% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/nBartlett.r
\name{nBartlett}
\alias{nBartlett}
\title{Bartlett, Anderson and Lawley Procedures to Determine the Number of Components/Factors}
\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}
}
\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.}
}
\description{
This function computes the Bartlett, Anderson and Lawley indices for determining the
number of components/factors to retain.
}
\details{
Note: the latex formulas are available only in the pdf version of this help file.

The hypothesis tested is: \cr

(1)  \eqn{\qquad \qquad H_k: \lambda_{k+1} = \ldots = \lambda_p} \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

(2) \eqn{\qquad \qquad V_k  =
  \prod\limits_{i = k + 1}^p
  \left\{ \frac{\displaystyle \lambda_i}{\frac{1}{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
}
\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=""))

}
\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}.New-York, NJ: Academic Press.

 Lawley, D. N. (1956). Tests of significance for the latent roots of covarianceand correlation matrix. \emph{Biometrika, 43}(1/2), 128-136.
}
\seealso{
\code{\link{plotuScree}}, \code{\link{nScree}}, \code{\link{plotnScree}}, \code{\link{plotParallel}}
}
\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}
}
\keyword{multivariate}