\HeaderA{factanal}{Factor Analysis}{factanal}
\aliasA{factanal.fit.mle}{factanal}{factanal.fit.mle}
\keyword{multivariate}{factanal}
\begin{Description}\relax
Perform maximum-likelihood factor analysis on a covariance matrix or
data matrix.
\end{Description}
\begin{Usage}
\begin{verbatim}
factanal(x, factors, data = NULL, covmat = NULL, n.obs = NA,
         subset, na.action,
         start = NULL, scores = c("none", "regression", "Bartlett"),
         rotation = "varimax", control = NULL, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] A formula or a numeric matrix or an object that can be
coerced to a numeric matrix.
\item[\code{factors}] The number of factors to be fitted.
\item[\code{data}] A data frame, used only if \code{x} is a formula.
\item[\code{covmat}] A covariance matrix, or a covariance list as returned by
\code{\LinkA{cov.wt}{cov.wt}}.  Of course, correlation matrices are covariance
matrices.
\item[\code{n.obs}] The number of observations, used if \code{covmat} is a
covariance matrix.
\item[\code{subset}] A specification of the cases to be used, if \code{x} is
used as a matrix or formula.
\item[\code{na.action}] The \code{na.action} to be used if \code{x} is
used as a formula.
\item[\code{start}] \code{NULL} or a matrix of starting values, each column
giving an initial set of uniquenesses.
\item[\code{scores}] Type of scores to produce, if any.  The default is none,
\code{"regression"} gives Thompson's scores, \code{"Bartlett"} given
Bartlett's weighted least-squares scores. Partial matching allows
these names to be abbreviated.
\item[\code{rotation}] character. \code{"none"} or the name of a function
to be used to rotate the factors: it will be called with first
argument the loadings matrix, and should return a list with component
\code{loadings} giving the rotated loadings, or just the rotated loadings.
\item[\code{control}] A list of control values,
\describe{
\item[nstart] The number of starting values to be tried if
\code{start = NULL}. Default 1.
\item[trace] logical. Output tracing information? Default \code{FALSE}.
\item[lower] The lower bound for uniquenesses during
optimization. Should be > 0. Default 0.005.
\item[opt] A list of control values to be passed to
\code{\LinkA{optim}{optim}}'s \code{control} argument.
\item[rotate] a list of additional arguments for the rotation function.
}

\item[\code{...}] Components of \code{control} can also be supplied as
named arguments to \code{factanal}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The factor analysis model is
\deqn{x = \Lambda f + e}{x = Lambda f + e}
for a \eqn{p}{}--element row-vector \eqn{x}{}, a \eqn{p \times k}{p x k}
matrix of \emph{loadings}, a \eqn{k}{}--element vector of \emph{scores}
and a \eqn{p}{}--element vector of errors.  None of the components
other than \eqn{x}{} is observed, but the major restriction is that the
scores be uncorrelated and of unit variance, and that the errors be
independent with variances \eqn{\Phi}{Phi}, the
\emph{uniquenesses}.  Thus factor analysis is in essence a model for
the covariance matrix of \eqn{x}{},
\deqn{\Sigma = \Lambda^\prime\Lambda + \Psi}{Sigma = Lambda'Lambda + Psi}
There is still some indeterminacy in the model for it is unchanged
if \eqn{\Lambda}{Lambda} is replaced by \eqn{G\Lambda}{G Lambda} for
any orthogonal matrix \eqn{G}{}.  Such matrices \eqn{G}{} are known as
\emph{rotations} (although the term is applied also to non-orthogonal
invertible matrices).

If \code{covmat} is supplied it is used.  Otherwise \code{x} is used if
it is a matrix, or a formula \code{x} is used with \code{data} to
construct a model matrix, and that is used to construct a covariance
matrix.  (It makes no sense for the formula to have a response,
and all the variables must be numeric.)  Once a covariance matrix is found or
calculated from \code{x}, it is converted to a correlation matrix for
analysis.  The correlation matrix is returned as component
\code{correlation} of the result.

The fit is done by optimizing the log likelihood assuming multivariate
normality over the uniquenesses.  (The maximizing loadings for given
uniquenesses can be found analytically: Lawley \& Maxwell (1971,
p. 27).)  All the starting values supplied in \code{start} are tried
in turn and the best fit obtained is used.  If \code{start = NULL}
then the first fit is started at the value suggested by Jöreskog
(1963) and given by Lawley \& Maxwell (1971, p. 31), and then
\code{control\$nstart - 1} other values are tried, randomly selected
as equal values of the uniquenesses.

