Mathematical Sciences: Exact Confidence Regions for Multi- dimensional Contingency Tables

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9510516 Lazzeroni Abstract Hardy-Weinberg disequilibrium and linkage disequilibrium are important concepts in population genetics. In practice, estimation of linkage disequilibrium in haplotype data is equivalent to estimation of the interactions in a large, sparse, multidimensional contingency table. Simultaneous estimation of Hardy-Weinberg and linkage disequilibrium on multilocus genotype data introduces the additional complications of missing information and symmetry constraints on marginal probabilities. To avoid large-sample approximations that can be unreliable for these data, one approach is to use "exact" statistical methods that condition inference on observed marginal totals. Unfortunately, computing exact confidence intervals and regions is presently infeasible because of the large number of tables consistent with the marginal totals of an observed table. The principal investigator designs and studies methods for exact inference for multidimensional contingency tables. These methods employ Markov chain Monte Carlo algorithms to simulate the conditional distributions required to construct exact confidence regions for the interaction parameters of a table. Topics addressed by this research include: development of useful statistical models of genetic disequilibrium; design, optimization, and convergence properties of the algorithms; and statistical properties of the resulting estimates. Genetic disequilibrium can point to the association of particular alleles with an increased risk of disease, reveal population substructure, or serve as a guide in positional cloning. This research provides geneticists with tools for estimating the degree and form of genetic disequilibrium evidenced by a set of data, and to evaluate the precision of those estimates. The methods will also be useful for estimating interactions in large, sparse, multidimensional contingency tables that arise in other settings.