Information Ratio Test for Model Misspecification in Quasi-Likelihood Inference

Abstract

In this article, we focus on the circumstances in quasi-likelihood inference that the estimation accuracy of mean structure parameters is guaranteed by correct specification of the first moment, but the estimation efficiency could be diminished due to misspecification of the second moment. We propose an information ratio (IR) statistic to test for model misspecification of the variance/covariance structure through a comparison between two forms of information matrix: the negative sensitivity matrix and the variability matrix. We establish asymptotic distributions of the proposed IR test statistics. We also suggest an approximation to the asymptotic distribution of the IR statistic via a perturbation resampling method. Moreover, we propose a selection criterion based on the IR test to select the best fitting variance/covariance structure from a class of candidates. Through simulation studies, it is shown that the IR statistic provides a powerful statistical tool to detect different scenarios of misspecification of the variance/covariance structures. In addition, the IR test as well as the proposed model selection procedure shows substantial improvement over some of the existing statistical methods. The IR-based model selection procedure is illustrated by analyzing the Madras Longitudinal Schizophrenia data. Appendices are included in the supplemental materials, which are available online.

KEY WORDS:

• Correlation structure selection
• Estimating functions
• Generalized estimating equations
• Homoscedasticity
• Information unbiasedness
• Perturbation

ACKNOWLEDGMENTS

The authors thank the two anonymous referees, the academic editor, and editor for their valuable comments and constructive suggestions that have led to an improvement in the theory presented in the article. The authors are grateful to Dr. Richard J. Cook and Dr. Grace Y. Yi from the University of Waterloo for valuable comments on this article. The authors appreciate the constructive suggestions from Dr. Tianxi Cai of the Harvard School of Public Health.

The work is supported by grants from the U.S. National Science Foundation to the second author (DMS 0904177) and Natural Science and Engineering Research Council of Canada to the third author.