\name{eigenBootParallel}
\alias{eigenBootParallel}
\title{ Bootstrapping of the Eigenvalues From a Data Frame}

\description{
 The \code{eigenBootParallel} function samples observations from a \code{data.frame}
 to produces correlation or covariance matrix from which eigenvalues are computed. The
 function returns statistics about these bootstrapped eigenvalues. Their means
 or their quantile could be used later to replace the eigenvalues inputed to
 a parallel analysis.  The \code{eigenBootParallel} can also computes random eigenvalues
 from empirical data by columns permutation (Buja and Eyuboglu, 1992).
 }

\usage{
 eigenBootParallel(x, quantile=0.95, nboot=30, option="permutation",
                   cor=TRUE, model="components", ...)
 }

\arguments{
  \item{x}{           data.frame: data from which a correlation matrix will be obtained}
  \item{quantile}{    numeric: eigenvalues quantile that will be reported }
  \item{nboot}{       numeric: number of bootstrap samples }
  \item{option}{      character: \code{"permutation"} or \code{"bootstrap"}}
  \item{cor}{         logical: if \code{TRUE} computes eigenvalues from a correlation
                      matrix, else from a covariance matrix (\code{eigenComputes})}
  \item{model}{       character: bootstraps from a principal component analysis
                      (\code{"components"}) or from a factor analysis (\code{"factors"}) }
  \item{...}{         variable: additionnal parameters to give to the \code{cor} or
                      \code{cov} functions}
 }

\value{
  \item{values}{ data.frame: mean, median, quantile, standard deviation,
                 minimum and maximum of bootstrapped eigenvalues }
 }

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

\references{
  Buja, A. and Eyuboglu, N. (1992). Remarks on parallel analysis.
   \emph{Multivariate Behavioral Research, 27}(4), 509-540.

  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 from the iris data
 eigenvalues <- eigenComputes(x=iris[,-5])
 
# Permutation parallel analysis distribution
 aparallel   <- eigenBootParallel(x=iris[,-5], quantile=0.95)$quantile

# Number of components to retain
 results     <- nScree(x = eigenvalues, aparallel = aparallel)
 results$Components
 plotnScree(results)
# ......................................................

# ......................................................
# Bootstrap distributions stude of the eigenvalues from iris data
# with different correlation methods
 eigenBootParallel(x=iris[,-5],quantile=0.05,
                   option="bootstrap",method="pearson")
 eigenBootParallel(x=iris[,-5],quantile=0.05,
                   option="bootstrap",method="spearman")
 eigenBootParallel(x=iris[,-5],quantile=0.05,
                   option="bootstrap",method="kendall")
 }

\keyword{ multivariate }