Identity-by-descent matrix decomposition using latent ancestral allele models.
Genetic linkage and association studies are empowered by proper modeling of relatedness among individuals. Such relatedness can be inferred from marker and/or pedigree information. In this study, the genetic relatedness among n inbred individuals at a particular locus is expressed as an n x n square matrix Q. The elements of Q are identity-by-descent probabilities, that is, probabilities that two individuals share an allele descended from a common ancestor. In this representation the definition of the ancestral alleles and their number remains implicit. For human inspection and further analysis, an explicit representation in terms of the ancestral allele origin and the number of alleles is desirable. To this purpose, we decompose the matrix Q by a latent class model with K classes (latent ancestral alleles). Let P be an n x K matrix with assignment probabilities of n individuals to K classes constrained such that every element is nonnegative and each row sums to 1. The problem then amounts to approximating Q by PP(T), while disregarding the diagonal elements. This is not an eigenvalue problem because of the constraints on P. An efficient algorithm for calculating P is provided. We indicate the potential utility of the latent ancestral allele model. For representative locus-specific Q matrices constructed for a set of maize inbreds, the proposed model recovered the known ancestry.
ter Braak
Cajo J F
CJ
Biometris, Wageningen University and Research Centre, Wageningen, The Netherlands. cajo.terbraak@wur.nl
Boer
Martin P
MP
Totir
L Radu
LR
Winkler
Christopher R
CR
Smith
Oscar S
OS
Bink
Marco C A M
MC
D000465:Algorithms; D000483:Alleles; D001499:Bayes Theorem; D008040:Genetic Linkage / Q000235:genetics*; D005819:Genetic Markers; D006801:Humans; D010375:Pedigree; D040641:Quantitative Trait Loci / Q000235:genetics*
D005819:Genetic Markers
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