\name{nBentler}
\alias{nBentler}

\title{ Bentler and Yuan's Procedure to Determine the Number of Components/Factors}

\description{
  This function computes the Bentler and Yuan's indices for determining the
  number of components/factors to retain.
 }
 

\usage{
 nBentler(x, N, log=TRUE, alpha=0.05, cor=TRUE, details=TRUE,
         minPar=c(min(lambda) - abs(min(lambda)) +.001, 0.001),
         maxPar=c(max(lambda),
                  lm(lambda ~ I(length(lambda):1))$coef[2]), ...)
 }


\arguments{
  \item{x}{          numeric: a \code{vector} of eigenvalues, a \code{matrix} of
                     correlations or of covariances or a \code{data.frame} of data}
  \item{N}{          numeric: number of subjects.}
  \item{log}{        logical: if \code{TRUE} does the maximization on the log values.}
  \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{minPar}{     numeric: minimums for the coefficient of the linear trend to maximize.}
  \item{maxPar}{     numeric: maximums for the coefficient of the linear trend to maximize.}
  \item{...}{        variable: additionnal parameters to give to the \code{cor} or
                     \code{cov} functions}
 }
 

\details{
  The implemented Bentler and Yuan's procedure must be used with care because
  the minimized function is not always stable, as Bentler and Yan (1996, 1998)
  already noted. In many
  cases, constraints must applied to obtain a solution,
  as the actual implementation did, but the user can modify
  these constraints.

  The hypothesis tested (Bentler and Yuan, 1996, equation 10) is: \cr \cr

 (1)  \eqn{\qquad \qquad H_k: \lambda_{k+i} = \alpha + \beta x_i, (i = 1, \ldots, q)} \cr

 The solution of the following simultaneous equations is needed to
 find \eqn{(\alpha, \beta) \in} \cr

 (2)  \eqn{\qquad \qquad f(x) = \sum_{i=1}^q \frac{ [ \lambda_{k+j} - N  \alpha + \beta x_j ]   x_j}{(\alpha + \beta x_j)^2}  = 0} \cr \cr
 and  \eqn{\qquad \qquad g(x) = \sum_{i=1}^q \frac{  \lambda_{k+j} - N  \alpha + \beta x_j   x_j}{(\alpha + \beta x_j)^2}  = 0} \cr

 The solution to this system of equations was implemented by minimizing the following equation: \cr

 (3)   \eqn{\qquad \qquad (\alpha, \beta) \in \inf{[h(x)]} = \inf{\log{[f(x)^2 + g(x)^2}}]} \cr

 The likelihood ratio test \eqn{LRT} proposed by Bentler and Yuan (1996, equation 7) follows a
 \eqn{\chi^2} probability distribution with \eqn{q-2} degrees of freedom and
 is equal to: \cr

 (4)  \eqn{\qquad \qquad LRT = N(k - p)\left\{ {\ln \left( {{n \over N}} \right) + 1} \right\}
       - N\sum\limits_{j = k + 1}^p {\ln \left\{ {{{\lambda _j } \over {\alpha  + \beta x_j }}} \right\}}
       + n\sum\limits_{j = k + 1}^p {\left\{ {{{\lambda _j } \over {\alpha  + \beta x_j }}} \right\}} } \cr

 With \eqn{p} beeing the number of eigenvalues, \eqn{k} the number of eigenvalues to test,
 \eqn{q} the \eqn{p-k} remaining eigenvalues, \eqn{N} the sample size, and \eqn{n = N-1}.
 Note that there is an error in the Bentler and Yuan equation, the variables
 \eqn{N} and \eqn{n} beeing inverted in the preceeding equation 4.

 A better strategy proposed by Bentler an Yuan (1998) is to used a minimized
 \eqn{\chi^2} solution. This strategy will be implemented in a future version
 of the \pkg{nFactors} package.
 }


\value{
  \item{nFactors}{ numeric: vector of the number of factors retained by the
                   Bentler and Yuan's procedure. }
  \item{details}{  numeric: matrix of the details of the computation.}
 }
 
\references{
 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.
 }

\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/} \cr \cr
    David Magis \cr
    Departement de mathematiques \cr
    Universite de Liege \cr
    \email{David.Magis@ulg.ac.be}
 }

\seealso{
 \code{\link{nBartlett}},
 \code{\link{bentlerParameters}}
 }

\examples{
## ................................................
## SIMPLE EXAMPLE OF THE BENTLER AND YUAN PROCEDURE

# Bentler (1996, p. 309) Table 2 - Example 2 .............
n=649
bentler2<-c(5.785, 3.088, 1.505, 0.582, 0.424, 0.386, 0.360, 0.337, 0.303,
            0.281, 0.246, 0.238, 0.200, 0.160, 0.130)

results  <- nBentler(x=bentler2, N=n)
results

plotuScree(x=bentler2, model="components",
    main=paste(results$nFactors,
    " factors retained by the Bentler and Yuan's procedure (1996, p. 309)",
    sep=""))
# ........................................................

# Bentler (1998, p. 140) Table 3 - Example 1 .............
n        <- 145
example1 <- c(8.135, 2.096, 1.693, 1.502, 1.025, 0.943, 0.901, 0.816, 0.790,
              0.707, 0.639, 0.543,
              0.533, 0.509, 0.478, 0.390, 0.382, 0.340, 0.334, 0.316, 0.297,
              0.268, 0.190, 0.173)
              
results  <- nBentler(x=example1, N=n)
results

plotuScree(x=example1, model="components",
   main=paste(results$nFactors,
   " factors retained by the Bentler and Yuan's procedure (1998, p. 140)",
   sep=""))
# ........................................................
 }

\keyword{ multivariate }