Hardy-Weinberg Equilibrium and the Ternary Plot
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The Hardy-Weinberg law, formulated about 100 years ago, states that under certain
assumptions, the three genotypes AA, AB and BB at a bi-allelic locus are expected to occur in
the proportions p2, 2pq, and q2 respectively, where p is the allele frequency of A, and q = 1-p.
There are many statistical tests being used to check whether empirical marker data obeys the
Hardy-Weinberg principle. Among these are the classical xi-square test (with or without
continuity correction), the likelihood ratio test, Fisher's Exact test, and exact tests in combination
with Monte Carlo and Markov Chain algorithms. Tests for Hardy-Weinberg equilibrium (HWE)
are numerical in nature, requiring the computation of a test statistic and a p-value.
There is however, ample space for the use of graphics in HWE tests, in particular for the ternary
plot. Nowadays, many genetical studies are using genetical markers known as Single
Nucleotide Polymorphisms (SNPs). SNP data comes in the form of counts, but from the counts
one typically computes genotype frequencies and allele frequencies. These frequencies satisfy
the unit-sum constraint, and their analysis therefore falls within the realm of compositional data
analysis (Aitchison, 1986). SNPs are usually bi-allelic, which implies that the genotype
frequencies can be adequately represented in a ternary plot. Compositions that are in exact
HWE describe a parabola in the ternary plot. Compositions for which HWE cannot be rejected in
a statistical test are typically “close" to the parabola, whereas compositions that differ
significantly from HWE are “far". By rewriting the statistics used to test for HWE in terms of
heterozygote frequencies, acceptance regions for HWE can be obtained that can be depicted in
the ternary plot. This way, compositions can be tested for HWE purely on the basis of their
position in the ternary plot (Graffelman & Morales, 2008). This leads to nice graphical
representations where large numbers of SNPs can be tested for HWE in a single graph. Several
examples of graphical tests for HWE (implemented in R software), will be shown, using SNP
data from different human populations
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