Longitudinal studies are often conducted to explore the cohort and age effects in many scientific areas. The within cluster correlation structure plays a very important role in longitudinal data analysis. This is because not only can an estimator be improved by incorporating the within cluster correlation structure into the estimation procedure, but also the within cluster correlation structure can sometimes provide valuable insights in practical problems. For example, it can reveal the correlation strengths among the impacts of various factors. Motivated by data typified by a set from Bangladesh pertinent to the use of contraceptives, we propose a random effect varying-coefficient model, and an estimation procedure for the within cluster correlation structure of the proposed model. The estimation procedure is optimization-free and the proposed estimators enjoy asymptotic normality under mild conditions. Simulations suggest that the proposed estimation is practicable for finite samples and resistent against mild forms of model misspecification. Finally, we analyze the data mentioned above with the new random effect varying-coefficient model together with the proposed estimation procedure, which reveals some interesting sociological dynamics.
"Estimation of the covariance matrix of random effects in longitudinal studies." Ann. Statist. 35 (6) 2795 - 2814, December 2007. https://doi.org/10.1214/009053607000000523