GOODNESS-OF-FIT TESTING FOR LATENT CLASS MODELS

Latent class models with sparse contingency tables can present problem s for model comparison and selection, because under these conditions t he distributions of goodness-of-fit indices are often unknown. This ca uses inaccuracies both in hypothesis testing and in model comparisons based on normed indices. In order to assess the extent of this problem , we carried out a simulation investigating the distributions of the l ikelihood ratio statistic G2, the Pearson statistic X2, and a new good ness-of-fit index suggested by Read and Cressie (1988). There were sub stantial deviations between the expectation of the chi-squared distrib ution and the means of the G2 and Read and Cressie distributions. In g eneral, the mean of the distribution of a statistic was closer to the expectation of the chi-squared distribution when the average cell expe ctation was large. there were fewer indicator items, and the latent cl ass measurement parameters were less extreme. It was found that the me an of the X2 distribution is generally closer to the expectation of th e chi-squared distribution than are the means of the other two indices we examined, but the standard deviation of the X2 distribution is con siderably larger than that of the other two indices and larger than th e standard deviation of the chi-squared distribution. We argue that a possible solution is to forgo reliance on theoretical distributions fo r expectations and quantiles of goodness-of-fit statistics. Instead, M onte Carlo sampling (Noreen, 1989) can be used to arrive at an empiric al central or noncentral distribution.