Estimating linkage disequilibrium from genotypes under Hardy-Weinberg equilibrium

Abstract

Background

Measures of linkage disequilibrium (LD) play a key role in a wide range of applications from disease association to demographic history estimation. The true population LD cannot be measured directly and instead can only be inferred from genetic samples, which are unavoidably subject to measurement error. Previous studies of r ² (a measure of LD), such as the bias due to finite sample size and its variance, were based on the special case that the true population-wise LD is zero. These results generally do not hold for non-zero \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {r}_{true}^2 $$\end{document} values, which are more common in real genetic data.

Results

This work generalises the estimation of r ² to all levels of LD, and for both phased and unphased data. First, we provide new formulae for the effect of finite sample size on the observed r ² values. Second, we find a new empirical formula for the variance of the observed r ², equals to 2 E[ r ²](1 −  E[ r ²])/ n, where n is the diploid sample size. Third, we propose a new routine, Constrained ML, a likelihood-based method to directly estimate haplotype frequencies and r ² from diploid genotypes under Hardy-Weinberg Equilibrium. While serving the same purpose as the pre-existing Expectation-Maximisation algorithm, the new routine can have better convergence and is simpler to use. A new likelihood-ratio test is also introduced to test for the absence of a particular haplotype. Extensive simulations are run to support these findings.

Conclusion

Most inferences on LD will benefit from our new findings, from point and interval estimation to hypothesis testing. Genetic analyses utilising r ² information will become more accurate as a result.