The interrelationships between time-to-event (survival time) variable and longitudinal covariates is often the primary research interest in medical and epidemiological studies. Due to the challenges encountered in some important clinical trials on AIDS and cancer research, recently the statisticians started modeling survival data and longitudinal data jointly via Cox's proportional hazards model. Such a joint modeling procedure or methodology has broad applications in many scientific research fields, but it is a considerably difficult problem due to censoring on the survival time and that the covariate process is only observed at some given time points. Up to now, statistical methods on this topic have not been fully or well developed, while the importance and needs for developing these methods have become more evident when the proposer and her collaborators recently encountered some more complicated problems which have not been studied in statistical literature; see examples listed below. Specifically, there have not been any modeling procedures that directly study the relationship between survival time and within-subject historic patterns of change in longitudinal covariates, nor have there been any works on joint modeling doubly censored or interval censored survival data together with (intensive or multi-phase intensive) longitudinal covariates, which is far more challenging than right censored data problem. In fact, there have been no published works on the Cox model with doubly censored data, not even for the case with time-independent covariates. In this research, asymptotic methods and simulations will be mainly used in the studies, and the issues under consideration include: (a) derivation of the empirical likelihood based MLE for the Cox model with longitudinal covariates for right censored, doubly censored and interval censored survival data, respectively; (b) computation algorithms for the MLE; (c) asymptotic properties of the MLE; (d) Wilk's theorem for the MLE; (e) goodness-of-fit tests for the Cox model; (f) comparison with alternative methods. At least two Ph.D. students of the proposer will be involved in and benefit from the proposed research.
The new statistical methodology to be developed in this project has direct impact to medical research, epidemiology, social and behavioral sciences, etc. For instance, the data examples which we have encountered and motivate the research of this project include the following problems on joint modeling survival time and longitudinal covariates. In a prostate cancer study on mice, part of the research focus is joint modeling interval censored survival time and longitudinal covariates. In a smoking cessation study, the research focus is joint modeling right censored survival time and intensive longitudinal covariates. In a recent study of child development, the research focus is joint modeling doubly censored survival time and multi-phase intensive longitudinal covariates.
