% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/iterativePrincipalAxis.r
\name{iterativePrincipalAxis}
\alias{iterativePrincipalAxis}
\title{Iterative Principal Axis Analysis}
\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{
values       numeric: variance of each component

varExplained numeric: variance explained by each component

varExplained numeric: cumulative variance explained by each component

loadings     numeric: loadings of each variable on each component

iterations   numeric: maximum number of iterations to obtain a solution

tolerance    numeric: minimal difference in the estimated communalities after a given iteration
}
\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
 based on the sum of loadings
 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 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
rRecovery(RU,fMultiple$loadings, diagCommunalities=FALSE)
# .......................................................

}
\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{principalAxis}}, \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}
\cr \cr David Magis \cr Departement de mathematiques \cr Universite de Liege
\cr \email{David.Magis@ulg.ac.be}
}
\keyword{multivariate}