Efficient test for deviation from Hardy-Weinberg equilibrium with known or ambiguous typing in highly polymorphic loci

Abstract

The Hardy-Weinberg equilibrium (HWE) assumption is essential to many population genetics models. Multiple tests were developed to test its applicability in observed genotypes. Current methods are divided into exact tests applicable to small populations and a small number of alleles, and approximate goodness-of-fit tests. Existing tests cannot handle ambiguous typing in multi-allelic loci. We here present a novel exact test Unambiguous Multi Allelic Test (UMAT) not limited to the number of alleles and population size, based on a perturbative approach around the current observations. We show its accuracy in the detection of deviation from HWE. We then propose an additional model to handle ambiguous typing using either sampling into UMAT or a goodness-of-fit test test with a variance estimate taking ambiguity into account, named Asymptotic Statistical Test with Ambiguity (ASTA). We show the accuracy of ASTA and the possibility of detecting the source of deviation from HWE. We apply these tests to the HLA loci to reproduce multiple previously reported deviations from HWE, and a large number of new ones.

Keywords: Gibbs sampling; Hardy–Weinberg equilibrium; imputation algorithms.

Figure 1

DC case results; (a). visual representation of the swap of two alleles in the UMAT test; first sample an allele according to the alleles distribution, then sample an allele according to the allele pairs distribution given allele and subtract one individual with this pair (it must exist), then, sample a pair of alleles according to the HWE alleles distribution and add one individual to it; then, in each iteration, sample an allele according to the allele pairs distribution given allele and subtract one individual from the pair ; sample an allele according to the allele marginal distribution and add one individual to the pair ; this way, only the marginal observations of the alleles are changed; and have one excessive marginal observation, as and had in the beginning and therefore take their place; note that we can randomly invert their order; similarly, and are missing one marginal observation, as and had in the beginning and therefore take their place in the next iteration; (b). 49 implementations of UMAT using the same observations in HWE, with ( represents the fit of the simulation to the HWE, as described in Materials and methods), and alleles; (c). 49 implementations of UMAT using the same observations with a slight deviation of HWE, with , and alleles; (d). P-value results of UMAT for different allele numbers and alpha values, with ; (e). P-value results of UMAT for different population sizes and alpha values, with 100 alleles; (f). elapsed time in seconds for running UMAT using different allele numbers and population sizes, with ; as one can see, the population size has no effect on the run time; (g). scatter plot showing for 100 randomly chosen SNPs, the P-value results obtained with Chi-Square and UMAT.

Figure 2

AC case results; (a). P-value results of UMAT with ambiguity for different values ( represents the probability for a sample to have ambiguous typing, as described in Materials and methods) and values, with 100 alleles and ; (b)–(d). scatter plot of real versus estimated variance for simulated data with 50 alleles, (respectively); each dot is an allele pair ; the x-axis represents the sampled variance over , the y-axis represents either the value of or the corrected denominator used in ASTA: ; (e). fraction of positive results for , and each test: raditional Chi-Squared, ASTA, and Chi-Squared with sampling (i.e. for each person sampling certain alleles given all his possible allele pair observations and then using a traditional Chi-Squared); the results are out of 300 simulations with five alleles and ; (f). we simulated data (see Methods) using alleles, ; here all the data are in HWE, except from the pairs containing the allele (top bar); each bar corresponds to a specific allele and shows the normalized statistic (the sum of Chi-Square term with correction only on pairs containing this allele, and divided by the Degree of Freedom (DOF): the number of pairs containing this allele minus 1), as well as the for this allele, here the is calculated using the statistic and DOF of the allele.

Figure 3

Deviation of alleles from HWE; each column is one of the five broad US populations; each row is a locus (A, B, C, DQB1, DRB1); within each subplot the bar is the ASTA score divided by the DOF; the color represents the log P value; deeper colors are more significant; the bars are ordered by decreasing significance; note that the scale of each plot is very different, but the coloring of the bar is consistent among all subplots.

Figure 4

Each subplot is a different locus (A,B,C,DQB1, DRB1); the different colors represent the log base 10 of: the scores divided by the DOF; the ASTA score (orange) is slightly higher than the classical Chi-Square (blue) and from the sampling (green); note that in this case, the ambiguity is limited; as such, the differences are not very large; the top populations are the sub-populations, followed by the broad populations, followed by the entire donor registry; each detailed population is colored according to the broad population.

Figure 5

Statistics of Chi-Squared, ASTA, and the inverse CDF of Chi-Squared using a 0.5 significance level; each statistic is calculated for different degrees of freedom (DOF) values, ranging from 1 to the number of allele pairs minus 1; the statistics are first calculated for the first two allele pairs (1 DOF), and subsequently, one additional pair is included to recalculate the statistics (increasing the DOF by 1) until all pairs are included; observations for each plot are generated with the following parameters: (a). alleles, population size, , ; (b). alleles, population size, , ; (c). alleles, population size, , ; (d). alleles, population size, , .