Statistical Modelling 12 (2012), 211–228
Process modelling for contingency tables with ordered categories
Matthew J Heaton
Department of Statistical Science,
Duke University
Durham, NC
USA
Simone C Gray
Environmental Protection Agency
Durham, NC
USA
Alan E Gelfand
Department of Statistical Science,
Duke University
Box 90251
Durham, NC 27708
USA
eMail: alan@stat.duke.edu
Abstract:
We consider the setting of a multi-way contingency table with ordinal classifications. The contribution of this paper is to propose a joint probability model for the uncensored variables that is apart from the imposed categorization. Specifically, for an m-way table, we assume that the cell counts arise as a binned point pattern over a bounded set in m-dimensional Euclidean space where the point pattern is a realization of a non-homogeneous Poisson process. The intensity which drives the point pattern is itself viewed as a realization of a log Gaussian process over the set.
With such an approach we achieve full inference regarding the underlying joint distribution, in particular, inference for familiar associations between the ordinal variables in the absence of interval censoring. Additionally, inference can be provided for any newly created cells where such creation is achieved through redefinition of the ordinal classifications. That is, rather than ad hoc reallocation, we achieve a fully model-based reallocation enabling quantification of uncertainty. For a contingency table with nominal classifications as well, our approach creates an intensity for the ordinal variables for each level of the nominal variables.
The methodology is detailed within a hierarchical framework, showing associated computation and convenient dimension reduction techniques to facilitate model fitting. We illustrate with both simulated data and a real census dataset.
Keywords:
Bayesian hierarchical model; block summing; Cox process;
Gaussian processes; nonhomogeneous Poisson process;
separable covariance functions; spatial interpolation
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