Mathematical properties of the r2 measure of linkage disequilibrium (2024)

2.1 $r_{\text{max}}^2$ (p_a, p_b)

Our first computation is of $r_{\text{max}}^2$(p_a,p_b), the maximum value of r² for given values of p_a and p_b, considering all possible values of p_ab. Given (p_a, p_b), the denominator of r² is fixed. Therefore, to maximize r², it suffices to choose the value for p_ab that maximizes the numerator. The possible values of p_ab are constrained by the fact that the frequency of a haplotype can be no more than the frequency of the least frequent allele that it contains and no less than 0 or the minimum overlap that can occur between two alleles based on their frequencies. It is at one of these extremes – the highest or lowest possible haplotype frequency – that the numerator is maximized. Thus, the maximum value of r² occurs either at p_ab = min(p_a, p_b) or at p_ab = max(0, p_a + p_b − 1), depending on the component, S_i, in which the given (p_a, p_b) is located. For S₁ and S₄, the maximum occurs at p_ab = p_a + p_b − 1, so

For S₂ and S₇, the maximum occurs at p_ab = p_a:

For S₃ and S₆, the maximum occurs at p_ab = p_b:

Finally, for S₅ and S₈, the maximum occurs at p_ab = 0:

Table 1 summarizes these results. Note that $r_{\text{max}}^2$ is continuous on the boundaries between components. An important consequence of equations 2–5 is that $r_{\text{max}}^2(p_a,p_b)=1$ if and only if p_b = p_a or p_b = 1−p_a.

Combining equations 2–5, Figure 2 shows a 3-dimensional plot of $r_{\text{max}}^2$ for all combinations (p_a, p_b). The X-shape of the figure illustrates the symmetries of r² as a function of p_a and p_b, as well as the property that r² can only equal 1 if the two loci have the same minor allele frequency. Additionally, the graph shows a very steep decay of $r_{\text{max}}^2$ moving away from the diagonals, indicating that even small differences in allele frequency between the two loci, especially if the frequencies are not near 1/2, can reduce the range of possible values for r² considerably.

We can quantify the effect of differences in minor allele frequency observed in Figure 2 by calculating the average $r_{\text{max}}^2$ value assuming independent Uniform(0,1) distributions for p_a and p_b. This computation amounts to evaluating the volume below $r_{\text{max}}^2$ over the unit square. Using symmetry, the total volume can be calculated by finding the volume over one of the eight components in Figure 1 and multiplying by eight. Denoting the volume of $r_{\text{max}}^2$ over component S₁ by V₁, we have

$$\begin{array}{ccc}{V}_1\hfill & =\hfill & {\displaystyle {\int }_0^{\frac{1}{2}}{\displaystyle {\int }_{1-p_a}^1\frac{(1-p_a)(1-p_b)}{p_a{p}_b}d{p}_{b}d{p}_a}}\hfill \\ \hfill & =\hfill & -{\displaystyle {\int }_0^{\frac{1}{2}}\frac{(1-p_a)}{p_a}[\text{ln}(1-p_a)+p_a]d{p}_a}\hfill \\ \hfill & =\hfill & \frac{1}{12}{\pi }^2-\frac{1}{2}{(\text{ln}\text{2})}^2+\frac{1}{2}\text{ln}2-\frac{7}{8}\hfill \\ \hfill & \approx \hfill & 0.05381.\hfill \end{array}$$

The last step uses the dilogarithm function ${\displaystyle {\int }_z^0}$ ln(1−t)/t dt = Li₂(z), where Li₂(0) = 0 and Li₂(1/2) = π²/12 − (ln 2)²/2 (Weisstein, 2003). Consequently the mean $r_{\text{max}}^2$ given p_a ~ Uniform(0, 1), p_b ~ Uniform(0, 1), and assuming p_a and p_b are independent is 8V₁ = 2π²/3−4(ln 2)²+4(ln 2)−7 ≈ 0.43051.

