\Header{varimax}{Rotation Methods for Factor Analysis}
\alias{promax}{varimax}
\keyword{multivariate}{varimax}
\begin{Description}\relax
These functions `rotate' loading matrices in factor analysis.\end{Description}
\begin{Usage}
\begin{verbatim}
varimax(x, normalize = TRUE, eps = 1e-5)
promax(x, m = 4)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] A loadings matrix, with \eqn{p}{} rows and \eqn{k < p}{} columns
\item[\code{m}] The power used the target for \code{promax}.  Values of 2 to
4 are recommended.
\item[\code{normalize}] logical. Should Kaiser normalization be performed?
If so the rows of \code{x} are re-scaled to unit length before
rotation, and scaled back afterwards.
\item[\code{eps}] The tolerance for stopping: the relative change in the sum
of singular values.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
These seek a `rotation' of the factors \code{x \%*\% T} that aims to
clarify the structure of the loadings matrix.  The matrix \code{T}
is a rotation (possibly with reflection) for \code{varimax}, but a
general linear transformation for \code{promax}, with the variance of
the factors being preserved.\end{Details}
\begin{Value}
A list with components
\begin{ldescription}
\item[\code{loadings}] The `rotated' loadings matrix, \code{x \%*\% rotmat}.
\item[\code{rotmat}] The `rotation matrix.
\end{ldescription}
\end{Value}
\begin{Author}\relax
B. D. Ripley\end{Author}
\begin{References}\relax
Hendrickson, A. E. and White, P. O. (1964) Promax: a quick method for
rotation to orthogonal oblique structure. \emph{British Journal of
Statistical Psychology}, \bold{17}, 65--70.

Horst, P. (1965) \emph{Factor Analysis of Data Matrices.} Holt,
Rinehart and Winston.  Chapter 10.

Kaiser, H. F. (1958) The varimax criterion for analytic rotation in
factor analysis. \emph{Psychometrika} \bold{23}, 187--200.

Lawley, D. N. and Maxwell, A. E. (1971) \emph{Factor Analysis as a
Statistical Method}. Second edition. Butterworths.\end{References}
\begin{SeeAlso}\relax
\code{\Link{factanal}}, \code{\Link{Harman74.cor}}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
data(swiss)
## varimax with normalize = T is the default
fa <- factanal( ~., 2, data = swiss)
varimax(fa$loadings, normalize = FALSE)
promax(fa$loadings)
\end{ExampleCode}
\end{Examples}

