% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/principalAxis.r
\name{principalAxis}
\alias{principalAxis}
\title{Principal Axis Analysis}
\usage{
principalAxis(R, nFactors = 2, communalities = "component")
}
\arguments{
\item{R}{numeric: correlation or covariance matrix}

\item{nFactors}{numeric: number of factors to retain}

\item{communalities}{character: initial values for communalities
(\code{"component", "maxr", "ginv" or "multiple"})}
}
\value{
\item{values}{ numeric: variance of each component/factor }
\item{varExplained}{ numeric: variance explained by each component/factor }
\item{varExplained}{ numeric: cumulative variance explained by each
component/factor } \item{loadings}{ numeric: loadings of each variable on
each component/factor }
}
\description{
The \code{PrincipalAxis} function returns a principal axis analysis without
iterated communalities estimates. Three different choices of communalities
estimates are given: maximum corelation, multiple correlation or estimates
based on the sum of the squared principal component analysis loadings.
Generally statistical packages initialize the the communalities at the
multiple correlation value (usual inverse or generalized inverse).
Unfortunately, this strategy cannot deal with singular correlation or
covariance matrices.  If a generalized inverse, the maximum correlation or
the estimated communalities based on the sum of loading are used instead,
then a solution can be computed.
}
\examples{

# .......................................................
# Example from Kim and Mueller (1978, p. 10)
# Population: upper diagonal
# Simulated sample: lower diagnonal
 R <- matrix(c( 1.000, .6008, .4984, .1920, .1959, .3466,
                .5600, 1.000, .4749, .2196, .1912, .2979,
                .4800, .4200, 1.000, .2079, .2010, .2445,
                .2240, .1960, .1680, 1.000, .4334, .3197,
                .1920, .1680, .1440, .4200, 1.000, .4207,
                .1600, .1400, .1200, .3500, .3000, 1.000),
                nrow=6, byrow=TRUE)

# Factor analysis: Principal axis factoring
# without iterated communalities -
# Kim and Mueller (1978, p. 21)
# Replace upper diagonal with lower diagonal
 RU <- diagReplace(R, upper=TRUE)
 principalAxis(RU, nFactors=2, communalities="component")
 principalAxis(RU, nFactors=2, communalities="maxr")
 principalAxis(RU, nFactors=2, communalities="multiple")
# Replace lower diagonal with upper diagonal
 RL <- diagReplace(R, upper=FALSE)
 principalAxis(RL, nFactors=2, communalities="component")
 principalAxis(RL, nFactors=2, communalities="maxr")
 principalAxis(RL, nFactors=2, communalities="multiple")
# .......................................................

}
\references{
Kim, J.-O. and Mueller, C. W. (1978). \emph{Introduction to
factor analysis. What it is and how to do it}. Beverly Hills, CA: Sage.

Kim, J.-O. and Mueller, C. W. (1987). \emph{Factor analysis. Statistical
methods and practical issues}. Beverly Hills, CA: Sage.
}
\seealso{
\code{\link{componentAxis}}, \code{\link{iterativePrincipalAxis}},
\code{\link{rRecovery}}
}
\author{
Gilles Raiche \cr Centre sur les Applications des Modeles de
Reponses aux Items (CAMRI) \cr Universite du Quebec a Montreal\cr
\email{raiche.gilles@uqam.ca}
}
\keyword{multivariate}