An Application of the Isometric Log-Ratio Transformation in Relatedness Research

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 187)

Abstract

Genetic marker data contains information on the degree of relatedness of a pair of individuals. Relatedness investigations are usually based on the extent to which alleles of a pair of individuals match over a set of markers for which their genotype has been determined. A distinction is usually drawn between alleles that are identical by state (IBS) and alleles that are identical by descent (IBD). Since any pair of individuals can only share 0, 1, or 2 alleles IBS or IBD for any marker, 3-way compositions can be computed that consist of the fractions of markers sharing 0, 1, or 2 alleles IBS (or IBD) for each pair. For any given standard relationship (e.g., parent–offspring, sister–brother, etc.) the probabilities \(k_0, k_1\) and \(k_2\) of sharing 0, 1 or 2 IBD alleles are easily deduced and are usually referred to as Cotterman’s coefficients. Marker data can be used to estimate these coefficients by maximum likelihood. This maximization problem has the 2-simplex as its domain. If there is no inbreeding, then the maximum must occur in a subset of the 2-simplex. The maximization problem is then subject to an additional nonlinear constraint (\(k_1^2 \ge 4 k_0 k_2\)). Special optimization routines are needed that do respect all constraints of the problem. A reparametrization of the likelihood in terms of isometric log-ratio (ilr) coordinates greatly simplifies the maximization problem. In isometric log-ratio coordinates the domain turns out to be rectangular, and maximization can be carried out by standard general-purpose maximization routines. We illustrate this point with some examples using data from the HapMap project.

Keywords

Genetic marker Identity-by-state Identity-by-descent Hardy–Weinberg equilibrium Composition Closure Ternary plot Isometric log-ratio transformation 

References

  1. 1.
    Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G., Barceló-Vidal, C.: Isometric logratio transformations for compositional data analysis. Math. Geol. 35(3), 279–300 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Epstein, M., Duren, W., Boehnke, M.: Improved inference of relationship for pairs of individuals. Am. J. Hum. Genet. 67, 1219–1231 (2000)CrossRefGoogle Scholar
  3. 3.
    Gazal, S., Sahbatou, M., Perdry, H., Letort, S., Gnin, E., Leutenegger, A.L.: Inbreeding coefficient estimation with dense snp data: comparison of strategies and application to hapmap III. Hum. Hered. 77(1–4), 49–62 (2014)CrossRefGoogle Scholar
  4. 4.
    Ghalanos, A., Theussl, S.: Rsolnp: general non-linear optimization using augmented lagrange multiplier method. R package version 1.15 (2014). http://CRAN.R-project.org/package=Rsolnp
  5. 5.
    Graffelman, J., Egozcue, J.J.: Hardy-weinberg equilibrium: a nonparametric compositional approach. In: Pawlowsky-Glahn, V., Buccianti, A. (eds.) Compositional Data Analysis: Theory and Applications, pp. 208–217. John Wiley & Sons, Ltd. (2011)Google Scholar
  6. 6.
    Pemberton, T.J., Wang, C., Li, J.Z., Rosenberg, N.A.: Inference of unexpected genetic relatedness among individuals in hapmap phase III. Am. J. Hum. Genet. 87, 457–464 (2010)CrossRefGoogle Scholar
  7. 7.
    R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2014). http://www.R-project.org/
  8. 8.
    Stevens, E.L., Baugher, J.D., Shirley, M.D., Frelin, L.P., Pevsner, J.: Unexpected relationships and inbreeding in hapmap phase III populations. PLoS ONE 7(11) (2012). doi:10.1371/journal.pone.0049575
  9. 9.
    The International HapMap Consortium: Integrating common and rare genetic variation in diverse human populations. Nature 467, 52–58 (2010)Google Scholar
  10. 10.
    Thompson, E.A.: The estimation of pairwise relationships. Ann. Hum. Genet. 39, 173–188 (1975)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Thompson, E.A.: Estimation of relationships from genetic data. In: Rao, C.R., Chakraborty, R. (eds.) Handb. Stat., vol. 8, pp. 255–269. Elsevier Science, Amsterdam (1991)Google Scholar
  12. 12.
    Weir, B.S., Anderson, A.D., Hepler, A.B.: Genetic relatedness analysis: modern data and new challenges. Nature Rev. Genet. 7(10), 771–780 (2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Department of Computer ScienceApplied Mathematics and Statistics, Universitat de GironaGironaSpain

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