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In longitudinal studies, we observe the subjects who are likely to progress to a new state during the study time. For example, in clinical trials the stage of a progressing disease is recorded at each follow-up visit. The primary goal is to estimate the relationship between the attributes and the subject’s progression state. In such studies, some subjects complete all their follow-up visits and their progression state are observed without any missingness. However, others miss their follow-up visits and when they come back, they learn that they have progressed to a new state. In this case, not only are their progression states at each follow-up interval-censored, but their time-dependent covariates are incomplete. In such studies, the observations are missing at random (MAR). The event of interest, i.e., progression, may have several possible patterns. In some studies, we might be studying progression to only one new state. For example, we are interested in studying the attributes that affect an individual’s progression from being a non-smoker to a frequent smoker. Another example would be the patients who are believed to have high risk for developing diabetes, are monitored for advancing to type 2 diabetes. In other studies, the event of interest involves multiple stages. Examples of these studies include several stages of cancer, or different stages of smoking (nonsmoker, light smoker, intermittent smoker, heavy smoker, etc.). These states are chronological. The times of observation, i.e., follow-up interview visits, are pre-specified for these studies. At each time point, the attributes are measured and recorded. Since the

Estimation of Association Between a Longitudinal Marker and Interval-Censored Progression Times