bentlerParameters function

Bentler and Yuan's Computation of the LRT Index and the Linear Trend Coefficients

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 use for graphing the linear trend function and to study it's behavior.

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 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.
• maxPar: numeric: maximums for the coefficient of the linear trend.
• resParx: numeric: restriction on the $\alpha$ coefficient (x) to graph the function to minimize.
• resPary: numeric: restriction on the $\beta$ coefficient (y) to graph the function to minimize.
• graphic: logical: if TRUE plots the minimized function "wireframe", "contourplot" or "levelplot".
• resolution: numeric: resolution of the 3D graph (number of points from $\alpha$ and from $\beta$).
• typePlot: character: plots 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.

Returns

• 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.

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) c("$\\qquad \\qquad H_k: \\lambda_{k+i} = \\alpha + \\beta x_i, (i = 1,\n$", "$\\ldots, q)$")

The solution of the following simultaneous equations is needed to find $(\alpha, \beta) \in$

(2) c("\\qquad \\qquad f(x) = \\sum_{i=1}^q \\frac{ [ \\lambda_{k+j} - N \\alpha\n", "+ \\beta x_j ] x_j}{(\\alpha + \\beta x_j)^2} = 0")

and c("\\qquad \\qquad g(x) = \\sum_{i=1}^q \\frac{ \\lambda_{k+j} - N \\alpha +\n", "\\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) c("\\qquad \\qquad (\\alpha, \\beta) \\in \\inf{[h(x)]} = \\inf{\\log{[f(x)^2\n", "+ 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) c("\\qquad \\qquad LRT = N(k - p)\\left\\{ {\\ln \\left( {{n \\over N}}\n", "\\right) + 1} \\right\\} - N\\sum\\limits_{j = k + 1}^p {\\ln \\left\\{ {{{\\lambda\n", "_j } \\over {\\alpha + \\beta x_j }}} \\right\\}} + n\\sum\\limits_{j = k + 1}^p\n", "${\\left\\{ {{{\\lambda _j } \\over {\\alpha + \\beta x_j }}} \\right\\}}$")

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 use a minimized $\chi^2$ solution. This strategy will be implemented in a future version of the 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,  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=""))
# ........................................................

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, nBentler

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