Association mapping via a class of haplotype-sharing statistics

Haplotype-sharing methods attempt to utilize insights from population genetics while maintaining the simplified statistical model used for association studies in genetic epidemiology. Coalescent models suggest that for some diseases, chromosomes of affected persons share a more recent common ancestor than a randomly selected pair of chromosomes. If a disease-causing mutation is relatively recent, haplotypes of affected persons may be identical by state (IBS) over a longer region near a risk locus than would be found among randomly selected haplotypes. Thus, haplotype sharing attempts association mapping by looking for regions where the patterns of similarity in IBS among haplotypes of affected persons differs from that found in random haplotypes.

In a recent paper, we derived the distribution of some previously proposed and novel haplotype-sharing tests [1]. Here, we give an overview of these results and apply them to the Genetic Analysis Workshop 15 (GAW15) Problem 3 data.

For the i^th of ncase-parent trios, let H_1iand H_2ibe the paternal transmitted and untransmitted haplotypes, while H_3iand H_4idenote the maternal transmitted and untransmitted haplotypes. Assume haplotypes having L loci, so that there are 2 ^L possible haplotypes. Let S _k (H₁, H₂) measure the similarity between haplotypes H₁ and H₂ at a fixed locus k. Many similarity metrics are possible; here we measure similarity by the maximum information length contrast, the number of loci H₁ and H₂ share IBS looking upstream and downstream from a fixed locus k. Let S _k be the matrix having (i, j)^th element S _k (H _i , H _j ). Let $\widehat{\pi }$, $\widehat{\rho }$, and $\widehat{p}$ denote vectors of haplotype frequency estimators for untransmitted, transmitted, and all haplotypes respectively, obtained under phase uncertainty.

It is possible to show that taking γ = $\widehat{p}$ yields the numerator of the haplotype-sharing statistics considered by each of van der Meulen and te Meerman [2], Bourgain et al. [3], Tzeng et al. [4], and Zhang et al. [5], though these statistics differ in the computation of their variances. Writing these "standard" haplotype sharing tests in the form Eq. (1) allows us to interpret them as looking for differences between vectors $\widehat{\rho }$ and $\widehat{\pi }$ that are in the direction of $\widehat{p}$ ^T S _k , i.e., in the direction of sharing with the parental haplotypes. The form of U _k (γ) also allows us to derive a simple formula for its variance. We make explicit the fact that γ is often a function of the data by writing $\widehat{\gamma }$. Using Slutsky's theorem [6, Section 1.5.4], as long as $\widehat{\gamma }\stackrel{p}{\to }{\gamma }_0\ne 0$ under the null hypothesis, Var{U _k ($\widehat{\gamma }$)} can be estimated by ${\widehat{\gamma }}^T{S}_k\widehat{\Sigma }{S}_k\widehat{\gamma }$, where $\widehat{\Sigma }$ is the empirical variance estimator of ($\widehat{\rho }$ - $\widehat{\pi }$). This variance estimator is considerably simpler than those previously proposed, and is valid even with phase uncertainty and for stratified populations [1]. Use of γ = $\widehat{p}$yields the statistic $T_{\widehat{p}}=U_k^2\left(\widehat{p}\right)/\text{Var}\left\{U_k\left(\widehat{p}\right)\right\}$, which we refer to as the p test. Another choice, γ = $\widehat{\rho }$, was used by Levinson et al. [7], who contrasted sharing in transmitted haplotypes, ${\widehat{\rho }}^T{S}_k\widehat{\rho }$, with the cross product ${\widehat{\rho }}^T{S}_k\widehat{\pi }$ to give ${\widehat{\rho }}^T{S}_k\widehat{\rho }-{\widehat{\rho }}^T{S}_k\widehat{\pi }={\widehat{\rho }}^T{S}_k\left(\widehat{\rho }-\widehat{\pi }\right)$. We call this the rho test.

An appealing choice of γ is ($\widehat{\rho }$ - $\widehat{\pi }$), as this direction weights differences in haplotypes by their differences in frequency (Gerard te Meerman, personal communication). However, Slutsky's theorem no longer applies as $\left(\widehat{\rho }-\widehat{\pi }\right)\stackrel{p}{\to }0$ under the null hypothesis. Instead, we use the fact that $U_k\left(\widehat{\rho }-\widehat{\pi }\right)={\left(\widehat{\rho }-\widehat{\pi }\right)}^T{S}_k\left(\widehat{\rho }-\widehat{\pi }\right)$ is a quadratic form whose distribution is a mixture of independent χ² variates, with weights given by the eigenvalues of the matrix $\widehat{\Sigma }$S _k . Following Imhof [8], we approximate this weighted χ² distribution using a three-moment approximation. We refer to the resulting test as the cross test.

Finally, we note that because the p test uses $\gamma =\widehat{p}=\frac{1}{2}\left(\widehat{\rho }+\widehat{\pi }\right)$, while the cross test uses γ = ($\widehat{\rho }$ - $\widehat{\pi }$), the two tests appear to be looking at sharing in orthogonal directions; hence, a combined test seems desirable. Thus, we seek the distribution of $T_{\widehat{p}}+U_k\left(\widehat{\rho }-\widehat{\pi }\right)={\left(\widehat{\rho }-\widehat{\pi }\right)}^T\left[\frac{{\widehat{p}}^T{S}_k{S}_k\widehat{p}}{{\widehat{p}}^T{S}_k\widehat{\Sigma }{S}_k\widehat{p}}+S_k\right]\left(\widehat{\rho }-\widehat{\pi }\right)$. Once again, this is a quadratic form whose distribution is a mixture of independent χ² variates, with weights given by the eigenvalues of the matrix $\widehat{\Sigma }\left[\frac{{\widehat{p}}^T{S}_k{S}_k\widehat{p}}{{\widehat{p}}^T{S}_k\widehat{\Sigma }{S}_k\widehat{p}}+S_k\right]$, and we approximate this distribution as in Imhof [8].

We presented an overview of a new framework for deriving haplotype-sharing statistics and applied four such statistics to the GAW15 simulated data. Our findings suggest that these haplotype-based statistics can result in greater power and better risk locus localization compared to the single-SNP (TDT) analysis. The framework presented allows visualization of relationships between tests and computation of simplified estimators of the asymptotic distribution of the test statistics. This second feature is quite important because previous estimators have been complex or have depended on permutation procedures, making systematic power studies difficult or impossible.