Measuring linkage disequilibrium by the partial correlation coefficient

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 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 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 and power of association tests to generalized linear models and discuss the potential use of 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 p_i, where i∈{A₁, A₂, B₁, B₂}, allele frequencies, and by p_j, where j∈{A₁B₁, A₁B₂, A₂B₁, A₂B₂}, haplotype frequencies. The widely used LD measure , where , 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 Y_A and Y_B the random variables of genotypes at loci A and B, respectively, and by S a vector of variables on the population structure. Regress Y_A and Y_B on S by the linear regression models Y_A=Sβ_A+ɛ_A and Y_B=Sβ_B+ɛ_B, respectively, where β_A and β_B are regression coefficients, and ɛ_A and ɛ_B are residuals. The partial correlation r_AB.S between Y_A and Y_B controlling for S is then defined as Pearson's correlation between the residual variables ɛ_A and ɛ_B (Yule, 1907). Alternatively, the partial correlation r_AB.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——between two loci controlling for the population structure, which is a direct extension of the original measure 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 r_AB.S in general is not equal to the conditional correlation r_AB|S. The former by definition is independent of S, whereas the latter is not necessarily free of S. Even if r_AB|S is free of S, there exists the inequality , where the equality holds when both the conditional variances and covariance of Y_A and Y_B given S are free of S (Lawrance, 1976). Below we performed a simulation study to show their subtle difference in case of r_AB|S being independent of S. Simulation settings I, III and V mimicked those by Mangin et al. (2012) (Table 1), except for replacing 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 , the partial correlation coefficient and the conditional correlation coefficient 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, was smaller than , 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 Y_A and Y_B 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.

	I	II	III	IV	V	VI
Population 1
	0.10	0.10	0.25	0.25	0.50	0.50
pA1	0.90	0.90	0.90	0.80	0.90	0.70
pB1	0.90	0.55	0.90	0.55	0.90	0.55
Population 2
	0.10	0.10	0.25	0.25	0.50	0.50
pA1	0.10	0.70	0.10	0.65	0.10	0.60
pB1	0.10	0.70	0.10	0.65	0.10	0.60
Mixed Population
	0.7227 (0.0188)	0.0438 (0.0131)	0.7925 (0.0160)	0.1980 (0.0247)	0.8757 (0.0125)	0.4716 (0.0281)
	0.1012 (0.0247)	0.0969 (0.0177)	0.2508 (0.0366)	0.2484 (0.0251)	0.5007 (0.0412)	0.4995 (0.0265)
	0.1014 (0.0247)	0.1030 (0.0186)	0.2511 (0.0366)	0.2515 (0.0253)	0.5011 (0.0412)	0.5005 (0.0264)

^aBased on 10 000 replicates.

^bFive hundred diploid subjects from each population.

Measuring LD between loci by 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 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 ɛ_B can be viewed as a surrogate variable for ɛ_A, and it is well known the asymptotic relative efficiency of a test using ɛ_B versus using ɛ_A 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 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 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.

The authors declare no conflict of interest.