This result and the shape of Figure 2 suggest that it is only possible to achieve high values of r² over relatively small portions of the space of possible values of p_a and p_b. For a constant c, 0 ≤ c ≤ 1, we can calculate the proportion of the allele frequency space where it is possible for r² to exceed c, p(c). Again using symmetry, we can restrict our attention to S₆. Using equation 4, the portion of S₆ in which $r_{\text{max}}^2$(p_a,p_b) ≥ c, whose area we denote by A₆, satisfies

$$p_b\ge \frac{c{p}_a}{1-p_a+c{p}_a}.$$

Considering the complement of the area of interest in S₆, we have

$$\begin{array}{ccc}\frac{1}{8}-A_6\hfill & =\hfill & {\displaystyle {\int }_0^{\frac{1}{2}}}{\displaystyle {\int }_0^{\frac{c{p}_a}{1-p_a+c{p}_a}}1d{p}_{b}d{p}_a}\hfill \\ \hfill & =\hfill & -\frac{c}{2(1-c)}-\frac{c\text{ln}(\frac{1}{2}+\frac{1}{2}c)}{{(1-c)}^2}.\hfill \end{array}$$

Thus, the proportion of the allele frequency space where it is possible for r² to exceed c is 8A₆, or

Figure 3 shows that the proportion of the allele frequency space where it is possible for r² to exceed c declines faster than linearly. For example, only over ~0.39709 of the allele frequency space is it possible for r² to exceed 1/2 and only over ~0.14232 of the space is it possible for r² to exceed 4/5.

2.2 $r_{\text{max}}^2(p_a,p_b)$ with p_a fixed

In contrast to the previous computations, in which we performed integrations over possible values of p_a and p_b, we now consider the case in which p_a is fixed. This computation enables us to identify the allele frequencies for a locus that is able to have high r² values across the broadest range of allele frequencies for a second locus. Assuming p_b has a Uniform(0,1) distribution, we can calculate $m(p_a)=E[r_{\text{max}}^2(p_a,p_b)|p_b\sim \text{Uniform(}0,1)]$, the mean maximum r² value as a function of p_a. We can assume that p_a ≤ 1/2 and then consider p_a > 1/2 by observing that m(p_a) = m(1−p_a). For p_a ≤ 1/2 we perform piecewise integration across components S₆, S₇, S₈, and S₁ using equations 2–5:

$$\begin{array}{ccc}m(p_a)\hfill & =\hfill & {\displaystyle {\int }_0^{p_a}\frac{(1-p_a)p_b}{p_a(1-p_b)}d{p}_b+{\displaystyle {\int }_{p_a}^{\frac{1}{2}}\frac{p_a(1-p_b)}{(1-p_a)p_b}d{p}_b}}\hfill \\ \hfill & \hfill & +{\displaystyle {\int }_{\frac{1}{2}}^{1-p_a}\frac{p_a{p}_b}{(1-p_a)(1-p_b)}d{p}_b+{\displaystyle {\int }_{1-p_a}^1\frac{(1-p_a)(1-p_b)}{p_a{p}_b}d{p}_b}}\hfill \\ \hfill & =\hfill & -\frac{2(1-p_a)}{p_a}\left[p_a+\text{ln}(1-p_a)\right]\hfill \\ \hfill & \hfill & +\frac{2p_a}{(1-p_a)}\left[\text{ln}\left(\frac{1}{2}\right)-\frac{1}{2}-\ln p_a+p_a\right].\hfill \end{array}$$

Figure 4 shows that the mean of $r_{\text{max}}^2(p_a,p_b)$, averaging over values of p_b, has an m-shape as a function of p_a. The maximum of this mean occurs at p_a ≈ 0.30131 and p_a ≈ 0.69869 and equals ~0.53091. Notice that the largest values of m(p_a) occur for intermediate minor allele frequencies rather than for minor allele frequencies close to 1/2. This finding can be explained by examining the contour plot of Figure 2, which suggests that slices through the graph made at intermediate frequencies for p_a contain more space with higher values of $r_{\text{max}}^2$ than do other slices.

