\name{generateStructure}
\alias{generateStructure}
\title{ Generate a Factor Structure Matrix.}

\description{
 The \code{generateStructure} function returns a \emph{mjc} factor structure matrix.
 The number of variables per major factor \emph{pmjc} is equal for each factor.
 The argument \emph{pmjc} must be divisible by \emph{nVar}.
 The arguments are strongly inspired from Zick and Velicer (1986, p. 435-436) methodology.
 }

\usage{
 generateStructure(var, mjc, pmjc, loadings, unique)
 }

\arguments{
  \item{var}{      numeric: number of variables}
  \item{mjc}{      numeric: number of major factors (factors with practical significance) }
  \item{pmjc}{     numeric: number of variables that load significantly on each major factor }
  \item{loadings}{ numeric: loadings on the significant variables on each major factor }
  \item{unique}{   numeric: loadings on the non significant variables on each major factor }
 }

\value{
  \item{values}{ numeric matrix: factor structure }
 }

\seealso{
 \code{\link{principalComponents}},
 \code{\link{iterativePrincipalAxis}},
 \code{\link{rRecovery}}
 }

\references{
  Zwick, W. R. and Velicer, W. F. (1986). Comparison of five rules for
   determining the number of components to retain. \emph{Psychological Bulletin,
   99}, 432-442.
 }

\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 inspired from Zwick and Velicer (1986, table 2, p. 437)
## ...................................................................
 unique=0.2; loadings=0.5
 zwick1 <- generateStructure(var=36, mjc=6, pmjc= 6, loadings=loadings,
                             unique=unique)
 zwick2 <- generateStructure(var=36, mjc=3, pmjc=12, loadings=loadings,
                             unique=unique)
 zwick3 <- generateStructure(var=72, mjc=9, pmjc= 8, loadings=loadings,
                             unique=unique)
 zwick4 <- generateStructure(var=72, mjc=6, pmjc=12, loadings=loadings,
                             unique=unique)
 sat=0.8
## ...................................................................
 zwick5 <- generateStructure(var=36, mjc=6, pmjc= 6, loadings=loadings,
                             unique=unique)
 zwick6 <- generateStructure(var=36, mjc=3, pmjc=12, loadings=loadings,
                             unique=unique)
 zwick7 <- generateStructure(var=72, mjc=9, pmjc= 8, loadings=loadings,
                             unique=unique)
 zwick8 <- generateStructure(var=72, mjc=6, pmjc=12, loadings=loadings,
                             unique=unique)
## ...................................................................

# nsubjects <- c(72, 144, 180, 360)
# require(psych)
# Produce an usual correlation matrix from a congeneric model
 nsubjects <- 72
 mzwick5   <- sim.structure(fx=as.matrix(zwick5), n=nsubjects)
 mzwick5$r

# Factor analysis: recovery of the factor structure
 iterativePrincipalAxis(mzwick5$model, nFactors=6,
                        communalities="ginv")$loadings
 iterativePrincipalAxis(mzwick5$r    , nFactors=6,
                        communalities="ginv")$loadings
 factanal(covmat=mzwick5$model,         factors=6)
 factanal(covmat=mzwick5$r    ,         factors=6)
 
# Number of components to retain
 eigenvalues  <- eigen(mzwick5$r)$values
 aparallel    <- parallel(var      = length(eigenvalues),
                          subject  = nsubjects,
                          rep      = 30,
                          quantile = 0.95,
                          model="components")$eigen$qevpea
 results <- nScree(x         = eigenvalues,
                   aparallel = aparallel)
 results$Components
 plotnScree(results)
 
# Number of factors to retain
 eigenvalues.fa  <- eigen(corFA(mzwick5$r))$values
 aparallel.fa    <- parallel(var      = length(eigenvalues.fa),
                             subject  = nsubjects,
                             rep      = 30,
                             quantile = 0.95,
                             model="factors")$eigen$qevpea
 results.fa <- nScree(x      = eigenvalues.fa,
                   aparallel = aparallel.fa,
                   model     ="factors")
 results.fa$Components
 plotnScree(results.fa)
# ......................................................
 }

\keyword{ multivariate }