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

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables of arbitrary dimensions. 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 consistent and asymptotically normal estimator. We show that when r is small (r = 2) the proposed statistics have more accurate empirical Type I errors and are more powerful than Pearson's X2 for a widely used item response model. Also, we show that the proposed statistics (but not X2 even for the maximum likelihood estimate) are asymptotically chi-squared under the null hypothesis when applied to subtables