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Heredity (Edinb). 2012 December; 109(6): 401–402.
Published online 2012 September 5. doi:  10.1038/hdy.2012.54
PMCID: PMC3499835

Measuring linkage disequilibrium by the partial correlation coefficient

C-Y Lin,1,2 G Xing,3 and C Xing1,4,*

Gametic phase disequilibrium, often referred to as linkage disequilibrium (LD), describes the non-independence of alleles at different loci on the same chromosome. There are various measures of LD proposed in the literature (Hedrick, 1987; Devlin and Risch, 1995) for the purposes of inferring population evolutionary history and mapping genes (Slatkin, 2008). In a recent paper in this journal, Mangin et al. (2012) proposed a new LD measure An external file that holds a picture, illustration, etc.
Object name is hdy201254e1.jpg aiming to correct the bias due to population structure by taking into account of the population structure matrix. In this letter, we point out that An external file that holds a picture, illustration, etc.
Object name is hdy201254e2.jpg is essentially the square of the partial correlation coefficient between two loci given the population structure, which was not explicitly explained in the paper. We also distinguish between the partial correlation and the conditional correlation, as the latter was ambiguously used in the paper. We further extend the result on the relationship between An external file that holds a picture, illustration, etc.
Object name is hdy201254e3.jpg and power of association tests to generalized linear models and discuss the potential use of An external file that holds a picture, illustration, etc.
Object name is hdy201254e4.jpg in human genetic mapping.

A natural way to measure LD is by the correlation coefficient. Consider two diallelic loci A and B, with alleles A1 and A2 at locus A and alleles B1 and B2 at locus B. Denote by pi, where i[set membership]{A1, A2, B1, B2}, allele frequencies, and by pj, where j[set membership]{A1B1, A1B2, A2B1, A2B2}, haplotype frequencies. The widely used LD measure An external file that holds a picture, illustration, etc.
Object name is hdy201254e5.jpg, where An external file that holds a picture, illustration, etc.
Object name is hdy201254e6.jpg, is the square of Pearson's correlation coefficient that measures the linear dependence between the two loci (Hill and Robertson, 1968). Suppose in a sample there exists population structure that can distort the correlation between the two loci. One way to measure LD controlling for confounding effects is by the partial correlation coefficient. Denote by YA and YB the random variables of genotypes at loci A and B, respectively, and by S a vector of variables on the population structure. Regress YA and YB on S by the linear regression models YA=SβA+epsilonA and YB=SβB+epsilonB, respectively, where βA and βB are regression coefficients, and epsilonA and epsilonB are residuals. The partial correlation rAB.S between YA and YB controlling for S is then defined as Pearson's correlation between the residual variables epsilonA and epsilonB (Yule, 1907). Alternatively, the partial correlation rAB.S can be calculated as a negative off-diagonal element of the inverse correlation matrix (Whittaker, 1990), which is exactly the square root of formula (1) in Mangin et al. (2012). Therefore, the new LD measure they proposed is the square of the partial correlation coefficient—An external file that holds a picture, illustration, etc.
Object name is hdy201254e7.jpg—between two loci controlling for the population structure, which is a direct extension of the original measure An external file that holds a picture, illustration, etc.
Object name is hdy201254e8.jpg that is used in the absence of population stratification.

