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FST and the triangle inequality for biallelic markers

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  • Arbisser, Ilana M.
  • Rosenberg, Noah A.

Abstract

The population differentiation statistic FST, introduced by Sewall Wright, is often treated as a pairwise distance measure between populations. As was known to Wright, however, FST is not a true metric because allele frequencies exist for which it does not satisfy the triangle inequality. We prove that a stronger result holds: for biallelic markers whose allele frequencies differ across three populations, FSTnever satisfies the triangle inequality. We study the deviation from the triangle inequality as a function of the allele frequencies of three populations, identifying the frequency vector at which the deviation is maximal. We also examine the implications of the failure of the triangle inequality for four-point conditions for placement of groups of four populations on evolutionary trees. Next, we study the extent to which FST fails to satisfy the triangle inequality in human genomic data, finding that some loci produce deviations near the maximum. We provide results describing the consequences of the theory for various types of data analysis, including multidimensional scaling and inference of neighbor-joining trees from pairwise FST matrices.

Suggested Citation

  • Arbisser, Ilana M. & Rosenberg, Noah A., 2020. "FST and the triangle inequality for biallelic markers," Theoretical Population Biology, Elsevier, vol. 133(C), pages 117-129.
    • Handle: RePEc:eee:thpobi:v:133:y:2020:i:c:p:117-129
    DOI: 10.1016/j.tpb.2019.05.003
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    References listed on IDEAS

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    1. Francis Cailliez, 1983. "The analytical solution of the additive constant problem," Psychometrika, Springer;The Psychometric Society, vol. 48(2), pages 305-308, June.
    2. Noah A Rosenberg & Saurabh Mahajan & Sohini Ramachandran & Chengfeng Zhao & Jonathan K Pritchard & Marcus W Feldman, 2005. "Clines, Clusters, and the Effect of Study Design on the Inference of Human Population Structure," PLOS Genetics, Public Library of Science, vol. 1(6), pages 1-12, December.
    3. Wang Chaolong & Szpiech Zachary A & Degnan James H & Jakobsson Mattias & Pemberton Trevor J & Hardy John A & Singleton Andrew B & Rosenberg Noah A, 2010. "Comparing Spatial Maps of Human Population-Genetic Variation Using Procrustes Analysis," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 9(1), pages 1-22, January.
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