2.3 $r_{\text{max}}^2(p_a,p_b)$ with p_a and p_a − p_b fixed

We now consider the situation in which p_a and the difference between allele frequencies |p_a − p_b| are known. This situation is similar to the scenario considered by Wray (2005) in which r² was assumed to exceed some known threshold, p_a was assumed to be known, and p_b = p_a + v was investigated.

Let p_a and p_b be minor allele frequencies (≤ 1/2) with p_a ≥ p_b, so that we are considering component S₆. Define d = p_a − p_b ≥ 0. Treating $r_{\text{max}}^2$ as a function of p_a and d, we can rewrite equation 4:

Figure 5 shows a 3-dimensional plot of $r_{\text{max}}^2$ as a function of the larger minor allele frequency (p_a) and the difference between minor allele frequencies (p_a − p_b). The twisted surface illustrates that for p_a fixed, $r_{\text{max}}^2$ decreases faster with the difference in minor allele frequency when p_a has smaller values. This observation corresponds to the steeper decline from the diagonals further from the center in Figure 2.

Holding d constant and nonnegative in equation 7, the maximum of $r_{\text{max}}^2$ for p_a ≤ 1/2 occurs at p_a = 1/2:

By rearranging this equation, we can calculate the maximum value of |p_a −p_b| possible given a known value of r²,

As equation 9 is based on the maximum of $r_{\text{max}}^2$ over all possible minor allele frequency values for p_a, it represents the broadest range possible for the difference in allele frequencies. For d to achieve the maximum value of (1 − r²)/[2(1 + r²)], p_a must equal 1/2.

The computation above assumes a known r² and determines the maximum for d. However, if we know p_a in addition to r², then we can solve exactly for the set of allowable values of p_b. Assuming again that p_a and p_b are minor allele frequencies (that is, at most 1/2), we must consider two cases: p_a ≥ p_b and p_a ≤ p_b. For the first case, (p_a, p_b) is in S₆. Rearranging equation 4,

so that r²p_a/(1 + r²p_a − p_a) ≤ p_b ≤ p_a. In the second case, (p_a, p_b) is in S₇ so we can rearrange equation 3 to obtain

Recalling our assumption that p_b ≤ 1/2 and combining equations 10 and 11, we find

$$\frac{r^2{p}_a}{1+r^2{p}_a-p_a}\le p_b\le \text{min}\left(\frac{1}{2},\frac{p_a}{r^2-r^2{p}_a+p_a}\right).$$

This result accords with the values that appear in Table 2 of Wray (2005).

2.4 $r_{\text{max}}^2$(p_a,p_b) and recombination

We have previously been examining r² with the assumption that it is possible for p_ab to take any value within its allowable range. This amounts to an assumption that we are not constraining the recombination history of the two loci under consideration. In this section, we consider a different situation: how does recombination affect r² for two loci that have not previously experienced recombination? This depends on the genealogical history of the loci.

Consider two possible genealogies (Figure 6). In Genealogy 1, a mutation at locus 1 arises later, but on the same side of the tree, as a mutation at locus 2. In Genealogy 2, the mutations at loci 1 and 2 arise on different sides of the tree so that no haplotypes carry both mutations. Assuming p_a ≤ p_b ≤ 1/2 so that the minor alleles are derived rather than ancestral, then without recombination, p_ab = p_a for Genealogy 1, and r²(p_a, p_b) = p_a (1− p_b)/[(1 − p_a)p_b] (Eberle et al., 2006). For Genealogy 2, without recombination p_ab = 0, so r²(p_a, p_b) = p_ap_b/[(1 − p_a)(1 − p_b)] (Eberle et al., 2006).

