\Header{biplot.princomp}{Biplot for Principal Components}
\keyword{multivariate}{biplot.princomp}
\keyword{hplot}{biplot.princomp}
\begin{Usage}
\begin{verbatim}
biplot.princomp(x, choices=1:2, scale=1, pc.biplot=FALSE, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\texttt{x}] an object of class \texttt{"princomp"}.
\item[\texttt{choices}] length 2 vector specifying the components to plot. Only the default
is a biplot in the strict sense.

\item[\texttt{scale}] The variables are scaled by \texttt{lambda $\,\hat{\,}$ scale} and the
observations are scaled by \texttt{lambda $\,\hat{\,}$ (1-scale)} where
\texttt{lambda} are the singular values as computed by
\texttt{\Link{princomp}}. Normally \texttt{0 <= scale <= 1}, and a warning 
will be issued if the specified \texttt{scale} is outside this range.

\item[\texttt{pc.biplot}] If true, use what Gabriel (1971) refers to as a "principal component
biplot", with \texttt{lambda = 1} and observations scaled up by sqrt(n) and
variables scaled down by sqrt(n).  Then inner products between
variables approximate covariances and distances between observations
approximate Mahalanobis distance.

\item[\texttt{...}] optional arguments to be passed to \texttt{\Link{biplot.default}}.
\end{ldescription}
\end{Arguments}
\begin{Description}
Produces a biplot (in the strict sense) from the output of
\texttt{\Link{princomp}(.)}.\end{Description}
\begin{Details}
This is a method for the generic function \texttt{biplot}.  There is
considerable confusion over the precise definitions: those of the
original paper, Gabriel (1971), are followed here.  Gabriel and
Odoroff (1990) use the same definitions, but their plots actually
correspond to \texttt{pc.biplot = TRUE}.\end{Details}
\begin{Section}{Side Effects}
a plot is produced on the current graphics device.\end{Section}
\begin{References}
Gabriel, K. R. (1971).
The biplot graphical display of matrices with applications to principal
component analysis.
\emph{Biometrika}, \bold{58} 453-467.

Gabriel, K. R. and Odoroff, C. L. (1990).
Biplots in biomedical research.
\emph{Statistics in Medicine}, \bold{9} 469-485.\end{References}
\begin{SeeAlso}
\texttt{\Link{biplot}}, \texttt{\Link{princomp}}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
data(crimes)
biplot(princomp(crimes))
\end{ExampleCode}
\end{Examples}

