Applying the Cox proportional hazards model when the change time of a binary time-varying covariate is interval censored

This paper develops methodology for estimation of the effect of a binary time-varying covariate on failure times when the change time of the covariate is interval censored. The motivating example is a study of cytomegalovirus (CMV) disease in patients with human immunodeficiency virus (HIV) disease. We are interested in determining whether CMV shedding predicts an increased hazard for developing active CMV disease. Since a clinical screening test is needed to detect CMV shedding, the time that shedding begins is only known to lie in an interval bounded by the patient's last negative and first positive tests. In a Cox proportional hazards model with a time-varying covariate for CMV shedding, the partial likelihood depends on the covariate status of every individual in the risk set at each failure time. Due to interval censoring, this is not always known. To solve this problem, we use a Monte Carlo EM algorithm with a Gibbs sampler embedded in the E-step. We generate multiple completed data sets by drawing imputed exact shedding times based on the joint likelihood of the shedding times and event times under the Cox model. The method is evaluated using a simulation study and is applied to the data set described above.