nBentler {nFactors}R Documentation

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

x numeric: a vector of eigenvalues, a matrix of correlations or of covariances or a data.frame of data
N numeric: number of subjects.
log logical: if TRUE does the maximization on the log values.
alpha numeric: statistical significance level.
cor logical: if TRUE computes eigenvalues from a correlation matrix, else from a covariance matrix
details logical: if TRUE also return detains about the computation for each eigenvalues.
minPar numeric: minimums for the coefficient of the linear trend to maximize.
maxPar numeric: maximums for the coefficient of the linear trend to maximize.
... variable: additionnal parameters to give to the cor or cov functions

Details

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

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

(1) qquad qquad H_k: λ_{k+i} = α + β x_i, (i = 1, ..., q)

The solution of the following simultaneous equations is needed to find (α, β) in

(2) qquad qquad f(x) = sum_{i=1}^q frac{ [ λ_{k+j} - N α + β x_j ] x_j}{(α + β x_j)^2} = 0

and qquad qquad g(x) = sum_{i=1}^q frac{ λ_{k+j} - N α + β x_j x_j}{(α + β x_j)^2} = 0

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

(3) qquad qquad (α, β) in inf{[h(x)]} = inf{log{[f(x)^2 + g(x)^2}}]

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

(4) qquad qquad LRT = N(k - p)<=ft{ {ln ( {{n over N}} ) + 1} right} - Nsumlimits_{j = k + 1}^p {ln <=ft{ {{{λ _j } over {α + β x_j }}} right}} + nsumlimits_{j = k + 1}^p {<=ft{ {{{λ _j } over {α + β x_j }}} right}}

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

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

Value

nFactors numeric: vector of the number of factors retained by the Bentler and Yuan's procedure.
details numeric: matrix of the details of the computation.

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca, http://www.er.uqam.ca/nobel/r17165/

David Magis
Research Group of Quantitative Psychology and Individual Differences
Katholieke Universiteit Leuven
David.Magis@psy.kuleuven.be, http://ppw.kuleuven.be/okp/home/

References

Bentler, P. M. and Yuan, K.-H. (1996). Test of linear trend in eigenvalues of a covariance matrix with application to data analysis. 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. Psychometrika, 63(2), 131-144.

See Also

nBartlett, 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=""))
# ........................................................
 

[Package nFactors version 2.3.1 Index]