Response to Graffelman: Tests of Hardy-Weinberg Equilibrium

To the Editor: Testing for Hardy-Weinberg equilibrium (HWE) is perhaps the most common quality-control procedure in all of human genetics. Although there are many potential explanations for departures from HWE, the prototypical causes of departure from HWE are genotyping error and differential missing-data rates among genotypes.1 These two are critically important because they can give rise to false positives in genetic association studies.2 Standard practice in association studies is to test for HWE in all samples (or control samples) and to reject any marker with a p value for HWE < α. For the HapMap project,3,4 α = 0.001, but other studies might elect different values. For large samples and common alleles, a convenient means of calculating these p values is to use a simple χ2 test. However, this χ2 test requires two simplifying assumptions that are never true: (1) that heterozygote counts are approximately normally distributed and (b) that these counts are continuous. In a Letter to the Editor, Graffelman suggests that a continuity correction mitigates problems associated with the second assumption. In our view, the best solution to the problems associated with using a χ2 test is the use of an exact test. A major impediment to exact tests is the associated computational burden, but that burden is greatly diminished with the use of the algorithm of Wigginton et al.5 for calculating exact probabilities and test statistics. Wigginton et al. note that with exact probabilities in hand, there are four possible tests of HWE. Specifically, they outline two one-tailed tests (Plow, Phigh) and two two-tailed tests (PHWE, P2α). They define PHWE as the probability of observing a genotype configuration at least as unlikely as that actually observed and P2α as min(1.0, 2Phigh, 2Plow). Wigginton et al. recommend that PHWE should be used in almost all circumstances and discard P2α as too conservative (i.e., as producing incorrect probability values). PDOST = min(2Phigh, 2Plow), the statistic proposed by Graffelman, is just an imperfect approximation of P2α. PDOST often takes values > 1.0 and still produces incorrect p values whenever allele frequencies are unequal. If we denote PDOST(ξ) as the value of PDOST associated with sample-configuration ξ, we can guarantee that, under the null hypothesis of HWE, P(PDOST ≤ PDOST(ξ)) < PDOST(ξ) whenever allele frequencies are unequal. In contrast, using PHWE, Plow, Phigh guarantees a properly calibrated test statistic so that, for example, P(PHWE ≤ PHWE(ξ)) = PHWE(ξ), regardless of allele frequency. A simple example is illustrative. Consider a sample of 100 individuals in whom two copies of the rare allele are present. Two configurations are possible, one with two heterozygotes and another with a single rare allele homozygote. The first configuration has probability of 198/199, and the second has a probability of 1/199. Suppose a single homozygote is observed. This gives a χ2 test p value of < 10−22 (without continuity correction) or < 10−6 (with continuity correction). Both are clearly wrong. Using PDOST is “better,” giving p = 2/199, but still incorrect. In contrast, PHWE correctly specifies that a configuration such as this occurs with p = 1/199. For rare alleles and unlikely configurations, PDOST and P2α are effectively equal to 2PHWE. For common alleles and large samples, the three statistics converge to more similar values, but, in those settings, χ2 test approximations can be conveniently used. The fact that PDOST and P2a detect fewer departures from HWE is not a virtue. It simply reflects that they are poorly calibrated statistics. If an investigator wishes to discard fewer SNPs due to HWE departures, it is more appropriate to lower α, the threshold for rejection. Just as we found no reason to recommend P2α originally, we find no reason to recommend PDOST now.