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)) + 0.001,
0.001), maxPar = c(max(lambda), lm(lambda ~
I(length(lambda):1))$coef[2]), ...)
Arguments
x |
numeric: a |
N |
numeric: number of subjects. |
log |
logical: if |
alpha |
numeric: statistical significance level. |
cor |
logical: if |
details |
logical: if |
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 |
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:
(1) \qquad \qquad H_k: \lambda_{k+i} = \alpha + \beta x_i, (i = 1,
\ldots, q)
The solution of the following simultaneous equations is needed to find
(\alpha, \beta) \in
(2) \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
and \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
The solution to this system of equations was implemented by minimizing the
following equation:
(3) \qquad \qquad (\alpha, \beta) \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)\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\}}
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
David Magis
Departement de mathematiques
Universite de Liege
David.Magis@ulg.ac.be
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
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=""))
# ........................................................