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parallel.Rd
\name{parallel}
\alias{parallel}

\title{ Parallel Analysis of a Correlation Matrix}

\description{
This function gives the distribution of the eigenvalues of correlation
matrices of random uncorrelated standardized normal variables. The mean
and a selected centile of this distribution are returned.
}

\usage{
parallel(subject = 100,
var     = 10,
rep     = 100,
cent    = 0.05)
}

\arguments{
\item{subject}{numeric: Nmber of subjects (default is 100)}
\item{var}{    numeric: Number of variables (default is 10) }
\item{rep}{    numeric: Number of replications of the correlation matrix (default is 100)}
\item{cent}{   numeric: Centile of the distribution on which the decision is made (default is 0.05)}
}

\details{
Note that if the decision is based on a centile value rather than on the mean, care must
be taken with the number of replications (\emph{rep}). In fact, the smaller the centile (\emph{cent}),
the bigger the number of replications.
}

\value{
\item{eigen}{         Data frame consisting of mean and the centile of the eigenvalues distribution }
\item{eigen$mevpea}{ Mean of the eigenvalues distribution} \item{eigen$sevpea}{  Standard deviation of the eigenvalues distribution}
\item{eigen$qevpea}{ Centile of the eigenvalues distribution} \item{eigen$sqevpea}{ Standard error of the centile of the eigenvalues distribution}
\item{subject}{       Number of subjects}
\item{variables}{     Number of variables}
\item{centile}{       Selected centile}
Otherwise, returns a summary of the parallel analysis.
}

\references{
Drasgow, F. and Lissak, R. (1983) Modified parallel analysis: a procedure for
examining the latent dimensionality of dichotomously
scored item responses. \emph{Journal of Applied Psychology, 68}(3), 363-373.

Hoyle, R. H. and Duvall, J. L. (2004). Determining the number of factors in
exploratory and confirmatory factor analysis.
In D. Kaplan (Ed.): \emph{The Sage handbook of quantitative methodology for
the social sciences}. Thousand Oaks, CA: Sage.

Horn, J. L. (1965). A rationale and test of the number of factors in factor
analysis. \emph{Psychometrika, 30}, 179-185.
}

\author{
Gilles Raiche, Universite du Quebec a Montreal
\email{raiche.gilles@uqam.ca}, \url{http://www.er.uqam.ca/nobel/r17165/}
}

\seealso{
}

\examples{
## SIMPLE EXAMPLE OF A PARALLEL ANALYSIS
## OF A CORRELATION MATRIX WITH ITS PLOT
data(dFactors)
eig      <- dFactors$Raiche$eigenvalues
subject  <- dFactors$Raiche$nsubjects
var      <- length(eig)
rep      <- 100
cent     <- 0.95
results  <- parallel(subject,var,rep,cent)

results

## IF THE DECISION IS BASED ON THE CENTILE USE qevpea INSTEAD
## OF mevpea ON THE FIRST LINE OF THE FOLLOWING CALL
plotuScree(eig,
main = "Parallel Analysis"
)
lines(1:var,
results$eigen$qevpea,
type="b",
col="green"
)

## ANOTHER SOLUTION IS SIMPLY TO
plotParallel(results)

## RESULTS
# $eigen # mevpea sevpea qevpea sevpea.1 # V1 1.5421626 0.09781869 1.4037201 0.020670924 # V2 1.3604323 0.05728471 1.2768656 0.012105332 # V3 1.2249034 0.04704870 1.1482431 0.009942272 # V4 1.1189148 0.03662555 1.0605407 0.007739666 # V5 1.0221635 0.04048780 0.9599296 0.008555832 # V6 0.9318382 0.04053704 0.8647949 0.008566237 # V7 0.8381154 0.04026090 0.7758708 0.008507883 # V8 0.7493151 0.04729122 0.6727706 0.009993521 # V9 0.6568985 0.04664676 0.5756055 0.009857334 # V10 0.5552561 0.04942935 0.4800394 0.010445348 #$subject
# [1] 100

# $variables # [1] 10 #$centile
# [1] 0.05

# attr(,"class")
# [1] "parallel"

}

\keyword{ multivariate }