src/library/stats/man/factanal.Rd - R - Git at Google

 % File src/library/stats/man/factanal.Rd
 % Part of the R package, https://www.R-project.org
 % Copyright 1995-2018 R Core Team
 % Distributed under GPL 2 or later

 \name{factanal}
 \alias{factanal}
 \encoding{UTF-8}
 \title{Factor Analysis}
 \description{
   Perform maximum-likelihood factor analysis on a covariance matrix or
   data matrix.
 }
 \usage{
 factanal(x, factors, data = NULL, covmat = NULL, n.obs = NA,
          subset, na.action, start = NULL,
          scores = c("none", "regression", "Bartlett"),
          rotation = "varimax", control = NULL, \dots)
 }
 \arguments{
   \item{x}{A formula or a numeric matrix or an object that can be
     coerced to a numeric matrix.}
   \item{factors}{The number of factors to be fitted.}
   \item{data}{An optional data frame (or similar: see
     \code{\link{model.frame}}), used only if \code{x} is a formula.  By
     default the variables are taken from \code{environment(formula)}.}
   \item{covmat}{A covariance matrix, or a covariance list as returned by
     \code{\link{cov.wt}}.  Of course, correlation matrices are covariance
     matrices.}
   \item{n.obs}{The number of observations, used if \code{covmat} is a
     covariance matrix.}
   \item{subset}{A specification of the cases to be used, if \code{x} is
     used as a matrix or formula.}
   \item{na.action}{The \code{na.action} to be used if \code{x} is
     used as a formula.}
   \item{start}{\code{NULL} or a matrix of starting values, each column
     giving an initial set of uniquenesses.}
   \item{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{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{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{\link{optim}}'s \code{control} argument.}
       \item{rotate}{a list of additional arguments for the rotation function.}
     }
   }
   \item{\dots}{Components of \code{control} can also be supplied as
     named arguments to \code{factanal}.}
 }
 \details{
   The factor analysis model is
   \deqn{x = \Lambda f + e}
   for a \eqn{p}--element vector \eqn{x}, a \eqn{p \times k}{p x k}
   matrix \eqn{\Lambda} of \emph{loadings}, a \eqn{k}--element vector
   \eqn{f} of \emph{scores} and a \eqn{p}--element vector \eqn{e} 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{\Psi}{Psi}, the \emph{uniquenesses}.  It is also common to
   scale the observed variables to unit variance, and done in this function.

   Thus factor analysis is in essence a model for the correlation matrix
   of \eqn{x},
   \deqn{\Sigma = \Lambda\Lambda^\prime + \Psi}{\Sigma = \Lambda \Lambda' + \Psi}
   There is still some indeterminacy in the model for it is unchanged
   if \eqn{\Lambda} is replaced by \eqn{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.\sspace{}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
   \enc{Jöreskog}{Joreskog} (1963) and given by Lawley & Maxwell
   (1971, p.\sspace{}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.\sspace{}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}.

   If \code{x} is a formula then the standard \code{NA}-handling is
   applied to the scores (if requested): see \code{\link{napredict}}.

   The \code{print} method (documented under \code{\link{loadings}})
   follows the factor analysis convention of drawing attention to the
   patterns of the results, so the default precision is three decimal
   places, and small loadings are suppressed.
 }
 \value{
   An object of class \code{"factanal"} with components
   \item{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.  This is of class \code{"loadings"}: see
     \code{\link{loadings}} for its \code{print} method.}
   \item{uniquenesses}{The uniquenesses computed.}
   \item{correlation}{The correlation matrix used.}
   \item{criteria}{The results of the optimization: the value of the
     criterion (a linear function of the negative log-likelihood) and information on the iterations used.}
   \item{factors}{The argument \code{factors}.}
   \item{dof}{The number of degrees of freedom of the factor analysis model.}
   \item{method}{The method: always \code{"mle"}.}
   \item{rotmat}{The rotation matrix if relevant.}
   \item{scores}{If requested, a matrix of scores.  \code{napredict} is
     applied to handle the treatment of values omitted by the \code{na.action}.}
   \item{n.obs}{The number of observations if available, or \code{NA}.}
   \item{call}{The matched call.}
   \item{na.action}{If relevant.}
   \item{STATISTIC, PVAL}{The significance-test statistic and P value, if
     it can be computed.}
 }

 \note{
   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 \sQuote{Heywood cases} (with one or more
   uniquenesses essentially zero) are much more common than most texts
   and some other programs would lead one to believe.
 }

 \references{
   Bartlett, M. S. (1937).
   The statistical conception of mental factors.
   \emph{British Journal of Psychology}, \bold{28}, 97--104.
   \doi{10.1111/j.2044-8295.1937.tb00863.x}.

   Bartlett, M. S. (1938).
   Methods of estimating mental factors.
   \emph{Nature}, \bold{141}, 609--610.
   \doi{10.1038/141246a0}.

   \enc{Jöreskog}{Joreskog}, 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.
 }

 \seealso{
   \code{\link{loadings}} (which explains some details of the
   \code{print} method), \code{\link{varimax}}, \code{\link{princomp}},
   \code{\link{ability.cov}}, \code{\link{Harman23.cor}},
   \code{\link{Harman74.cor}}.

