Testing for Hardy-Weinberg equilibrium at biallelic genetic markers on the X chromosome

Abstract

Testing genetic markers for Hardy-Weinberg equilibrium (HWE) is an important tool for detecting genotyping errors in large-scale genotyping studies. For markers at the X chromosome, typically the χ(2) or exact test is applied to the females only, and the hemizygous males are considered to be uninformative. In this paper we show that the males are relevant, because a difference in allele frequency between males and females may indicate HWE not to hold. The testing of markers on the X chromosome has received little attention, and in this paper we lay down the foundation for testing biallelic X-chromosomal markers for HWE. We develop four frequentist statistical test procedures for X-linked markers that take both males and females into account: the χ(2) test, likelihood ratio test, exact test and permutation test. Exact tests that include males are shown to have a better Type I error rate. Empirical data from the GENEVA project on venous thromboembolism is used to illustrate the proposed tests. Results obtained with the new tests differ substantially from tests that are based on female genotype counts only. The new tests detect differences in allele frequencies and seem able to uncover additional genotyping error that would have gone unnoticed in HWE tests based on females only.

Figure 1

HWE for a biallelic marker on the X chromosome. (a) Evolution of male and female allele frequencies over time after an initial difference of 1. (b) Simultaneous evolution of the female genotype frequencies over time.

Figure 2

Hardy–Weinberg (dis)equilibrium for a biallelic marker on the X chromosome. (a) Equilibrium. (b) Disequilibrium due to deviating female genotype frequencies. (c) Disequilibrium due to nonhomogeneous allele frequencies. (d) Disequilibrium due to deviating female genotype frequencies and nonhomogeneous allele frequencies.

Figure 3

Joint distribution of m_A and f_AB for given sample size, minor allele count and number of males and females. Example for n=20, n_m=10, n_f=10 and n_A=6.

Figure 4

Type I error rate of X-chromosomal tests for HWP as a function of sex ratio and MAF. The Type I error rate of an all-individual test for HWP is plotted for the exact test with the standard P-value (red), the exact test with the mid P-value (green) and the χ² test without continuity correction (blue). n, sample size (100); nf, number of females; nm, number of males.

Figure 5

Type I error rate of tests for HWP as a function of sex ratio and MAF. The Type I error rate of an all-individual test for HWP on the X chromosome is plotted for the exact test with the standard P-value (red), and for the exact test that excluded the males (orange). n, sample size (100); nf, number of females; nm, number of males.

Figure 6

Scatter plots of P-values in original and -log10 scale for χ² tests (a, b) and exact tests (c, d) for HWE using females only and using both males and females for 4158 SNPs at the X chromosome of the venous thrombosis database. The horizontal and vertical black lines in (b) and (d) correspond to a significance level of 5%. Points colored according to their significance level in Fisher's test for equality of allele frequencies (range 0–1 from red to green).

Figure 7

Q–Q plots of −log₁₀ transformed P-values of χ² and exact tests for HWE for 4158 SNPs of the venous thrombosis database. (a, c) Females only and (b, d) all individuals.

Figure 8

Cluster plots of allele intensities of four SNPs of the venous thrombosis database. (a and b) are significant in both the female-only (P=0.0025, P=0.0010) and all-individual test (P=0.0005, P=0.0023). (c) is non-significant in the female-only test (P=0.4261) but highly significant in the all-individual test (P=0.0012). (d) is non-significant in the female-only test (P=0.8732) and close to significant in the all-individual test (P=0.0914).