% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/nBentler.r
\name{nBentler}
\alias{nBentler}
\title{Bentler and Yuan's Procedure to Determine the Number of Components/Factors}
\usage{
nBentler(x, N, log = TRUE, alpha = 0.05, cor = TRUE,
  details = TRUE, minPar = c(min(lambda) - abs(min(lambda)) + 0.001,
  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}
}
\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.}
}
\description{
This function computes the Bentler and Yuan's indices for determining the
number of components/factors to retain.
}
\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.
}
\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=""))
# ........................................................

}
\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.
}
\seealso{
\code{\link{nBartlett}}, \code{\link{bentlerParameters}}
}
\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}
\cr \cr David Magis \cr Departement de mathematiques \cr Universite de Liege
\cr \email{David.Magis@ulg.ac.be}
}
\keyword{multivariate}