Analysis of Partially Incomplete Tables of Breast Cancer Characteristics with an Ordinal Variable

Our goal is to model the joint distribution of a series of 4×2×2×2 contingency tables for which some of the data are partially collapsed (i.e., aggregated in as few as two dimensions). More specifically, the joint distribution of 4 clinical characteristics in breast cancer patients is estimated. These characteristics include estrogen receptor status (positive/negative), nodal involvement (positive/negative), HER2-neu expression (positive/negative), and stage of disease (I, II, III, IV). The joint distribution of the first three characteristics is estimated conditional on stage of disease and we propose a dynamic model for the conditional probabilities that let them evolve as the stage of disease progresses. The dynamic model is based on a series of Dirichlet distributions whose parameters are related by a Markov prior structure (called dynamic Dirichlet prior). This model makes use of information across disease stage (known as "borrowing strength") and provides a way of estimating the distribution of patients with particular tumor characteristics. In addition, since some of the data sources are aggregated, a data augmentation technique is proposed to carry out a meta-analysis of the different datasets.