Abstract
More than 25 years ago, Aitchison showed that the logratio principal component analysis of multiple samples of a biallelic polymorphism can evidentiate the Hardy-Weinberg law. However, hitherto compositional data analysis, that is, the logratio approach, has had little impact in population genetics. This article extends Aitchison's work to multiallelic polymorphisms showing how the Hardy-Weinberg law manifests itself in a logratio based statistical analysis with larger genotypic compositions. Excellent visualisations of equilibrium and disequilibrium are achieved by using compositional biplots based on allele and genotype frequencies taken across multiple populations. Some fundamental relationships between allelic and genotypic compositions are derived, and the close relationships between the logratio principal component analysis of allelic and genotypic compositions and the corresponding compositional biplots are established. Simulations and practical genetic data analysis are used to explore the implications of Hardy-Weinberg equilibrium for the logratio principal component analysis of genotypic compositions. A general multiallelic compositional measure for disequilibrium is presented, and shown to relate to the classical inbreeding coefficient. The proposed compositional analysis is illustrated with biallelic glyoxalase genotypes and with two multiallelic loci from the 1000 Genomes project, the forensic microsatellite D2S441 and the ABO locus. For the latter, a haplotype based approach is used and generates predictions of the three-allele ABO genotypes for the individuals of the expanded 1000 Genomes project.