Multivariate contingency tables and the analysis of exchangeability

There are settings in the social and health sciences where it is natural to question whether a collection of discrete random variables is exchangeable. In this paper the inter-relationships between parameter symmetry, parameter invariance, and exchangeable discrete random variables are investigated within the log-linear models framework. We demonstrate how log-linear models can be used to formulate and test hypotheses of various forms of exchangeability, and to characterize departures from exchangeability. Conditions under which the observed cross-classification collapses into a lower dimensional cross-classification, while preserving the essential probability structure of the higher dimensional cross-classification, and the model structure of this lower dimensional cross-classification are presented. The development is sufficiently general to allow for subsetting the variables into classes, which is important for some applications. For example, in studying the spatial clustering of periodontal disease, there is interest in studying differences among disease patterns between the upper and lower arches in terms of parameter symmetry, parameter invariance, and exchangeability. Cross-sectional periodontal disease data from a study of Pima Indians residing in the Gila River Indian Community are used to illustrate how log-linear models may be used to examine for exchangeability, and for specific departures from exchangeability.