Exact Tests of Hardy-Weinberg Equilibrium in Case-Parent Triad Design and Algebraic Statistics

• Activity Number: 515
• Type: Contributed
• Date/Time: Wednesday, August 3, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistics in Epidemiology
• Author(s): Subramanyam Kasala*+ and Marepalli B. Rao and Marepalli B. Rao
• Companies: The University of North Carolina at Wilmington and University of Cincinnati and University of Cincinnati
• Keywords: Triad Design ; Alleles ; Hardy-Weinberg Equilibrium ; p value ; Markov Base ; MCMC

Abstract

Case-Parent Triad Designs are useful in genetic studies. Typically, in a triad desgin, association between a bi-allelic marker and a complex disease will be under focus. A triad consists of a child and child's parents. Triad data consist of genotypes of the members of the triads in the sample. The focus of the presentation is developing an exact test of Hardy-Weinberg Equilibrium (HWE) at the marker a la Fisher. In the literature, tests of HWE are conducted obliquely. Here, we postulate the null hypothesis of HWE directly and the hypothesis is composiste with one nuisance parameter. Our exact test involves algebraic statistics, Markov bases, and MCMC. The key idea is to look at the fiber of all possible triad data sets with allele frequencies equal to those of the given triad data set. It turns out that the fiber is a certain collection of 2x15 matrices with non-negative integer entries and prescribed row sums. The size of a fiber is usually very large. We will build a Markov base, which is small, for the fiber. The given triad data set and the Markov base will generate the entire fiber. We will then develop an MCMC algorithm to estimate the p value under the equilibrium.