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Mathematical properties of the r2 measure of linkage disequilibrium

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  • VanLiere, Jenna M.
  • Rosenberg, Noah A.

Abstract

Statistics for linkage disequilibrium (LD), the non-random association of alleles at two loci, depend on the frequencies of the alleles at the loci under consideration. Here, we examine the r2 measure of LD and its mathematical relationship to allele frequencies, quantifying the constraints on its maximum value. Assuming independent uniform distributions for the allele frequencies of two biallelic loci, we find that the mean maximum value of r2 is ∼0.43051, and that r2 can exceed a threshold of 4/5 in only ∼14.232% of the allele frequency space. If one locus is assumed to have known allele frequencies–the situation in an association study in which LD between a known marker locus and an unknown trait locus is of interest–we find that the mean maximum value of r2 is greatest when the known locus has a minor allele frequency of ∼0.30131. We find that in 1/4 of the space of allowed values of minor allele frequencies and haplotype frequencies at a pair of loci, the unconstrained maximum r2 allowing for the possibility of recombination between the loci exceeds the constrained maximum assuming that no recombination has occurred. Finally, we use rmax2 to examine the connection between r2 and the D′ measure of linkage disequilibrium, finding that r2/rmax2=D′2 for ∼72.683% of the space of allowed values of (pa,pb,pab). Our results concerning the properties of r2 have the potential to inform the interpretation of unusual LD behavior and to assist in the design of LD-based association-mapping studies.

Suggested Citation

  • VanLiere, Jenna M. & Rosenberg, Noah A., 2008. "Mathematical properties of the r2 measure of linkage disequilibrium," Theoretical Population Biology, Elsevier, vol. 74(1), pages 130-137.
    • Handle: RePEc:eee:thpobi:v:74:y:2008:i:1:p:130-137
    DOI: 10.1016/j.tpb.2008.05.006
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    References listed on IDEAS

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    1. Vincent Plagnol & Jeffrey D Wall, 2006. "Possible Ancestral Structure in Human Populations," PLOS Genetics, Public Library of Science, vol. 2(7), pages 1-8, July.
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    Cited by:

    1. Stanley Luck, 2019. "Factoring a 2 x 2 contingency table," PLOS ONE, Public Library of Science, vol. 14(10), pages 1-23, October.
    2. Edge, Michael D. & Rosenberg, Noah A., 2014. "Upper bounds on FST in terms of the frequency of the most frequent allele and total homozygosity: The case of a specified number of alleles," Theoretical Population Biology, Elsevier, vol. 97(C), pages 20-34.
    3. Smith, Reginald D., 2020. "The nonlinear structure of linkage disequilibrium," Theoretical Population Biology, Elsevier, vol. 134(C), pages 160-170.
    4. Garud, Nandita R. & Rosenberg, Noah A., 2015. "Enhancing the mathematical properties of new haplotype homozygosity statistics for the detection of selective sweeps," Theoretical Population Biology, Elsevier, vol. 102(C), pages 94-101.

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