Heterogeneous quantile regression for longitudinal data with structures

Subgroup analysis for modeling longitudinal data with heterogeneity across all individuals has drawn attention in the modern statistical learning. In this paper, we focus on heterogeneous quantile regression model and propose to achieve variable selection, heterogeneous subgrouping and parameter estimation simultaneously, by using the smoothed generalized estimating equations in conjunction with the multi-directional separation penalty. The proposed method allows individuals to be divided into multiple subgroups for different heterogeneous covariates such that estimation efficiency can be gained through incorporating individual correlation structure and sharing information within subgroups. A data -driven procedure based on a modified BIC is applied to estimate the number of subgroups. Theoretical properties of the oracle estimator given the underlying true subpopulation information are firstly provided and then it is shown that the proposed estimator is equivalent to the oracle estimator under some conditions. The finitesample performance of the proposed estimators is studied through simulations and an application to an AIDS dataset is also presented.