The uniquenesses are technically constrained to lie in \eqn{[0, 1]}{},
but near-zero values are problematical, and the optimization is
done with a lower bound of \code{control\$lower}, default 0.005
(Lawley \& Maxwell, 1971, p. 32).

Scores can only be produced if a data matrix is supplied and used.
The first method is the regression method of Thomson (1951), the
second the weighted least squares method of Bartlett (1937, 8).
Both are estimates of the unobserved scores \eqn{f}{}.  Thomson's method
regresses (in the population) the unknown \eqn{f}{} on \eqn{x}{} to yield
\deqn{\hat f = \Lambda^\prime \Sigma^{-1} x}{hat f = Lambda' Sigma^-1 x}
and then substitutes the sample estimates of the quantities on the
right-hand side.  Bartlett's method minimizes the sum of squares of
standardized errors over the choice of \eqn{f}{}, given (the fitted)
\eqn{\Lambda}{Lambda}.

If \code{x} is a formula then the standard NA-handling is applied to
the scores (if requested): see \code{\LinkA{napredict}{napredict}}.
\end{Details}
\begin{Value}
An object of class \code{"factanal"} with components
\begin{ldescription}
\item[\code{loadings}] A matrix of loadings, one column for each factor.  The
factors are ordered in decreasing order of sums of squares of
loadings, and given the sign that will make the sum of the loadings
positive.
\item[\code{uniquenesses}] The uniquenesses computed.
\item[\code{correlation}] The correlation matrix used.
\item[\code{criteria}] The results of the optimization: the value of the
negative log-likelihood and information on the iterations used.
\item[\code{factors}] The argument \code{factors}.
\item[\code{dof}] The number of degrees of freedom of the factor analysis model.
\item[\code{method}] The method: always \code{"mle"}.
\item[\code{scores}] If requested, a matrix of scores.
\item[\code{n.obs}] The number of observations if available, or \code{NA}.
\item[\code{call}] The matched call.
\item[\code{na.action}] If relevant.
\item[\code{STATISTIC, PVAL}] The significance-test statistic and P value, if
if can be computed.
\end{ldescription}
\end{Value}
\begin{Note}\relax
There are so many variations on factor analysis that it is hard to
compare output from different programs.  Further, the optimization in
maximum likelihood factor analysis is hard, and many other examples we
compared had less good fits than produced by this function.  In
particular, solutions which are Heywood cases (with one or more
uniquenesses essentially zero) are much often common than most texts
and some other programs would lead one to believe.
\end{Note}
\begin{References}\relax
Bartlett, M. S. (1937) The statistical conception of mental factors.
\emph{British Journal of Psychology}, \bold{28}, 97--104.

Bartlett, M. S. (1938) Methods of estimating mental
factors. \emph{Nature}, \bold{141}, 609--610.

Jöreskog, K. G. (1963)
\emph{Statistical Estimation in Factor Analysis.}  Almqvist and Wicksell.

Lawley, D. N. and Maxwell, A. E. (1971) \emph{Factor Analysis as a
Statistical Method.} Second edition. Butterworths.

Thomson, G. H. (1951) \emph{The Factorial Analysis of Human Ability.}
London University Press.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{print.loadings}{print.loadings}},
\code{\LinkA{varimax}{varimax}}, \code{\LinkA{princomp}{princomp}},
\code{\LinkA{ability.cov}{ability.cov}}, \code{\LinkA{Harman23.cor}{Harman23.cor}},
\code{\LinkA{Harman74.cor}{Harman74.cor}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
# A little demonstration, v2 is just v1 with noise,
# and same for v4 vs. v3 and v6 vs. v5
# Last four cases are there to add noise
# and introduce a positive manifold (g factor)
v1 <- c(1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,5,6)
v2 <- c(1,2,1,1,1,1,2,1,2,1,3,4,3,3,3,4,6,5)
v3 <- c(3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,5,4,6)
v4 <- c(3,3,4,3,3,1,1,2,1,1,1,1,2,1,1,5,6,4)
v5 <- c(1,1,1,1,1,3,3,3,3,3,1,1,1,1,1,6,4,5)
v6 <- c(1,1,1,2,1,3,3,3,4,3,1,1,1,2,1,6,5,4)
m1 <- cbind(v1,v2,v3,v4,v5,v6)
cor(m1)
factanal(m1, factors=3) # varimax is the default
factanal(m1, factors=3, rotation="promax")
# The following shows the g factor as PC1
prcomp(m1)

## formula interface
factanal(~v1+v2+v3+v4+v5+v6, factors = 3,
         scores = "Bartlett")$scores

## a realistic example from Barthlomew (1987, pp. 61-65)
example(ability.cov)
\end{ExampleCode}
\end{Examples}