In typical settings, recombination reduces linkage disequilibrium, as recombination separates new alleles from the haplotypic background on which they arose. For Genealogy 1 in Figure 6, the unconstrained maximum r² allowing p_ab to take on any possible value is precisely $r_{\text{max}}^2(p_a,p_b)$ (equation 3), the value taken when p_ab = p_a and no recombination has occurred. Thus, any recombination events that reduce p_ab below p_a will lead to a decrease in r². However, with Genealogy 2 we can see that situations do exist in which recombination can lead to an increase in LD. Consider Genealogy 2 and suppose recombination occurs such that the frequency of the recombinant haplotype (ab) becomes p_ab > 0. This haplotype can arise through recombination events between Ab haplotypes and aB haplotypes. Is it possible for r², with recombination events allowed, to be greater than r² in the absence of recombination? Solving the inequality

$$\frac{{(p_{ab}-p_a{p}_b)}^2}{p_a(1-p_a)p_b(1-p_b)}>\frac{p_a{p}_b}{(1-p_a)(1-p_b)},$$

we obtain

$$p_{ab}>2p_a{p}_b.$$

For each p_a ≤ 1/2 and p_b ≤ 1/2 it is possible to choose a value of p_ab that satisfies p_ab ≥ 2p_ap_b. Recall our assumption that p_a ≤ p_b, which restricts p_ab ≤ p_a. Thus, if the fraction of recombinant haplotypes satisfies

then the occurrence of recombination produces an increase in r² compared to the maximum possible value had no recombination occurred on the genealogy. Figure 7 shows an example of the variation in r² as a function of p_ab for p_a = 0.3 and p_b = 0.4. Once p_ab exceeds 2p_ap_b = 0.24, the value of r² between loci increases above the initial value in the absence of recombination.

Using inequality 12, we can determine the fraction of the space of allowed values for (p_a, p_b, p_ab) in which the unconstrained maximum r² permitting recombination (p_ab not necessarily equal to 0) exceeds the maximum under the assumption that no recombination occurs (p_ab = 0). The volume of the region where recombination inflates r² is

$${\displaystyle {\int }_0^{\frac{1}{2}}{\displaystyle {\int }_0^{p_b}{\displaystyle {\int }_{2p_a{p}_b}^{p_a}1d{p}_{ab}d{p}_{a}d{p}_b=\frac{1}{192}.}}}$$

The volume of the region of allowed values for (p_a, p_b, p_ab), assuming p_ab ≤ p_a ≤ p_b ≤ 1/2, is

$${\displaystyle {\int }_0^{\frac{1}{2}}{\displaystyle {\int }_0^{p_b}{\displaystyle {\int }_0^{p_a}1d{p}_{ab}d{p}_{a}d{p}_b=\frac{1}{48}.}}}$$

Taking the quotient of these results, the fraction of the space of possible values in which recombination inflates r² is 1/4. Thus, averaging over possible values for (p_a,p_b) with p_a ≤ p_b ≤ 1/2, on average 1/4 of possible values for p_ab lead to r²(p_a, p_b, p_ab) > r²(p_a, p_b, 0).

2.5 The relationship between r²(p_a, p_b, p_ab) and D′(p_a, p_b, p_ab)

So far, we have focused on the r² measure of LD and on various properties of its maximum value. A second LD statistic, namely D′, is defined based on maxima and minima. Our computations with $r_{\text{max}}^2$ provide a basis for examining the connection between r² and D′.

D′ is defined as

where D = p_ab−p_a−p_b, D_max = min(p_ap_b, (1−p_a)(1−p_b)) if D < 0, and D_max = min(p_a(1−p_b), (1−p_a)p_b) if D > 0 (Lewontin, 1964). Given any values for p_a and p_b, D′ can take on any value from −1 to 1, thus differing from r² in that its range is not frequency-dependent (Hedrick, 1987; Lewontin, 1988).

