,keep.source=TRUE} 

<<setup, echo = FALSE, results = hide>>=
rm(list = ls())
s <- search()[-1]
s <- s[-match(c("package:base", "package:stats", "package:graphics", "package:grDevices",
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a <- Sys.setlocale("LC_ALL", "C")
book <- TRUE
refs <- cbind(c("AItR", "SI", "CI", "ANOVA", "MLR", "GLM", 
                "DE", "RP", "SA", "ALDI", "ALDII", "MA", "PCA", 
                "MDS", "CA"), 1:15)
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@

\pagestyle{headings}

\chapter[Analysis of Variance]{Analysis of Variance: Weight Gain, Foster
Feeding in Rats, Water Hardness and Male Egyptian Skulls \label{ANOVA}}

\section{Introduction}


\section{Analysis of Variance}


\section{Analysis Using \R{}}

\subsection{Weight Gain in Rats \label{ANOVA:rats}}

Before applying analysis of variance to the data in Table~\ref{ANOVA-weightgain-tab} 
we should try to summarise the main features of the data 
by calculating means and standard deviations and by producing some hopefully 
informative graphs. The data is available in the \Rclass{data.frame} 
\Robject{weightgain}. The following \R{} code produces the 
required summary statistics 
<<ANOVA-weightgain-mean-var, echo = TRUE>>=
data("weightgain", package = "HSAUR")
tapply(weightgain$weightgain, 
       list(weightgain$source, weightgain$type), mean)
tapply(weightgain$weightgain, 
       list(weightgain$source, weightgain$type), sd)
@

\begin{figure}
\begin{center}
<<ANOVA-weightgain-plot, echo = TRUE, fig = TRUE>>=
plot.design(weightgain)
@
\caption{Plot of mean weight gain for each level of the two factors.
\label{ANOVA-weightgain-fig}}
\end{center}
\end{figure}


To apply analysis of variance to the data we can use the \Rcmd{aov} 
function in \R{} and then the \Rcmd{summary} method to give 
us the usual analysis of variance table. The model \Rclass{formula}
specifies a two-way layout with interaction terms, where the first
factor is \Robject{source}, and the second factor is \Robject{type}.
<<ANOVA-weightgain-aov, echo = TRUE>>=
wg_aov <- aov(weightgain ~ source * type, data = weightgain)
@

\renewcommand{\nextcaption}{\R{} output of the ANOVA fit  
                            for the \Robject{weightgain} data.
                            \label{ANOVA-weightgain-output}}
\SchunkLabel
<<ANOVA-weightgain-aov-summary, echo = TRUE>>=
summary(wg_aov)
@
\SchunkRaw


\begin{figure}
\begin{center}
<<ANOVA-weightgain-iplot, echo = TRUE, fig = FALSE, eval = FALSE>>=
interaction.plot(weightgain$type, weightgain$source, 
                 weightgain$weightgain) 
@
<<ANOVA-weightgain-iplot-nice, echo = FALSE, fig = TRUE>>=
interaction.plot(weightgain$type, weightgain$source, weightgain$weightgain,
legend = FALSE)
legend(1.5, 95, legend = levels(weightgain$source), title = "weightgain$source", 
       lty = c(2,1), bty = "n")
@
\caption{Interaction plot of type $\times$ source. \label{ANOVA-weightgain-fig2}}
\end{center}
\end{figure}

The estimates of the intercept and the main and interaction effects can be
extracted from the model fit by
<<ANOVA-weightgain-coef, echo = TRUE>>=
coef(wg_aov)
@
Note that the model was fitted with the restrictions $\gamma_1 = 0$ 
(corresponding to \Rlevel{Beef}) and
$\beta_1 = 0$ (corresponding to \Rlevel{High}) because treatment contrasts
were used as default as can be seen from
<<ANOVA-weightgain-contrasts, echo = TRUE>>=
options("contrasts")
@
Thus, the coefficient for \Robject{source} of $\Sexpr{coef(wg_aov)[2]}$ 
can be interpreted as an estimate of the difference $\gamma_2 - \gamma_1$.
Alternatively, we can use the restriction $\sum_i \gamma_i = 0$ by
<<ANOVA-weightgain-coef-sum, echo = TRUE>>=
coef(aov(weightgain ~ source + type + source:type, 
    data = weightgain, contrasts = list(source = contr.sum)))
@

