A goodness-of-fit test for the latent class model when expected frequencies are small

In this paper, a goodness-of-fit test for the latent class model is presented. The test uses only the limited information in the second-order marginal distributions from a set of k dichotomous variables, and it is intended for use when k is large and the sample size, n, is moderate or small. In that situation, a 2^k contingency table formed by the full cross-classification of k variables will be sparse in the sense that a high proportion of cell frequencies will be equal to zero or 1, and the chi-square approximation for traditional goodness-of-fit statistics such as the likelihood ratio will not be valid. The second-order marginal frequencies, which correspond to the bivariate distributions, are rarely sparse even when the joint frequencies have a high proportion of zero cells. Results from Monte Carlo experiments are presented that compare the rates of Type I and Type II errors for the proposed test to the rates for traditional goodness-of-fit tests. Results show that under commonly encountered conditions, a test of fit based on the limited information in the second-order marginals has a Type II error rate that is no higher than the error rate found for full-information test statistics, and that the test statistic given in this paper does not suffer from ill effects of sparseness in the joint frequencies.