Semiparametric Mean–Covariance Regression Analysis for Longitudinal Data

Efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure. Existing approaches usually focus on modeling the mean with specification of certain covariance structures, which may lead to inefficient or biased estimators of parameters in the mean if misspecification occurs. In this article, we propose a data-driven approach based on semiparametric regression models for the mean and the covariance simultaneously, motivated by the modified Cholesky decomposition. A regression spline-based approach using generalized estimating equations is developed to estimate the parameters in the mean and the covariance. The resulting estimators for the regression coefficients in both the mean and the covariance are shown to be consistent and asymptotically normally distributed. In addition, the nonparametric functions in these two structures are estimated at their optimal rate of convergence. Simulation studies and a real data analysis show that the proposed approach yields highly efficient estimators for the parameters in the mean, and provides parsimonious estimation for the covariance structure. Supplemental materials for the article are available online.