D′ is equal to D normalized by its maximum given the allele frequencies; r² can similarly be normalized by its maximum to obtain $r^2/r_{\text{max}}^2$. This quotient is the squared correlation coefficient between allelic indicator variables at two loci, standardized by the maximum squared correlation possible given the frequencies of the alleles at the two loci. As D′² and $r^2/r_{\text{max}}^2.$ both have numerator D², it is natural to compare their different normalization procedures to determine if they represent the same quantity. We can rewrite $r^2/r_{\text{max}}^2.$ as

Here, $D_{r_{\text{max}}^2}$ is defined as p_ab − p_ap_b evaluated at the value of p_ab that produces the maximum of D² as a function of p_a and p_b. This quantity differs across components of the allele frequency space, as described in Section 2.1. Comparing equation 14 to equation 13, we find that $D{\prime }^2=r^2/r_{\text{max}}^2\text{if}{D}_{\text{max}}^2=D_{r_{\text{max}}^2}^2.$

The sign of D determines how D_max is computed. Thus, whether $D_{\text{max}}=D_{r_{\text{max}}^2}$, and consequently $D{\prime }^2=r^2/r_{\text{max}}^2$ depends on the sign of D. For example, consider S₁, in which p_a < 1/2, p_b > 1/2, and p_a + p_b > 1. In this component, D_max equals $D_{r_{\text{max}}^2}=(1-p_a)(1-p_b)$ only when D is less than 0 (p_ab < p_ap_b). In each of the eight components, $D_{\text{max}}^2=D_{r_{\text{max}}^2}^2$ either when D < 0 or when D > 0, but not in both cases (Table 1). Thus, the region in which $D{\prime }^2=r^2/r_{\text{max}}^2$ includes some but not all of the space of possible values of p_a, p_b, and p_ab. When $D{\prime }^2\ne r^2/r_{\text{max}}^2$ D′² is always greater than $r^2/r_{\text{max}}^2$

As D′² and $r^2/r_{\text{max}}^2$ are functions of p_a, p_b, and p_ab, we can fix one of these three variables and examine the relationship between D′² and $r^2/r_{\text{max}}^2$ as a function of the other two variables. If we fix p_ab, then the domain for (p_a, p_b) is a triangle, as p_a ≥ p_ab, p_b ≥ p_ab, and p_a + p_b − 1 ≤ p_ab. Inside this triangle, Figure 8 shows the values of (p_a, p_b) where $D{\prime }^2=r^2/r_{\text{max}}^2$ for p_ab=0.1, 0.4, and 0.7. The three graphs represent the three qualitatively different patterns observed for such graphs as p_ab varies from 0 to 1. For p_ab = 0.1, the domain spans all eight components, S₁ to S₈. For p_ab = 0.4, the domain spans all eight components, but in two of these components there is no region in which $D{\prime }^2=r^2/r_{\text{max}}^2$ and in two other components there is no region in which $D{\prime }^2\ne r^2/r_{\text{max}}^2$. Finally, for p_ab = 0.7, the domain spans only two components, S₂ and S₃. The transition points between the three cases occur at p_ab = 1/4, where the boundary line p_ab = p_ap_b crosses the point (1/2, 1/2), and at p_ab = 1/2 where the space of allowable (p_a, p_b) becomes restricted to the upper right quadrant.

