##### https://github.com/cran/nFactors
Tip revision: 923d0cc
nBartlett.rd
\name{nBartlett}
\alias{nBartlett}

\title{ Bartlett, Anderson and Lawley Procedures to Determine the Number of Components/Factors}

\description{
This function computes the Bartlett, Anderson and Lawley indices for determining the
number of components/factors to retain.
}

\usage{
nBartlett(x, N, alpha=0.05, cor=TRUE, details=TRUE, correction=TRUE, ...)
}

\arguments{
\item{x}{          numeric: a \code{vector} of eigenvalues, a \code{matrix} of
correlations or of covariances or a \code{data.frame} of data (eigenFrom)}
\item{N}{          numeric: number of subjects}
\item{alpha}{      numeric: statistical significance level }
\item{cor}{        logical: if \code{TRUE} computes eigenvalues from a correlation
matrix, else from a covariance matrix}
\item{details}{    logical: if \code{TRUE} also returns detains about the
computation for each eigenvalue}
\item{correction}{ logical: if \code{TRUE} uses a correction for the degree
of freedom after the first eigenvalue}
\item{...}{        variable: additionnal parameters to give to the \code{cor} or
\code{cov} functions}
}

\details{
The hypothesis tested is: \cr \cr

This hypothesis is verified by the application of different version of a
\eqn{\chi^2} test with different values for the degrees of freedom.
Each of these tests shares the compution of a \eqn{V_k} value: \cr

\prod\limits_{i = k + 1}^p {\left\{ {{{\lambda _i }
\over {{\raise0.7ex\hbox{$1$} \!\mathord{\left/
{\vphantom {1 q}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$q$}}\sum\limits_{i = k + 1}^p {\lambda _i } }}} \right\}}
}

\eqn{p} is the number of eigenvalues, \eqn{k} the number of eigenvalues to test,
and \eqn{q} the \eqn{p-k} remaining eigenvalues. \eqn{n} is equal to the sample size
minus 1 (\eqn{n = N-1}). \cr

The Anderson statistic is distributed as a \eqn{\chi^2} with \eqn{(q + 2)(q - 1)/2} degrees
of freedom and is equal to: \cr

(3) \eqn{\qquad \qquad - n\log (V_k ) \sim \chi _{(q + 2)(q - 1)/2}^2 } \cr

An improvement of this statistic from Bartlett (Bentler, and Yuan, 1996, p. 300;
Horn and Engstrom, 1979, equation 8) is distributed as a \eqn{\chi^2}
with \eqn{(q)(q - 1)/2} degrees of freedom and is equal to: \cr

(4) \eqn{\qquad \qquad - \left[ {n - k - {{2q^2 q + 2} \over {6q}}}
\right]\log (V_k ) \sim \chi _{(q + 2)(q - 1)/2}^2 }  \cr

Finally, Anderson (1956) and James (1969) proposed another statistic. \cr

(5) \eqn{\qquad \qquad - \left[ {n - k - {{2q^2 q + 2} \over {6q}}
+ \sum\limits_{i = 1}^k {{{\bar \lambda _q^2 } \over {\left( {\lambda _i
- \bar \lambda _q } \right)^2 }}} } \right]\log (V_k ) \sim \chi _{(q + 2)(q - 1)/2}^2 } \cr

Bartlett (1950, 1951) proposed a correction to the degrees of freedom of these \eqn{\chi^2} after the
first significant test: \eqn{(q+2)(q - 1)/2}. \cr
}

\value{
\item{nFactors}{ numeric: vector of the number of factors retained by the
Bartlett, Anderson and Lawley procedures. }
\item{details}{  numeric: matrix of the details for each index.}
}

\references{

Anderson, T. W. (1963). Asymptotic theory for principal component analysis.
\emph{Annals of Mathematical Statistics, 34}, 122-148.

Bartlett, M. S. (1950). Tests of significance in factor analysis.
\emph{British Journal of Psychology, 3}, 77-85.

Bartlett, M. S. (1951). A further note on tests of significance. \emph{British
Journal of Psychology, 4}, 1-2.

Bentler, P. M. and Yuan, K.-H. (1996). Test of linear trend in eigenvalues of
a covariance matrix with application to data analysis.
\emph{British Journal of Mathematical and Statistical Psychology, 49}, 299-312.

Bentler, P. M. and Yuan, K.-H. (1998). Test of linear trend in the smallest
eigenvalues of the correlation matrix. \emph{Psychometrika, 63}(2), 131-144.

Horn, J. L. and Engstrom, R. (1979). Cattell's scree test in relation to
Bartlett's chi-square test and other observations on the number of factors
problem. \emph{Multivariate Behavioral Reasearch, 14}(3), 283-300.

James, A. T. (1969). Test of equality of the latent roots of the covariance
matrix. \emph{In} P. K. Krishna (Eds): \emph{Multivariate analysis, volume 2}.

Lawley, D. N. (1956). Tests of significance for the latent roots of covariance
and correlation matrix. \emph{Biometrika, 43}(1/2), 128-136.
}

\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 A BARTLETT PROCEDURE

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

results  <- nBartlett(x=eig, N= 100, alpha=0.05, details=TRUE)
results

plotuScree(eig, main=paste(results$nFactors[1], ", ", results$nFactors[2], " or ",
results\$nFactors[3],
" factors retained by the LRT procedures",
sep=""))
}

\keyword{ multivariate }