Bounding measures of genetic similarity and diversity using majorization.

*(English)*Zbl 1417.92102To further advance the study of mathematical properties of population-genetic statistics describing similarity and diversity of alleles in a population, the authors investigate an application of the theory of majorization to these statistics. They exploit the majorization theory to study the relationship of the frequency of the most frequent allele to various homozygosity-related statistics as well as to the Shannon-Wearer entropy statistic for genetic diversity. The method introduced by the authors provide simpler derivations of results reported by
N. A. Rosenberg and M. Jakobsson [“The relationship between homozygosity and the frequency of the most frequent allele”, Genetics 179, 2027–2036 (2008; doi:10.1534/genetics.107.084772)] and S. B. Reddy and N. A. Rosenberg [J. Math. Biol. 64, No. 1–2, 87–108 (2012; Zbl 1284.92058)]; these derivations naturally lead to consider a larger family of homozygosity-related statistics called \(\alpha\)-homozygosities, where extreme values occur at the same allele frequency vectors as the standard homozygosity. The constraints are illustrated on the statistics using data from human populations.

Reviewer: József Sándor (Cluj-Napoca)

##### MSC:

92D10 | Genetics and epigenetics |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

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\textit{A. J. Aw} and \textit{N. A. Rosenberg}, J. Math. Biol. 77, No. 3, 711--737 (2018; Zbl 1417.92102)

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##### References:

[1] | Edge, MD; Rosenberg, NA, Upper bounds on \(F_{ST}\) in terms of the frequency of the most frequent allele and total homozygosity: the case of a specified number of alleles, Theor Popul Biol, 97, 20-34, (2014) · Zbl 1303.92070 |

[2] | Garud, NR; Rosenberg, NA, Enhancing the mathematical properties of new haplotype homozygosity statistics for the detection of selective sweeps, Theor Popul Biol, 102, 94-101, (2015) · Zbl 1342.92126 |

[3] | Hedrick, PW, Highly variable loci and their interpretation in evolution and conservation, Evolution, 53, 313-318, (1999) |

[4] | Hedrick, PW, A standardized genetic differentiation measure, Evolution, 59, 1633-1638, (2005) |

[5] | Ho S-W, Verdú S (2015) Convexity/concavity of Rényi entropy and \(\alpha \)-mutual information. In: Proceedings of the 2015 IEEE international symposium on information theory (ISIT). IEEE, pp 745-749 |

[6] | Jakobsson, M.; Edge, MD; Rosenberg, NA, The relationship between \(F_{ST}\) and the frequency of the most frequent allele, Genetics, 193, 515-528, (2013) |

[7] | Karamata, J., Sur une inégalité relative aux fonctions convexes, Publ l’Inst Math, 1, 145-147, (1932) · JFM 58.0211.01 |

[8] | Legendre P, Legendre L (1998) Numerical ecology. Second English edition. Elsevier, Amsterdam · Zbl 1033.92036 |

[9] | Lewontin, RC, The apportionment of human diversity, Evol Biol, 6, 381-398, (1972) |

[10] | Long, JC; Kittles, RA, Human genetic diversity and the nonexistence of biological races, Hum Biol, 75, 449-471, (2003) |

[11] | Marshall AW, Olkin I, Arnold B (2010) Inequalities: theory of majorization and its applications. Springer, New York · Zbl 1219.26003 |

[12] | Maruki, T.; Kumar, S.; Kim, Y., Purifying selection modulates the estimates of population differentiation and confounds genome-wide comparisons across single-nucleotide polymorphisms, Mol Biol Evol, 29, 3617-3623, (2012) |

[13] | Reddy, SB; Rosenberg, NA, Refining the relationship between homozygosity and the frequency of the most frequent allele, J Math Biol, 64, 87-108, (2012) · Zbl 1284.92058 |

[14] | Rosenberg, NA; Jakobsson, M., The relationship between homozygosity and the frequency of the most frequent allele, Genetics, 179, 2027-2036, (2008) |

[15] | Rosenberg, NA; Kang, JTL, Genetic diversity and societally important disparities, Genetics, 201, 1-12, (2015) |

[16] | Rosenberg, NA; Mahajan, S.; Ramachandran, S.; Zhao, C.; Pritchard, JK; Feldman, MW, Clines, clusters, and the effect of study design on the inference of human population structure, PLoS Genet, 1, e70, (2005) |

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