Generalized linear mixed models (GLMMs) are commonly used to analyze longitudinal categorical data. In these models, we typically assume that the random effects covariance matrix is constant across the subject and is restricted because of its high dimensionality and its positive definiteness. However, the covariance matrix may differ by measured covariates in many situations, and ignoring this heterogeneity can result in biased estimates of the fixed effects. In this paper, we propose a heterogenous random effects covariance matrix, which depends on covariates, obtained using the modified Cholesky decomposition. This decomposition results in parameters that can be easily modeled without concern that the resulting estimator will not be positive definite. The parameters have a sensible interpretation. We analyze metabolic syndrome data from a Korean Genomic Epidemiology Study using our proposed model.
Bibliographical noteFunding Information:
The authors are grateful to three referees for many helpful comments. For the corresponding author Jae Keun Yoo, this work was supported by Basic Science Research Program through the National Research Foundation of Korea (KRF) funded by the Ministry of Education, Science and Technology ( 2011-0005581 ).
- Cholesky decomposition
- Longitudinal data