Exact Tests of Hardy-Weinberg Equilibrium a la Fisher in Multi-Allelic Biomarkers and Algebraic Statistics

• Activity Number: 463
• Type: Contributed
• Date/Time: Wednesday, August 3, 2011 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistics in Epidemiology
• Author(s): Marepalli B. Rao*+ and Subramanyam Kasala and Marepalli B. Rao
• Companies: University of Cincinnati and The University of North Carolina at Wilmington and University of Cincinnati
• Keywords: Exact Test ; Hardy-Weinberg Law ; MCMC ; p value ; Permutation test ; Alleles

Abstract

In the context of large contingency tables with sparse data and/or highly imbalanced entries, algebraic statistics methodology is successfully implemented to conduct a test of independence a la Fisher. In this presentation, we want to rope in algebraic statistics to develop an exact test of Hardy-Weinberg Equilibrium a la Fisher in multi-allelic biomarkers. The test is a combination of fibers, Markov bases, and MCMC. The key idea is to look at the fiber of all possible genotype data sets with allele frequencies the same as those of the given genotype data set. It turns out that the fiber can be characterized as the collection of all symmetric matrices with non-negative integer entries, even diagonals, and row sums the same as the allele frequencies of the given genotype data set. Even for moderate number of alleles and sample sizes, the fiber could be humongous. We will construct a Markov base which with the given data set can generate the entire fiber. The size of a Markov base is much, much smaller than the size of the fiber. We then develop an MCMC algorithm to estimate the p value under the equilibrium.