Empirical likelihood based longitudinal studies

In longitudinal data analysis, our primary interest is in the regression parameters for \nthe marginal expectations of the longitudinal responses; the longitudinal correlation \nparameters are of secondary interest. The joint likelihood function for longitudinal \ndata is challenging, particularly for correlated discrete outcome data. Marginal modeling \napproaches such as generalized estimating equations (GEEs) have received much \nattention in the context of longitudinal regression. These methods are based on the \nestimates of the first two moments of the data and the working correlation structure. \nThe confidence regions and hypothesis tests are based on the asymptotic normality. \nThe methods are sensitive to misspecification of the variance function and the \nworking correlation structure. Because of such misspecifications, the estimates can \nbe inefficient and inconsistent, and inference may give incorrect results. To overcome \nthis problem, we propose an empirical likelihood (EL) procedure based on a set of \nestimating equations for the parameter of interest and discuss its characteristics and \nasymptotic properties. We also provide an algorithm based on EL principles for the \nestimation of the regression parameters and the construction of a confidence region \nfor the parameter of interest. We extend our approach to variable selection for highdimensional \nlongitudinal data with many covariates. In this situation it is necessary \nto identify a submodel that adequately represents the data. Including redundant \nvariables may impact the model’s accuracy and efficiency for inference. We propose a \npenalized empirical likelihood (PEL) variable selection based on GEEs; the variable \nselection and the estimation of the coefficients are carried out simultaneously. We \ndiscuss its characteristics and asymptotic properties, and present an algorithm for optimizing \nPEL. Simulation studies show that when the model assumptions are correct, \nour method performs as well as existing methods, and when the model is misspecified, \nit has clear advantages. We have applied the method to two case examples.