Informative Estimation and Selection of Correlation Structure for Longitudinal Data

Abstract

Identifying an informative correlation structure is important in improving estimation efficiency for longitudinal data. We approximate the empirical estimator of the correlation matrix by groups of known basis matrices that represent different correlation structures, and transform the correlation structure selection problem to a covariate selection problem. To address both the complexity and the informativeness of the correlation matrix, we minimize an objective function that consists of two parts: the difference between the empirical information and a model approximation of the correlation matrix, and a penalty that penalizes models with too many basis matrices. The unique feature of the proposed estimation and selection of correlation structure is that it does not require the specification of the likelihood function, and therefore it is applicable for discrete longitudinal data. We carry out the proposed method through a groupwise penalty strategy, which is able to identify more complex structures. The proposed method possesses the oracle property and selects the true correlation structure consistently. In addition, the estimator of the correlation parameters follows a normal distribution asymptotically. Simulation studies and a data example confirm that the proposed method works effectively in estimating and selecting the true structure in finite samples, and it enables improvement in estimation efficiency by selecting the true structures.

KEY WORDS:

• Correlation structure
• Longitudinal data
• Oracle property
• Quadratic inference function

Acknowledgments

Zhou’s research was supported by the National Science Foundation (DMS-0906665). Qu’s research was supported by the National Science Foundation (DMS-0906660). The authors are grateful to the reviewers, the associate editor, and the coeditors for their insightful comments and suggestions that have significantly improved the article.