nSeScree.rd
\name{nSeScree}
\alias{nSeScree}

\title{ Standard Error Scree and Coefficient of Determination Procedures to
Determine the Number of Components/Factors}

\description{
This function computes the \emph{seScree} (\eqn{S_{Y \bullet X}}) indices
(Zoski and Jurs, 1996) and the
coefficient of determination indices of Nelson (2005) \eqn{R^2} for determining the
number of components/factors to retain.
}

\usage{
nSeScree(x, cor=TRUE, model="components", details=TRUE, r2limen=0.75, ...)
}

\arguments{
\item{x}{       numeric: eigenvalues.}
\item{cor}{     logical: if \code{TRUE} computes eigenvalues from a correlation
matrix, else from a covariance matrix}
\item{model}{   character: \code{"components"} or \code{"factors"} }
\item{details}{ logical: if \code{TRUE} also returns details about the computation for each eigenvalue.}
\item{r2limen}{ numeric: criterion value retained for the coefficient of determination indices.}
\item{...}{     variable: additionnal parameters to give to the \code{eigenComputes}
and \code{cor} or \code{cov} functions}
}

\details{
The Zoski and Jurs \eqn{S_{Y \bullet X}} index is the standard error of the estimate
(predicted) eigenvalues by the regression from the \eqn{(k+1, \ldots, p)} subsequent
ranks of the eigenvalues. The standard error is computed as:

\sqrt{ \frac{(\lambda_k - \hat{\lambda}_k)^2} {p-2} } } \cr

A value of \eqn{1/p} is choosen as the criteria to determine the number of
components or factors to retain, \emph{p} corresponding to the number of
variables.

The Nelson \eqn{R^2} index is simply the multiple regresion coefficient of
determination for the \eqn{k+1, \ldots, p} eigenvalues.
Note that Nelson didn't give formal prescriptions for the criteria for this
index. He only suggested that a value of 0.75 or more must be considered. More
is to be done to explore adequate values.
}

\value{
\item{nFactors}{ numeric: number of components/factors retained by the seScree procedure. }
\item{details}{  numeric: matrix of the details for each index.}
}

\references{
Nasser, F. (2002). The performance of regression-based variations of the visual
scree for determining the number of common factors. \emph{Educational and
Psychological Measurement, 62(3)}, 397-419.

Nelson, L. R. (2005). Some observations on the scree test, and on coefficient
alpha. \emph{Thai Journal of Educational Research and Measurement, 3(1)}, 1-17.

Zoski, K. and Jurs, S. (1993). Using multiple regression to determine the
number of factors to retain in factor analysis. \emph{Multiple Linear Regression
Viewpoints, 20}(1), 5-9.

Zoski, K. and Jurs, S. (1996). An objective counterpart to the visuel scree
test for factor analysis: the standard error scree. \emph{Educational and
Psychological Measurement, 56}(3), 443-451.
}

\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/}
}

\seealso{
}

\examples{
## SIMPLE EXAMPLE OF SESCREE AND R2 ANALYSIS

data(dFactors)
eig      <- dFactors$Raiche$eigenvalues

results  <- nSeScree(eig)
results

plotuScree(eig, main=paste(results$nFactors[1], " or ", results$nFactors[2],
" factors retained by the sescree and R2 procedures",
sep=""))
}

\keyword{ multivariate }