Multivariate random length and missing data

In many experiments a random number of severity measurements are collected on an experimental unit. We refer to data arising from such experiments as multivariate random length data. Often, in data of this type, both the number of measurements as well as the severity of each measurement inform on treatment effect. Consider two examples: In a cardiovascular clinical trial both the number of arterial blockages as well as the degree of occlusion of each blockage is informative on the performance of a cholesterol lowering medication; In a reproductive toxicology study each fetus from a litter of random size is assessed for developmental abnormalities, both the number of fetuses as well as the developmental status of each fetus can inform on the effect of a toxin. We present both parametric and semiparametric models for such "random length" data and use these models to estimate measures of treatment effect that effectively incorporate information from both event frequency and severity. In toxicology studies the ordering of the fetuses in a litter is unimportant, and thus the multivariate severity measures have an exchangeable structure. We develop a matrix representation to model general multivariate exchangeable categorical data. We present a nonparametric estimator for such data, and discuss parameterizations of the representation. We formulate the problem of estimating treatment effect from random length data as an unique type of missing data problem. We develop theory for this formulation and discuss the limitations of semiparametric estimation in this case. The models for multivariate exchangeable categorical data are then utilized to model random length data in the context of a missing data problem. We conduct simulation studies and apply these methodologies to data examples from cardiology and toxicology studies.