As the formula of partial covariance was referred to as the one for ‘conditional covariance' in the paper (p 286), it is worth pointing out that these two are equivalent only in special situations, such as when variables follow a multivariate normal distribution. The partial correlation rAB.S in general is not equal to the conditional correlation rAB|S. The former by definition is independent of S, whereas the latter is not necessarily free of S. Even if rAB|S is free of S, there exists the inequality An external file that holds a picture, illustration, etc.
Object name is hdy201254e9.jpg, where the equality holds when both the conditional variances and covariance of YA and YB given S are free of S (Lawrance, 1976). Below we performed a simulation study to show their subtle difference in case of rAB|S being independent of S. Simulation settings I, III and V mimicked those by Mangin et al. (2012) (Table 1), except for replacing An external file that holds a picture, illustration, etc.
Object name is hdy201254e10.jpg by 0.1; in settings II, IV and VI, the allele frequencies in the second population were also changed. In these six settings, the two loci were in the same degree of LD in the two populations but with different minor allele frequencies. In each population 1000 haplotypes were simulated and randomly assigned into 500 pairs. The genotypes were then scored in an additive fashion. The crude sample correlation coefficient An external file that holds a picture, illustration, etc.
Object name is hdy201254e11.jpg, the partial correlation coefficient An external file that holds a picture, illustration, etc.
Object name is hdy201254e12.jpg and the conditional correlation coefficient An external file that holds a picture, illustration, etc.
Object name is hdy201254e13.jpg were estimated based on the genotypic scores. Ten thousand replicates were simulated, and the mean and standard error of the estimates were recorded in Table 1. In all settings, An external file that holds a picture, illustration, etc.
Object name is hdy201254e14.jpg was smaller than An external file that holds a picture, illustration, etc.
Object name is hdy201254e15.jpg, but the difference between them was small. Theoretically, in settings I, III and V they should be equal because the minor allele and the major allele at each diallelic locus were simply flipped between the two populations, and thus both the conditional variances and covariance of YA and YB given S are free of S; the small differences (~10−4) were due to sampling errors. In settings II, IV and VI, the differences between them (~10−3) were one order of magnitude greater than that in settings I, III and V.

Table 1
Mean (and its s.e.a) of correlation coefficient estimates of a mixture of two populationsb

Measuring LD between loci by An external file that holds a picture, illustration, etc.
Object name is hdy201254e16.jpg in the case of population stratification is in the same spirit as measuring correlation between covariate-adjusted phenotypes and genotypes in genetic association studies (Price et al., 2006; Xing et al., 2011). Suppose an allele at locus A is the causal variant for a trait. Mangin et al. (2012) derived in a linear regression setting that the power to detect association between the trait and locus A would be reduced by a factor of An external file that holds a picture, illustration, etc.
Object name is hdy201254e17.jpg when locus B was examined instead. As a matter of fact, this conclusion holds in general when modeling phenotype–genotype association by a generalized linear model, as epsilonB can be viewed as a surrogate variable for epsilonA, and it is well known the asymptotic relative efficiency of a test using epsilonB versus using epsilonA equals the square of their correlation coefficient (Lagakos, 1988; Tosteson and Tsiatis, 1988).

Characterizing LD structure is instructive in designing genetic association studies, which is a major goal of the International HapMap Consortium (2005) and the 1000 Genomes Project Consortium (2010). These projects focus on genetically homogeneous populations to document population-specific parameters. However, in reality, a study sample can be genetically heterogeneous with substructure even though the recruiting criterion requires a specific ethnic group. Imagine the diversity of African Americans in a metropolitan area. Considering that a lot of genome-wide association studies have been carried out, it will be valuable to use these available genome-wide genotypes to document ethnic- and geographic-specific An external file that holds a picture, illustration, etc.
Object name is hdy201254e18.jpg for the purpose of facilitating future genetic studies conducted in the same population.

Finally, we also want to point out that the other LD measure An external file that holds a picture, illustration, etc.
Object name is hdy201254e19.jpg proposed by Mangin et al. (2012) for the purpose of correcting the bias due to relatedness is the square of the correlation coefficient of two loci modeled by a linear regression—the coefficient of determination—using generalized least squares given the kinship matrix instead of using ordinary least squares and assuming independence between subjects as when calculating the usual correlation coefficient.

Acknowledgments

We thank Dr Robert Elston for critically reading, commenting and editing the paper. This study is supported by the American Heart Association Scientist Development Grant 10SDG4220051 to CX.

Notes

The authors declare no conflict of interest.

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