Dynamic copula-based methods for estimating rank-tracking probabilities with longitudinal data

The rank-tracking probability (RTP) is a useful statistical index for measuring the 'tracking ability' of longitudinal disease risk factors in biomedical studies. Two existing unstructured smoothing methods for estimating the RTP in literature require sufficient observations at any two design time points, which may be violated in practice. We consider the dynamic estimation methods based on semiparametric copula modeling and smoothing method and compare the corresponding estimators under three smoothing ways: parametric-smoothing (C-PAS), probability-smoothing (C-PRS) and RTP-smoothing (C-RS). The proposed estimators are consistent under some mild assumptions when the true copula model is known. Among them, the C-PAS, though performs best under the true copula model, is not robust for the copula model selection. The C-PRS and C-RS estimators are preferred under the selected copula model, which have much smaller mean squared errors than unstructured smoothing methods. Moreover, the C-RS estimator performs slightly better than C-PRS estimator. Some numerical simulations and an application of a longitudinal epidemiological study are carried out to illustrate the performances of the proposed methods.