Likelihood Ratio Tests With Three-Way Tables

• Published in: Journal of the American Statistical Association Vol. 105; no. 490; pp. 740 - 749
• Main Authors: Cheng, Philip E., Liou, Michelle, Aston, John A. D.
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.06.2010; American Statistical Association; Taylor & Francis Ltd
• Subjects: Applications; Breslow-Day test; Cochran-Mantel-Haenszel test; Entropy; Estimators; Exact sciences and technology; General topics; Inference; Interval estimators; Mantels; Mathematics; Multivariate analysis; Mutual information; Parametric inference; Pearson test; Probability and statistics; Ratio test; Ratios; Sciences and techniques of general use; Social interaction; Statistical variance; Statistics; Tests; Theory and Methods; Three-way interaction; Invariant; Cochran-Mantel-Haenszel test; Decomposition method; Non parametric estimation; Three-way interaction; Statistical association; Decomposition; Statistical estimation; Multivariate analysis; Statistical method; Hypothesis test; Statistical test; Pearson test; Relative entropy; Likelihood ratio test; Independence test; Contingency table; Conditional test; Omnibus test; Breslow-Day test; Mutual information
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/jasa.2010.tm09061

Likelihood ratio (LR) tests for association and for interaction are examined for three-way contingency tables, particularly the widely used 2 × 2 × K table. Mutual information identities are used to characterize the information decomposition and the logical relationship between the omnibus LR test for conditional independence across K strata and its two independent components, LR tests for interaction and for uniform association. The latter two tests are logically connected to formulating a natural two-step test for conditional independence. The proposed two-step test with reduced nominal levels is suggested instead of the Breslow-Day test and the Cochran-Mantel-Haenszel test. This yields efficient interval estimation for both the interaction parameter and the common odds ratio compared with using the Mantel-Haenszel estimate. This allows the development of power analysis for testing general hypotheses of varied interactions, using an invariant Pythagorean law of relative entropy.