\subsection{Foster Feeding of Rats of Different Genotype}

As in the previous subsection we will begin the analysis 
of the foster feeding data in Table~\ref{ANOVA-foster-tab} 
with a plot of the mean litter weight for the different genotypes 
of mother and litter (see Figure~\ref{ANOVA-foster-fig}). 
The data are in the \Rclass{data.frame}
\Robject{foster}
<<ANOVA-foster, echo = TRUE>>=
data("foster", package = "HSAUR")
@

\begin{figure}
\begin{center}
<<ANOVA-foster-plot, echo = TRUE, fig = TRUE>>=
plot.design(foster)
@
\caption{Plot of mean litter weight for each level of the two factors for
the \Robject{foster} data. \label{ANOVA-foster-fig}}
\end{center}
\end{figure}


We can derive the two analyses of variance tables for the 
foster feeding example by applying the \R{} code
<<ANOVA-foster-aov-one, echo = TRUE, eval = FALSE>>=
summary(aov(weight ~ litgen * motgen, data = foster))
@
to give
<<ANOVA-foster-aov-one, echo = FALSE>>=
summary(aov(weight ~ litgen * motgen, data = foster))
@
and then the code
<<ANOVA-foster-aov-two, echo = TRUE, eval = FALSE>>=
summary(aov(weight ~ motgen * litgen, data = foster))
@
to give
<<ANOVA-foster-aov-two, echo = FALSE>>=
summary(aov(weight ~ motgen * litgen, data = foster))
@
There are (small) differences 
in the sum of squares for the two main effects and, consequently, 
in the associated $F$-tests and $p$-values. 
\index{F-tests@$F$-tests}
This would not be true 
if in the previous example in Subsection~\ref{ANOVA:rats} 
we had used the code
<<ANOVA-weightgain-again, echo = TRUE, eval = FALSE>>=
summary(aov(weightgain ~ type * source, data = weightgain))
@
instead of the code which produced Figure~\ref{ANOVA-weightgain-output}
(readers should confirm that this is the case).


We can investigate the effect of genotype B on litter weight
in more detail by the use of \stress{multiple comparison procedures}
\index{Multiple comparison procedures|(}
\citep[see][]{HSAUR:Everitt1996}. Such procedures allow
a comparison of all pairs of levels of a factor whilst maintaining
the nominal significance level at its selected value and producing
adjusted confidence intervals for mean differences. One such
procedure is called \stress{Tukey honest significant differences} 
\index{Tukey honest significant differences}
suggested 
by \cite{HSAUR:Tukey1953}, see \cite{HSAUR:HochbergTamhane1987} also. Here,
we are interested in simultaneous confidence intervals for the weight
differences between all four genotypes of the mother.
First, an ANOVA model is fitted
<<ANOVA-foster-aov, echo = TRUE>>=
foster_aov <- aov(weight ~ litgen * motgen, data = foster)
@
which serves as the basis of the multiple comparisons, here with allpair
differences by
<<ANOVA-foster-tukeyHSD, echo = TRUE>>=
foster_hsd <- TukeyHSD(foster_aov, "motgen")
foster_hsd
@
A convenient \Rcmd{plot} method exists for this object and we can get a
graphical representation of the multiple confidence intervals as shown in
Figure~\ref{ANOVA-foster-mc}.
It appears that there  is only evidence for a difference in the 
B and J genotypes. 

\begin{figure}
\begin{center}
<<ANOVA-foster-tukeyHSDplot, echo = TRUE, fig = TRUE>>= 
plot(foster_hsd)
@
\caption{Graphical presentation of multiple comparison
         results for the \Robject{foster} feeding data. \label{ANOVA-foster-mc}}
\end{center}
\end{figure}
\index{Multiple comparison procedures|)}

\subsection{Water Hardness and Mortality}

The water hardness and mortality data for $61$ large towns in England and
Wales (see 
Table~2.3)
was analysed in 
Chapter~2
and here
we will extend the analysis by an assessment of the differences of both
hardness and mortality in the North or South.  The hypothesis that the
two-dimensional mean-vector of water hardness and mortality is the same for
cities in the North and the South can be tested by \stress{Hotelling-Lawley}
test in a multivariate analysis of variance framework. The \R{} function
\Rcmd{manova} can be used to fit such a model and the corresponding
\Rcmd{summary} method performs the test specified by the \Rcmd{test}
argument
<<ANOVA-water-manova, echo = TRUE>>=
data("water", package = "HSAUR")
summary(manova(cbind(hardness, mortality) ~ location, 
    data = water), test = "Hotelling-Lawley")
@
The \Rcmd{cbind} statement in the left hand side of the formula indicates
that a \stress{multivariate} response variable is to be modelled.
\index{cbind function in formula@\texttt{cbind} function in \textit{formula}}
The $p$-value associated with the \stress{Hotelling-Lawley} statistic is very small and
there is strong evidence that the mean vectors of the two variables are not
the same in the two regions. Looking at the sample means
<<ANOVA-water-means, echo = TRUE>>=
tapply(water$hardness, water$location, mean)
tapply(water$mortality, water$location, mean)
@
we see large differences in the two regions both in water hardness and
mortality, where low mortality is associated with hard water in the South
and high mortality with soft water in the North (see Figure~\ref{SI-water-sp} also).

