\HeaderA{proj}{Projections of Models}{proj}
\methaliasA{proj.aov}{proj}{proj.aov}
\methaliasA{proj.aovlist}{proj}{proj.aovlist}
\methaliasA{proj.default}{proj}{proj.default}
\methaliasA{proj.lm}{proj}{proj.lm}
\keyword{models}{proj}
\begin{Description}\relax
\code{proj} returns a matrix or list of matrices giving the projections
of the data onto the terms of a linear model.  It is most frequently
used for \code{\LinkA{aov}{aov}} models.
\end{Description}
\begin{Usage}
\begin{verbatim}
proj(object, ...)  

## S3 method for class 'aov':
proj(object, onedf = FALSE, unweighted.scale = FALSE, ...)  

## S3 method for class 'aovlist':
proj(object, onedf = FALSE, unweighted.scale = FALSE, ...)  

## Default S3 method:
proj(object, onedf = TRUE, ...)  

## S3 method for class 'lm':
proj(object, onedf = FALSE, unweighted.scale = FALSE, ...)  
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{object}] An object of class \code{"lm"} or a class inheriting from
it, or an object with a similar structure including in particular
components \code{qr} and \code{effects}.
\item[\code{onedf}] A logical flag. If \code{TRUE}, a projection is returned for all
the columns of the model matrix. If \code{FALSE}, the single-column
projections are collapsed by terms of the model (as represented in
the analysis of variance table).
\item[\code{unweighted.scale}] If the fit producing \code{object} used
weights, this determines if the projections correspond to weighted or
unweighted observations.
\item[\code{...}] Swallow and ignore any other arguments.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
A projection is given for each stratum of the object, so for \code{aov}
models with an \code{Error} term the result is a list of projections.
\end{Details}
\begin{Value}
A projection matrix or (for multi-stratum objects) a list of
projection matrices.

Each projection is a matrix with a row for each observations and
either a column for each term (\code{onedf = FALSE}) or for each
coefficient (\code{onedf = TRUE}). Projection matrices from the
default method have orthogonal columns representing the projection of
the response onto the column space of the Q matrix from the QR
decomposition.  The fitted values are the sum of the projections, and
the sum of squares for each column is the reduction in sum of squares
from fitting that column (after those to the left of it).

The methods for \code{lm} and \code{aov} models add a column to the
projection matrix giving the residuals (the projection of the data
onto the orthogonal complement of the model space).

Strictly, when \code{onedf = FALSE} the result is not a projection,
but the columns represent sums of projections onto the columns of the
model matrix corresponding to that term. In this case the matrix does
not depend on the coding used.
\end{Value}
\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{aov}{aov}}, \code{\LinkA{lm}{lm}}, \code{\LinkA{model.tables}{model.tables}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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)
npk.aov <- aov(yield ~ block + N*P*K, npk)
proj(npk.aov)

## as a test, not particularly sensible
options(contrasts=c("contr.helmert", "contr.treatment"))
npk.aovE <- aov(yield ~  N*P*K + Error(block), npk)
proj(npk.aovE)
\end{ExampleCode}
\end{Examples}

