Limited Information Goodness-of-fit Testing in Multidimensional Contingency Tables

Abstract

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order r . They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when r is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson’s X 2 . For this model in nonsparse situations, the proposed statistics are also more powerful than X 2 . In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.