\subsection{Male Egyptian Skulls}
\index{Multivariate analysis of variance (MANOVA)|(}
We can begin by looking at a table of mean values for the 
four measurements within each of the five epochs. The measurements 
are available in the \Rclass{data.frame} \Robject{skulls} and we can compute
the means over all epochs by
<<ANOVA-skulls-data, echo = TRUE>>=
data("skulls", package = "HSAUR")
means <- aggregate(skulls[,c("mb", "bh", "bl", "nh")], 
                   list(epoch = skulls$epoch), mean)
means
@
It may also be useful to look at these means graphically 
and this could be done in a variety of ways. Here we construct 
a scatterplot matrix of the means using the code attached to 
Figure~\ref{ANOVA-skulls-fig}.

\begin{figure}
\begin{center}
<<ANOVA-skulls-fig, echo = TRUE, fig = TRUE>>=
pairs(means[,-1],
    panel = function(x, y) {
        text(x, y, abbreviate(levels(skulls$epoch)))
    })
@
\caption{Scatterplot matrix of epoch means for Egyptian
         \Robject{skulls} data. \label{ANOVA-skulls-fig}}
\end{center}
\end{figure}

There appear to be quite large differences between the epoch 
means, at least on some of the four measurements. We can now 
test for a difference more formally by using MANOVA with the 
following \R{} code to apply each of the four possible test criteria 
mentioned earlier;
<<ANOVA-skulls-manova, echo = TRUE>>=
skulls_manova <- manova(cbind(mb, bh, bl, nh) ~ epoch, 
                        data = skulls)
summary(skulls_manova, test = "Pillai")
summary(skulls_manova, test = "Wilks") 
summary(skulls_manova, test = "Hotelling-Lawley")
summary(skulls_manova, test = "Roy")
@
The $p$-value associated 
with each four test criteria is very small and there is strong 
evidence that the skull measurements differ between the five 
epochs. We might now move on to investigate which epochs differ 
and on which variables. We can look at the univariate $F$-tests 
\index{F-tests@$F$-tests}
for each of the four variables by using the code
<<ANOVA-skulls-manova2, echo = TRUE>>=
summary.aov(skulls_manova)
@
We see that the results for the maximum breadths (\Robject{mb}) and
basialiveolar length (\Robject{bl}) are highly significant, with those for the other two 
variables, in particular for nasal heights (\Robject{nh}), 
suggesting little evidence of a difference. To look at the pairwise multivariate tests
(any of the four test criteria are equivalent in the case of a one-way
layout with two levels only) we can use the \Rcmd{summary} method and
\Rcmd{manova} function as follows:
<<ANOVA-skulls-manova3, echo = TRUE>>=
summary(manova(cbind(mb, bh, bl, nh) ~ epoch, data = skulls, 
               subset = epoch %in% c("c4000BC", "c3300BC")))
summary(manova(cbind(mb, bh, bl, nh) ~ epoch, data = skulls, 
               subset = epoch %in% c("c4000BC", "c1850BC")))
summary(manova(cbind(mb, bh, bl, nh) ~ epoch, data = skulls, 
               subset = epoch %in% c("c4000BC", "c200BC")))
summary(manova(cbind(mb, bh, bl, nh) ~ epoch, data = skulls, 
               subset = epoch %in% c("c4000BC", "cAD150")))
@
To keep the overall significance level for the set of all 
pairwise multivariate tests under some control (and still maintain 
a reasonable power), \cite{HSAUR:Stevens2001} recommends setting
the nominal level $\alpha = 0.15$ 
and carrying out each test at the $\alpha / m$ level 
where $m$ s the number of tests performed. The 
results of the four pairwise tests suggest that as the epochs become 
further separated in time the four skull measurements become 
increasingly distinct.
\index{Multivariate analysis of variance (MANOVA)|)}


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\end{document}
