Power comparisons for testing goodness of fit for constrained models on high-dimensional cross-classified tables

In applications of models for cross-classified categorical variables, goodness of fit (GOF) is commonly assessed by either Pearson's $X^2$ statistic or the likelihood ratio statistic with reference to an asymptotic chi-square approximation. However, if the number of cross-classified variables is large, difficulties are encountered in obtaining a valid test using the traditional statistics due to sparseness and dilution of power. Because of these difficulties, several asymptotic chi-square GOF statistics that focus on lower-order marginal distributions have been proposed. Another method that generally does not encounter the difficulties caused by sparseness and dilution is the likelihood ratio difference test of a nested model. In this paper, Monte Carlo simulations are used to compare the power of a test of fit on lower-order marginals to the power of a test using the likelihood ratio difference statistic for detecting lack of fit in a constrained latent variable model. Results show that the likelihood ratio difference test has substantially higher power. We recommend using a statistic based on lower-order marginals to test the unconstrained model before applying the likelihood ratio difference test. Then, if the general model has an adequate fit, we recommend testing the constrained model using a likelihood ratio difference statistic.