A Compositional Approach to Allele Sharing Analysis

  • I. Galván-Femenía
  • J. Graffelman
  • C. Barceló-i-Vidal
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 187)


Relatedness is of great interest in population-based genetic association studies. These studies search for genetic factors related to disease. Many statistical methods used in population-based genetic association studies (such as standard regression models, t-tests, and logistic regression) assume that the observations (individuals) are independent. These techniques can fail if independence is not satisfied. Allele sharing is a powerful data analysis technique for analyzing the degree of dependence in diploid species. Two individuals can share 0, 1, or 2 alleles for any genetic marker. This sharing may be assessed for alleles identical by state (IBS) or identical by descent (IBD). Starting from IBS alleles, it is possible to detect the type of relationship of a pair of individuals by using graphical methods. Typical allele sharing analysis consists of plotting the fraction of loci sharing 2 IBS alleles versus the fraction of sharing 0 IBS alleles. Compositional data analysis can be applied to allele sharing analysis because the proportions of sharing 0, 1 or 2 IBS alleles (denoted by \(p_0\), \(p_1\), and \(p_2\)) form a 3-part-composition. This chapter provides a graphical method to detect family relationships by plotting the isometric log-ratio transformation of \(p_0\), \(p_1\), and \(p_2\). On the other hand, the probabilities of sharing 0, 1, or 2 IBD alleles (denoted by \(k_0, k_1, k_2\)), which are termed Cotterman’s coefficients, depend on the relatedness: monozygotic twins, full-siblings, parent-offspring, avuncular, first cousins, etc. It is possible to infer the type of family relationship of a pair of individuals by using maximum likelihood methods. As a result, the estimated vector \({\hat{\varvec{k}}}=(\hat{k}_0, \hat{k}_1,\hat{k}_2)\) for each pair of individuals forms a 3-part-composition and can be plotted in a ternary diagram to identify the degree of relatedness. An R package has been developed for the study of genetic relatedness based on genetic markers such as microsatellites and single nucleotide polymorphisms from human populations, and is used for the computations and graphics of this contribution.


Allele sharing Identical by state Identical by descent Cotterman’s coefficients Ternary diagram Isometric log-ratio transformation 



We thank the referees and the editors for their comments on the manuscript. This study was supported by grant CODARSS MTM2012-33236 (2013–2015) of the Spanish Ministry of Education and Science.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • I. Galván-Femenía
    • 1
  • J. Graffelman
    • 2
  • C. Barceló-i-Vidal
    • 1
  1. 1.Department of Computer Science, Applied Mathematics and StatisticsUniversitat de GironaGironaSpain
  2. 2.Department of Statistics and Operations ResearchUniversitat Politècnica de CatalunyaBarcelonaSpain

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