As a function of p_ab, we can calculate the fraction of the space of possibilities where $D{\prime }^2=r^2/r_{\text{max}}^2$. For a given p_ab, the space of possible values of (p_a, p_b) is bounded by p_a = p_ab, p_b = p_ab, and p_ab = p_a + p_b − 1, producing a triangle of area (1 − p_ab)²/2. For 0 ≤ p_ab ≤ 1/4, we calculate the area where $D{\prime }^2=r^2/r_{\text{max}}^2$ by subtracting the area where the two quantities are not equal from the total area possible, yielding

For 1/4 ≤ p_ab ≤ 1/2, we calculate the area where $D{\prime }^2=r^2/r_{\text{max}}^2$ by summing areas in each quadrant and noting that the upper left and lower right quadrants have the same area. This area is

For 1/2 ≤ p_ab ≤ 1, the calculation of the area where $D{\prime }^2=r^2/r_{\text{max}}^2$ is simplified due to the restriction of the space of possible values of (p_a, p_b) to the upper right quadrant. This area is

Computing the integrals in equations 15–17 and then dividing by the area of the space of possible values of (p_a, p_b), we find that the fraction of the space where $D{\prime }^2=r^2/r_{\text{max}}^2$ is

Figure 9 shows a plot of this function. The minimum fraction of the space where $D{\prime }^2=r^2/r_{\text{max}}^2$ is 0.31357, which occurs at p_ab ≈ 0.37162. The fraction is generally large for large p_ab; when p_ab is large the probability is quite high that D > 0. In S₂ and S₃, positive D leads to $D{\prime }^2=r^2/r_{\text{max}}^2$.

By integrating the function in equation 18 from 0 to 1, we can obtain the fraction of the space of all three variables – p_a, p_b, and p_ab – in which $D{\prime }^2=r^2/r_{\text{max}}^2$. Again using the dilogarithm, we obtain

$$\frac{1}{3}{\pi }^2-4{(\ln 2)}^2+\frac{3}{2}-8{\text{Li}}_2\left(\frac{1}{4}\right)$$

for the probability that a set of values of p_a, p_b and p_ab chosen from the space of possible values leads to $r^2/r_{\text{max}}^2=D{\prime }^2$ Numerically, this probability is ~0.72683.

