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Comparative Study
. 2019;84(3):127-143.
doi: 10.1159/000504171. Epub 2020 Feb 11.

Mathematical Properties of Linkage Disequilibrium Statistics Defined by Normalization of the Coefficient D = pAB - pApB

Jonathan T L Kang  1 , Noah A Rosenberg  2
Affiliations
Comparative Study

Mathematical Properties of Linkage Disequilibrium Statistics Defined by Normalization of the Coefficient D = pAB - pApB

Jonathan T L Kang et al. Hum Hered. 2019.

Abstract

Background: Many statistics for measuring linkage disequilibrium (LD) take the form of a normalization of the LD coefficient D. Different normalizations produce statistics with different ranges, interpretations, and arguments favoring their use.

Methods: Here, to compare the mathematical properties of these normalizations, we consider 5 of these normalized statistics, describing their upper bounds, the mean values of their maxima over the set of possible allele frequency pairs, and the size of the allele frequency regions accessible given specified values of the statistics.

Results: We produce detailed characterizations of these properties for the statistics d and ρ, analogous to computations previously performed for r2. We examine the relationships among the statistics, uncovering conditions under which some of them have close connections.

Conclusion: The results contribute insight into LD measurement, particularly the understanding of differences in the features of different LD measures when computed on the same data.

Keywords: Allele frequencies; Linkage disequilibrium; Population genetics; Statistical genetics; Statistics.

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Conflict of interest statement

Statement of ethics

The authors have no ethical conflicts to disclose.

Disclosure statement

The authors have no conflicts of interest to declare.

Figures

Figure 1:
Figure 1:
A unit square showing all possible combinations of the frequencies pA and pB. The region is subdivided into eight octants S1,...,S8.
Figure 2:
Figure 2:
rmax2 and |d|max as functions of pA and pB. (A) Contour plot of rmax2. (B) Contour plot of |d|max. The plots consider the maximum over all possible values of pAB. The functions plotted appear in Table 1. |d| is defined only in octants S1, S2, S5, and S6.
Figure 3:
Figure 3:
The portion of the permissible allele frequency space where r2 and |d| can exceed a specific value of c, as a function of c. pr2(c) is taken from eq. 6, p|d|(c) is taken from eq. 14, and both functions appear in Table 4.
Figure 4:
Figure 4:
Mean maximum value of r2 and |d|, if pA and pB are drawn from independent Beta-(α, α) distributions. The dotted line indicates α = 1, which gives values that are identical to the case in which pA and pB are drawn from independent Uniform-(0, 1) distributions.
Figure 5:
Figure 5:
Values of linkage disequilibrium statistics as functions of the haplotype frequency pAB, for fixed values of pA and pB in S6.
Figure 6:
Figure 6:
Mean value of three linkage disequilibrium statistics in S6 as functions of pA and pB, assuming pAB ~ Uniform-(0,pB).(A)r2.(B)|d|.(C)r2/rmax2.
Figure 7:
Figure 7:
The distributions of r2, |d|, and r2/rmax2 values calculated from data in S6, when |D′| = 1. (A) r2 values if pAB = pB, lying on the surface r2 = (1 − pA)pB/[pA(1 − pB)]. (B) r2 values if pAB = 0, lying on the surface r2 = pApB/[(1 − pA)(1 − pB)]. (C) |d| values if pAB = pB, lying on the surface |d| = (1 − pA)/(1 − pB). (D) |d| values if pAB = 0, lying on the surface |d|=(1−pA)/(1−pB). (E) r2/rmax2 values if pAB = pB, lying on the surface r2/rmax2=1.(F)r2/rmax2 values if pAB = 0, lying on the surface r2/rmax2=[pA/(1−pA)]2.
Figure 8:
Figure 8:
The distributions of values of linkage disequilibrium statistics calculated from data in S6, given specific ranges of values for pA and pB. For each of four windows for pA and four windows for pB, we divide points into bins based on their values of each of four statistics (|D′|,r2,|d|,r2/rmax2). The number of locus pairs falling into a pair of bins appears in the top right corner of the group of four histograms associated with the bin pair.
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