\HeaderA{aov}{Fit an Analysis of Variance Model}{aov}
\aliasA{Error}{aov}{Error}
\aliasA{print.aov}{aov}{print.aov}
\aliasA{print.aovlist}{aov}{print.aovlist}
\keyword{models}{aov}
\keyword{regression}{aov}
\begin{Description}\relax
Fit an analysis of variance model by a call to \code{lm} for each stratum.
\end{Description}
\begin{Usage}
\begin{verbatim}
aov(formula, data = NULL, projections = FALSE, qr = TRUE,
    contrasts = NULL, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{formula}] A formula specifying the model.
\item[\code{data}] A data frame in which the variables specified in the
formula will be found. If missing, the variables are searched for in
the standard way.
\item[\code{projections}] Logical flag: should the projections be returned?
\item[\code{qr}] Logical flag: should the QR decomposition be returned?
\item[\code{contrasts}] A list of contrasts to be used for some of the factors
in the formula. These are not used for any \code{Error} term, and
supplying contrasts for factors only in the \code{Error} term will give
a warning.
\item[\code{...}] Arguments to be passed to \code{lm}, such as \code{subset}
or \code{na.action}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
This provides a wrapper to \code{lm} for fitting linear models to
balanced or unbalanced experimental designs.

The main difference from \code{lm} is in the way \code{print},
\code{summary} and so on handle the fit: this is expressed in the
traditional language of the analysis of variance rather than that of
linear models.

If the formula contains a single \code{Error} term, this is used to
specify error strata, and appropriate models are fitted within each
error stratum.

The formula can specify multiple responses.

Weights can be specified by a \code{weights} argument, but should not
be used with an \code{Error} term, and are incompletely supported
(e.g., not by \code{\LinkA{model.tables}{model.tables}}).
\end{Details}
\begin{Value}
An object of class \code{c("aov", "lm")} or for multiple responses
of class \code{c("maov", "aov", "mlm", "lm")} or for multiple error
strata of class \code{"aovlist"}.  There are
\code{\LinkA{print}{print}} and \code{\LinkA{summary}{summary}} methods available for these.
\end{Value}
\begin{Note}\relax
\code{aov} is designed for balanced designs, and the results can be
hard to interpret without balance: beware that missing values in the
response(s) will likely lose the balance.  If there are two or more
error strata, the methods used are statistically inefficient without
balance, and it may be better to use \code{\LinkA{lme}{lme}}.

Balance can be checked with the \code{\LinkA{replications}{replications}} function.

The default \sQuote{contrasts} in \R{} are not orthogonal contrasts, and
\code{aov} and its helper functions will work better with such
contrasts: see the examples for how to select these.
\end{Note}
\begin{Author}\relax
The design was inspired by the S function of the same name described
in Chambers \emph{et al.} (1992).
\end{Author}
\begin{References}\relax
Chambers, J. M., Freeny, A and Heiberger, R. M. (1992)
\emph{Analysis of variance; designed experiments.}
Chapter 5 of \emph{Statistical Models in S}
eds J. M. Chambers and T. J. Hastie, Wadsworth \& Brooks/Cole.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{lm}{lm}}, \code{\LinkA{summary.aov}{summary.aov}},
\code{\LinkA{replications}{replications}}, \code{\LinkA{alias}{alias}},
\code{\LinkA{proj}{proj}}, \code{\LinkA{model.tables}{model.tables}}, \code{\LinkA{TukeyHSD}{TukeyHSD}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## From Venables and Ripley (2002) p.165.
N <- c(0,1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,1,0,1,1,0,0)
P <- c(1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,0)
K <- c(1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0,1,1,1,0,1,0)
yield <- c(49.5,62.8,46.8,57.0,59.8,58.5,55.5,56.0,62.8,55.8,69.5,55.0,
           62.0,48.8,45.5,44.2,52.0,51.5,49.8,48.8,57.2,59.0,53.2,56.0)
npk <- data.frame(block=gl(6,4), N=factor(N), P=factor(P),
                  K=factor(K), yield=yield)

## Set orthogonal contrasts.
op <- options(contrasts=c("contr.helmert", "contr.poly"))
( npk.aov <- aov(yield ~ block + N*P*K, npk) )
summary(npk.aov)
coefficients(npk.aov)

## to show the effects of re-ordering terms contrast the two fits
aov(yield ~ block + N * P + K, npk)
aov(terms(yield ~ block + N * P + K, keep.order=TRUE), npk)

## as a test, not particularly sensible statistically
npk.aovE <- aov(yield ~  N*P*K + Error(block), npk)
npk.aovE
summary(npk.aovE)
options(op)# reset to previous
\end{ExampleCode}
\end{Examples}

