iterativePrincipalAxis.rd
\name{iterativePrincipalAxis}
\alias{iterativePrincipalAxis}
\title{ Iterative Principal Axis Analysis }

\description{
The \code{iterativePrincipalAxis} function returns a principal axis analysis with
iterated communality estimates. Four different choices of initial communality
estimates are given: maximum correlation, multiple correlation (usual and
generalized inverse) or estimates based
on the sum of the squared principal component analysis loadings. Generally, statistical
packages initialize the communalities at the multiple correlation value.
Unfortunately, this strategy cannot always deal with singular correlation or
covariance matrices.
If a generalized inverse, the maximum correlation or the estimated communalities
are used instead, then a solution can be computed.
}

\usage{
iterativePrincipalAxis(R,
nFactors=2,
communalities="component",
iterations=20,
tolerance=0.001)
}

\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"})}
\item{iterations}{    numeric: maximum number of iterations to obtain a solution}
\item{tolerance}{     numeric: minimal difference in the estimated
communalities after a given iteration}
}

\value{
\item{values}{       numeric: variance of each component }
\item{varExplained}{ numeric: variance explained by each component }
\item{varExplained}{ numeric: cumulative variance explained by each component }
\item{iterations}{   numeric: maximum number of iterations to obtain a solution}
\item{tolerance}{    numeric: minimal difference in the estimated communalities
after a given iteration}
}

\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{
}

\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}, \url{http://www.er.uqam.ca/nobel/r17165/}
}

\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 with iterated communalities
# Kim and Mueller (1978, p. 23)
# Replace upper diagonal with lower diagonal
RU         <- diagReplace(R, upper=TRUE)
nFactors   <- 2
fComponent <- iterativePrincipalAxis(RU, nFactors=nFactors,
communalities="component")
fComponent
rRecovery(RU,fComponent$loadings, diagCommunalities=FALSE) fMaxr <- iterativePrincipalAxis(RU, nFactors=nFactors, communalities="maxr") fMaxr rRecovery(RU,fMaxr$loadings, diagCommunalities=FALSE)

fMultiple  <- iterativePrincipalAxis(RU, nFactors=nFactors,
communalities="multiple")
fMultiple