bentlerParameters {nFactors} | R Documentation |
This function computes the Bentler and Yuan's (1996, 1998) LRT index
for the linear trend in eigenvalues of a covariance matrix.
The related chi^2 and p-value are also computed.
This function is generally called from the nBentler
function.
But it could be of interest to use it for graphing the linear trend function to
minimize and so to study it's behavior.
bentlerParameters(x, N, nFactors, log=TRUE, cor=TRUE, minPar=c(min(lambda) - abs(min(lambda)) + .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", ...)
x |
numeric: a vector of eigenvalues, a matrix of
correlations or of covariances or a data.frame of data |
N |
numeric: number of subjects. |
nFactors |
numeric: number of components to test. |
log |
logical: if TRUE the minimization is applied on the log values. |
cor |
logical: if TRUE computes eigenvalues from a correlation
matrix, else from a covariance matrix |
minPar |
numeric: minimums for the coefficient of the linear trend to minimize. |
maxPar |
numeric: maximums for the coefficient of the linear trend to minimize. |
resParx |
numeric: restriction on the α coefficient (x) to graph the function to minimize. |
resPary |
numeric: restriction on the β coefficient (y) to graph the function to minimize. |
graphic |
logical: if TRUE plot the minimized function "wireframe" ,
"contourplot" or "levelplot" . |
resolution |
numeric: resolution of the 3D graph (number of points from α and from β). |
typePlot |
character: plot the minimized function according to a 3D plot: "wireframe" ,
"contourplot" or "levelplot" . |
... |
variable: additionnal parameters from the "wireframe" ,
"contourplot" or "levelplot" lattice
functions. Also additionnal parameters for the eigenFrom function. |
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 use a minimized chi^2 solution. This strategy will be implemented in a future version of the nFactors package.
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. |
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/
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.
## ................................................ ## 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="")) # ........................................................