Factors Influencing the Mixture Index of Model Fit in Contingency Tables Showing Independence

Abstract

This study: (a) evaluated several competing techniques for dealing with sampling zeros for the two-point mixture model index, * π , in contingency tables when the independence assumption holds; and (b) investigated the performance of the estimate, * π , in various combinations of conditions, as a function of different sizes of tables, different marginal distributions and different sample sizes. Furthermore, the standard error of π ∗ estimated by use of a method proposed by Rudas, Clogg and Lindsay (1994) and the “true” standard error based on empirical simulations in various scenarios especially when encountering small sample sizes and π ∗ close to 0 were compared. These goals were achieved by Monte Carlo methods that simulate a variety of scenarios and then were applied to two real data examples. BACKGROUND AND DEVELOPMENT Traditional methods for evaluating contingency table models based on chi square statistics or quantities derived from them are not appealing in many applied research settings. According to Rudas (1998), “First, when the model is not true, a comparison of the data to what could only be expected if it were is of very little meaning; second, the actual distribution of the statistic may be very different from the reference distribution if some of the underlying assumptions are violated.” In addition, conventional methods are sensitive to sample size; often a model is rejected when fitted to a large data set even though the model may represent a reasonable summary of the data for practical purposes. In sharp contrast to chi-squared tests of fit methods, which rely heavily on size of the table, sample size and actual true probabilities, the mixture index of fit proposed by Rudas,