\Header{contrast}{Contrast Matrices}
\alias{contr.helmert}{contrast}
\alias{contr.poly}{contrast}
\alias{contr.sum}{contrast}
\alias{contr.treatment}{contrast}
\keyword{design}{contrast}
\keyword{regression}{contrast}
\keyword{array}{contrast}
\begin{Description}\relax
Return a matrix of contrasts.\end{Description}
\begin{Usage}
\begin{verbatim}
contr.helmert(n, contrasts = TRUE)
contr.poly(n, contrasts = TRUE)
contr.sum(n, contrasts = TRUE)
contr.treatment(n, base = 1, contrasts = TRUE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{n}] a vector of levels for a factor, or the number of levels.
\item[\code{contrasts}] a logical indicating whether contrasts should be
computed.
\item[\code{base}] an integer specifying which group is considered the
baseline group. Ignored if \code{contrasts} is \code{FALSE}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
These functions are used for creating contrast matrices for use in
fitting analysis of variance and regression models.  The columns of
the resulting matrices contain contrasts which can be used for coding
a factor with \code{n} levels.  The returned value contains the
computed contrasts.  If the argument \code{contrasts} is \code{FALSE}
then a square indicator matrix is returned.

Note that as from \R{} version 0.62.2, \code{contr.poly} returns
contrasts based on orthogonal (rather than raw) polynomials.\end{Details}
\begin{Value}
A matrix with \code{n} rows and \code{k} columns, with \code{k=n-1} if
\code{contrasts} is \code{TRUE} and \code{k=n} if \code{contrasts} is
\code{FALSE}.\end{Value}
\begin{SeeAlso}\relax
\code{\Link{contrasts}},
\code{\Link{C}},
and
\code{\Link{aov}},
\code{\Link{glm}},
\code{\Link{lm}}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
(cH <- contr.helmert(4))
apply(cH, 2,sum) # column sums are 0!
crossprod(cH) # diagonal -- columns are orthogonal
contr.helmert(4, contrasts = FALSE) # just the 4 x 4 identity matrix

(cT <- contr.treatment(5))
all(crossprod(cT) == diag(4)) # TRUE: even orthonormal

(cP <- contr.poly(3)) # Linear and Quadratic
zapsmall(crossprod(cP), dig=15) # orthonormal up to fuzz
\end{ExampleCode}
\end{Examples}

