Limited- and Full-Information Estimation and Goodness-of-Fit Testing in 2~n Contingency Tables: A Unified Framework

摘要

High-dimensional contingency tables tend to be sparse, and standard goodness-of-fit statistics such as X~2 cannot be used without pooling categories. As an improvement on arbitrary pooling, for goodness of fit of large 2~n contingency tables, we propose classes of quadratic form statistics based on the residuals of margins or multivariate moments up to order r. These classes of test statistics are asymptotically chi-squared distributed under the null hypothesis. Further, the marginal residuals are useful for diagnosing lack of fit of parametric models. We show that when r is small (r = 2,3), the proposed statistics have better small-sample properties and are asymptotically more powerful than X~2 for some useful multivariate binary models. Related to these test statistics is a class of limited-information estimators based on low-dimensional margins. We show that these estimators have high efficiency for one commonly used latent trait model for binary data.