1. Amos CI. Successful design and conduct of genome-wide association studies. Human Molecular Genetics. 2007;16:R220–R225. doi: 10.1093/hmg/ddm161. [DOI] [PMC free article] [PubMed] [Google Scholar]
2. Carlson CS, Eberle MA, Rieder MJ, Yi Q, Kruglyak L, Nickerson DA. Selecting a maximally informative set of single-nucleotide polymorphisms for association analyses using linkage disequilibrium. American Journal of Human Genetics. 2004;74:106–120. doi: 10.1086/381000. [DOI] [PMC free article] [PubMed] [Google Scholar]
3. de Bakker PIW, Yelensky R, Pe’er I, Gabriel SB, Daly MJ, Altshuler D. Efficiency and power in genetic association studies. Nature Genetics. 2005;37:1217–1223. doi: 10.1038/ng1669. [DOI] [PubMed] [Google Scholar]
4. Devlin B, Risch N. A comparison of linkage disequilibrium measures for fine-scale mapping. Genomics. 1995;29:311–322. doi: 10.1006/geno.1995.9003. [DOI] [PubMed] [Google Scholar]
5. Eberle MA, Ng PC, Kuhn K, Zhou L, Peiffer DA, Galver L, Viaud-Martinez KA, Taylor Lawley C, Gunderson KL, Shen R, Murray SS. Power to detect risk alleles using genome-wide tag SNP panels. PLoS Genetics. 2007;3:1827–1837. doi: 10.1371/journal.pgen.0030170. [DOI] [PMC free article] [PubMed] [Google Scholar]
6. Eberle MA, Rieder MJ, Kruglyak L, Nickerson DA. Allele frequency matching between SNPs reveals an excess of linkage disequilibrium in genic regions of the human genome. PLoS Genetics. 2006;2:1319–1327. doi: 10.1371/journal.pgen.0020142. [DOI] [PMC free article] [PubMed] [Google Scholar]
7. Hedrick PW. Gametic disequilibrium measures: proceed with caution. Genetics. 1987;117:331–341. doi: 10.1093/genetics/117.2.331. [DOI] [PMC free article] [PubMed] [Google Scholar]
8. Hill WG, Robertson A. Linkage disequilibrium in finite populations. Theoretical and Applied Genetics. 1968;38:226–231. doi: 10.1007/BF01245622. [DOI] [PubMed] [Google Scholar]
9. Hudson RR. Linkage disequilibrium and recombination. In: Balding DJ, Bishop M, Cannings C, editors. Handbook of Statistical Genetics. Chichester, UK: Wiley; 2001. pp. 309–324. chapter 11. [Google Scholar]
10. Jorgenson E, Witte JS. Coverage and power in genomewide association studies. American Journal of Human Genetics. 2006;78:884–888. doi: 10.1086/503751. [DOI] [PMC free article] [PubMed] [Google Scholar]
11. Kruglyak L. Prospects for whole-genome linkage disequilibrium mapping of common disease genes. Nature Genetics. 1999;22:139–144. doi: 10.1038/9642. [DOI] [PubMed] [Google Scholar]
12. Lewontin RC. The interaction of selection and linkage. I. General considerations; heterotic models. Genetics. 1964;49:49–67. doi: 10.1093/genetics/49.1.49. [DOI] [PMC free article] [PubMed] [Google Scholar]
13. Lewontin RC. On measures of gametic disequilibrium. Genetics. 1988;120:849–852. doi: 10.1093/genetics/120.3.849. [DOI] [PMC free article] [PubMed] [Google Scholar]
14. Morton NE, Zhang W, Taillon-Miller P, Ennis S, Kwok P-Y, Collins A. The optimal measure of allelic association. Proceedings of the National Academy of Sciences USA. 2001;98:5217–5221. doi: 10.1073/pnas.091062198. [DOI] [PMC free article] [PubMed] [Google Scholar]
15. Plagnol V, Wall JD. Possible ancestral structure in human populations. PLoS Genetics. 2006;2:972–979. doi: 10.1371/journal.pgen.0020105. [DOI] [PMC free article] [PubMed] [Google Scholar]
16. Pritchard JK, Przeworski M. Linkage disequilibrium in humans: models and data. American Journal of Human Genetics. 2001;69:1–14. doi: 10.1086/321275. [DOI] [PMC free article] [PubMed] [Google Scholar]
17. Slatkin M. Linkage disequilibrium – understanding the evolutionary past and mapping the medical future. Nature Reviews Genetics. 2008:9. doi: 10.1038/nrg2361. [DOI] [PMC free article] [PubMed] [Google Scholar]
18. Terwilliger JD, Hiekkalinna T. An utter refutation of the ‘Fundamental Theorem of the HapMap’. European Journal of Human Genetics. 2006;14:426–437. doi: 10.1038/sj.ejhg.5201583. [DOI] [PubMed] [Google Scholar]
19. Tishkoff SA, Dietzsch E, Speed W, Pakstis AJ, Kidd JR, Cheung K, Bonné-Tamir B, Santachiara-Benerecetti AS, Moral P, Krings M, Pääbo S, Watson E, Risch N, Jenkins T, Kidd KK. Global patterns of linkage disequilibrium at the CD4 locus and modern human origins. Science. 1996;271:1380–1387. doi: 10.1126/science.271.5254.1380. [DOI] [PubMed] [Google Scholar]
20. Weisstein EW. CRC Concise Encyclopedia of Mathematics. 2nd edition. Boca Raton: Chapman & Hall/CRC; 2003. [Google Scholar]
21. Wray NR. Allele frequencies and the r2 measure of linkage disequilibrium: impact on design and interpretation of association sutdies. Twin Research and Human Genetics. 2005;8:87–94. doi: 10.1375/1832427053738827. [DOI] [PubMed] [Google Scholar]
22. Zondervan KT, Cardon LR. The complex interplay among factors that influence allelic association. Nature Reviews Genetics. 2004;5:89–100. doi: 10.1038/nrg1270. [DOI] [PubMed] [Google Scholar]