bentlerParameters {nFactors} | R Documentation |
Bentler and Yuan's Computation of the LRT Index and the Linear Trend Coefficients
Description
This function computes the Bentler and Yuan's (1996, 1998) LRT index
for the linear trend in eigenvalues of a covariance matrix. The related
and p-value are also computed. This function is
generally called from the
nBentler
function. But it could be of use
for graphing the linear trend function and to study it's behavior.
Usage
bentlerParameters(x, N, nFactors, log = TRUE, cor = TRUE,
minPar = c(min(lambda) - abs(min(lambda)) + 0.001, 0.001),
maxPar = c(max(lambda), lm(lambda ~ I(length(lambda):1))$coef[2]),
resParx = c(0.01, 2), resPary = c(0.01, 2), graphic = TRUE,
resolution = 30, typePlot = "wireframe", ...)
Arguments
x |
numeric: a |
N |
numeric: number of subjects. |
nFactors |
numeric: number of components to test. |
log |
logical: if |
cor |
logical: if |
minPar |
numeric: minimums for the coefficient of the linear trend. |
maxPar |
numeric: maximums for the coefficient of the linear trend. |
resParx |
numeric: restriction on the |
resPary |
numeric: restriction on the |
graphic |
logical: if |
resolution |
numeric: resolution of the 3D graph (number of points from
|
typePlot |
character: plots the minimized function according to a 3D
plot: |
... |
variable: additionnal parameters from the |
Details
The implemented Bentler and Yuan's procedure must be used with care because the minimized function is not always stable. In many cases, constraints must applied to obtain a solution. The actual implementation did, but the user can modify these constraints.
The hypothesis tested (Bentler and Yuan, 1996, equation 10) is:
(1)
The solution of the following simultaneous equations is needed to find
(2)
and
The solution to this system of equations was implemented by minimizing the
following equation:
(3)
The likelihood ratio test proposed by Bentler and Yuan (1996,
equation 7) follows a
probability distribution with
degrees of freedom and is equal to:
(4)
With beeing the number of eigenvalues,
the number of
eigenvalues to test,
the
remaining eigenvalues,
the sample size, and
. Note that there is an error in the
Bentler and Yuan equation, the variables
and
beeing inverted
in the preceeding equation 4.
A better strategy proposed by Bentler an Yuan (1998) is to use a minimized
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, details=TRUE)
results
# Two different figures to verify the convergence problem identified with
# the 2th component
bentlerParameters(x=bentler2, N=n, nFactors= 2, graphic=TRUE,
typePlot="contourplot",
resParx=c(0,9), resPary=c(0,9), cor=FALSE)
bentlerParameters(x=bentler2, N=n, nFactors= 4, graphic=TRUE, drape=TRUE,
resParx=c(0,9), resPary=c(0,9),
scales = list(arrows = FALSE) )
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, details=TRUE)
results
# Two different figures to verify the convergence problem identified with
# the 10th component
bentlerParameters(x=example1, N=n, nFactors= 10, graphic=TRUE,
typePlot="contourplot",
resParx=c(0,0.4), resPary=c(0,0.4))
bentlerParameters(x=example1, N=n, nFactors= 10, graphic=TRUE, drape=TRUE,
resParx=c(0,0.4), resPary=c(0,0.4),
scales = list(arrows = FALSE) )
plotuScree(x=example1, model="components",
main=paste(results$nFactors,
" factors retained by the Bentler and Yuan's procedure (1998, p. 140)",
sep=""))
# ........................................................