A Canonical Measure of Allelic Association

Abstract

The measurement of biallelic pair-wise association called linkage disequilibrium (LD) is an important issue in order to understand the genomic architecture. A large variety of such measures of association have been proposed in the literature. We propose and justify six biometrical postulates which should be fulfilled by a canonical measure of LD. In short, LD measures are defined as a mapping of probability tables to the set of real numbers. They should be zero in case of independence and extremal if one of the entries approaches zero while the marginals are positively bounded. They should reflect the symmetry group of the tables and be invariant under certain transformations of the marginals (selection invariance). There scale should have maximum entropy relative to a calibrating symmetric distribution. None of the established measures fulfil all of these properties in general. We prove that there is a unique canonical measure of LD for each choice of a calibrating distribution. We compa- re the canonical LD measures with other candidates from the literature. We recommend the canonical measure derived from Jeffreys' non-informative prior distribution when assessing linkage disequilibrium of SNP array data. In a second part, we consider various estimators for the theoretical LD measures discussed and compare them in a simulation study.