   Other rotation methods are available in various contributed packages,
   including \CRANpkg{GPArotation} and \CRANpkg{psych}.
 }

 \examples{
 # 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
 \donttest{factanal(m1, factors = 3, rotation = "promax")}
 # The following shows the g factor as PC1
 \donttest{prcomp(m1) # signs may depend on platform}

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

 \donttest{## a realistic example from Bartholomew (1987, pp. 61-65)
 utils::example(ability.cov)
 }}
 \keyword{multivariate}
	% File src/library/stats/man/factanal.Rd
	% Part of the R package, https://www.R-project.org
	% Copyright 1995-2018 R Core Team
	% Distributed under GPL 2 or later

	\name{factanal}
	\alias{factanal}
	\encoding{UTF-8}
	\title{Factor Analysis}
	\description{
	Perform maximum-likelihood factor analysis on a covariance matrix or
	data matrix.
	}
	\usage{
	factanal(x, factors, data = NULL, covmat = NULL, n.obs = NA,
	subset, na.action, start = NULL,
	scores = c("none", "regression", "Bartlett"),
	rotation = "varimax", control = NULL, \dots)
	}
	\arguments{
	\item{x}{A formula or a numeric matrix or an object that can be
	coerced to a numeric matrix.}
	\item{factors}{The number of factors to be fitted.}
	\item{data}{An optional data frame (or similar: see
	\code{\link{model.frame}}), used only if \code{x} is a formula. By
	default the variables are taken from \code{environment(formula)}.}
	\item{covmat}{A covariance matrix, or a covariance list as returned by
	\code{\link{cov.wt}}. Of course, correlation matrices are covariance
	matrices.}
	\item{n.obs}{The number of observations, used if \code{covmat} is a
	covariance matrix.}
	\item{subset}{A specification of the cases to be used, if \code{x} is
	used as a matrix or formula.}
	\item{na.action}{The \code{na.action} to be used if \code{x} is
	used as a formula.}
	\item{start}{\code{NULL} or a matrix of starting values, each column
	giving an initial set of uniquenesses.}
	\item{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{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{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{\link{optim}}'s \code{control} argument.}
	\item{rotate}{a list of additional arguments for the rotation function.}
	}
	}
	\item{\dots}{Components of \code{control} can also be supplied as
	named arguments to \code{factanal}.}
	}
	\details{
	The factor analysis model is
	\deqn{x = \Lambda f + e}
	for a \eqn{p}--element vector \eqn{x}, a \eqn{p \times k}{p x k}
	matrix \eqn{\Lambda} of \emph{loadings}, a \eqn{k}--element vector
	\eqn{f} of \emph{scores} and a \eqn{p}--element vector \eqn{e} 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{\Psi}{Psi}, the \emph{uniquenesses}. It is also common to
	scale the observed variables to unit variance, and done in this function.

	Thus factor analysis is in essence a model for the correlation matrix
	of \eqn{x},
	\deqn{\Sigma = \Lambda\Lambda^\prime + \Psi}{\Sigma = \Lambda \Lambda' + \Psi}
	There is still some indeterminacy in the model for it is unchanged
	if \eqn{\Lambda} is replaced by \eqn{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.\sspace{}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
	\enc{Jöreskog}{Joreskog} (1963) and given by Lawley & Maxwell
	(1971, p.\sspace{}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.\sspace{}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}.

	If \code{x} is a formula then the standard \code{NA}-handling is
	applied to the scores (if requested): see \code{\link{napredict}}.

	The \code{print} method (documented under \code{\link{loadings}})
	follows the factor analysis convention of drawing attention to the
	patterns of the results, so the default precision is three decimal
	places, and small loadings are suppressed.
	}
	\value{
	An object of class \code{"factanal"} with components
	\item{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. This is of class \code{"loadings"}: see
	\code{\link{loadings}} for its \code{print} method.}
	\item{uniquenesses}{The uniquenesses computed.}
	\item{correlation}{The correlation matrix used.}
	\item{criteria}{The results of the optimization: the value of the
	criterion (a linear function of the negative log-likelihood) and information on the iterations used.}
	\item{factors}{The argument \code{factors}.}
	\item{dof}{The number of degrees of freedom of the factor analysis model.}
	\item{method}{The method: always \code{"mle"}.}
	\item{rotmat}{The rotation matrix if relevant.}
	\item{scores}{If requested, a matrix of scores. \code{napredict} is
	applied to handle the treatment of values omitted by the \code{na.action}.}
	\item{n.obs}{The number of observations if available, or \code{NA}.}
	\item{call}{The matched call.}
	\item{na.action}{If relevant.}
	\item{STATISTIC, PVAL}{The significance-test statistic and P value, if
	it can be computed.}
	}

	\note{
	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 \sQuote{Heywood cases} (with one or more
	uniquenesses essentially zero) are much more common than most texts
	and some other programs would lead one to believe.
	}

	\references{
	Bartlett, M. S. (1937).
	The statistical conception of mental factors.
	\emph{British Journal of Psychology}, \bold{28}, 97--104.
	\doi{10.1111/j.2044-8295.1937.tb00863.x}.

	Bartlett, M. S. (1938).
	Methods of estimating mental factors.
	\emph{Nature}, \bold{141}, 609--610.
	\doi{10.1038/141246a0}.

	\enc{Jöreskog}{Joreskog}, 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.
	}

	\seealso{
	\code{\link{loadings}} (which explains some details of the
	\code{print} method), \code{\link{varimax}}, \code{\link{princomp}},
	\code{\link{ability.cov}}, \code{\link{Harman23.cor}},
	\code{\link{Harman74.cor}}.

	Other rotation methods are available in various contributed packages,
	including \CRANpkg{GPArotation} and \CRANpkg{psych}.
	}

	\examples{
	# 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
	\donttest{factanal(m1, factors = 3, rotation = "promax")}
	# The following shows the g factor as PC1
	\donttest{prcomp(m1) # signs may depend on platform}

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

	\donttest{## a realistic example from Bartholomew (1987, pp. 61-65)
	utils::example(ability.cov)
	}}
	\keyword{multivariate}