Regression Analysis When Covariates Are Regression Parameters of A Random Effects Model for Observed Longitudinal Measurements

Abstract

We consider regression analysis when covariate variables are the underlying regression coefficients of another linear mixed model. A naive approach is to use each subject's repeated measurements, which follow the linear mixed model, and obtain subject-specific estimated coefficients to replace the covariate variables. However, directly replacing the unobserved covariates in the primary regression by these estimated coefficients may result in a significantly biased estimator. The aforementioned problem can be evaluated as a generalization of the classical additive error model where repeated measurements are stationary. To correct for these biases, we investigate a pseudo expected estimating equation (EEE) estimator, a regression calibration (RC) estimator and a refined version of the RC estimator. The pseudo EEE estimator is a pseudo maximum likelihood estimator if the complete data score is from a likelihood function, whereas the RC estimator directly replaces unobserved covariates by certain conditional expectations. For linear regression, the two estimators are identical under certain conditions. However, when the primary regression model is a generalized linear model, the RC estimator is usually biased. We thus consider a refined regression calibration estimator whose performance is close to that of the pseudo EEE estimator but does not require numerical integration. Special cases such as logit, probit and log links are examined. The RC estimator is also extended to the proportional hazards regression model. In addition to the distribution theory, we evaluate the methods through finite sample simulations and a real data application on a child growth study.