In some clinical trials, data are gathered longitudinally on both the frequency of an event and its severity. Oftentimes, it is not feasible to obtain the exact time of the events, and the events are collected over fixed follow-up intervals. We refer to this type of data as longitudinal random length data, since the subjects are observed repeatedly and, at each assessment time, the data can be viewed as vectors of severities with lengths determined by the number of events experienced during the assessment.

Suppose the interest is in comparing two treatments, and the treatments are evaluated at multiple points in time. Treatment effect is reflected in simultaneous changes in both the number of events and the severity of each event. Consequently, one needs to jointly model the two outcomes to better evaluate treatment effects. The main objective of this dissertation is to introduce a framework for longitudinal random length data.

We propose two multiple population models for such data. We parameterize the models such that, at each measurement time, both the distribution of the random lengths and the distributional mean of each component of the severity vectors depend on the underlying parameter reflecting the treatment effect at that time. Given the random lengths, we assume the distribution of the severities to be multivariate normal. Conditional on the number of events, the dependence in the vector of severities recorded at a single measurement time is modeled using compound symmetry.

The first model assumes the numbers of events for a subject at different time points to be independent Poisson random variables and dependence over time is built into the severity measures. The second model generalizes the first one, by adding another layer of dependence over time. We further assume the numbers of the events experienced by a subject across time to be dependent and use a multivariate Poisson distribution to model them. For each model we describe the maximum likelihood estimation procedure and provide the asymptotic properties for the estimators. We apply both models to analyze a data set containing stressful life events in adolescents with major depressive disorder.

Keywords. longitudinal random length, repeated measurements, informative cluster size, clustered data, multivariate Poisson distribution.