Interpreting Hardy-Weinberg Failures

What a Hardy-Weinberg Departure Can—and Cannot—Tell You

A Hardy-Weinberg (HWE) test asks whether observed genotype counts are compatible with the proportions expected under a specified population model. In genomic quality control, a departure can be a useful warning: genotyping error, population structure, or other departures from the model may be present. But the test does not identify which explanation applies. Nor does a small p-value, by itself, show that a variant's genotype calls are bad.

The central task is therefore interpretation, not simply labelling a variant as “failed.” Begin with the observed and expected counts, the size and direction of the departure, the sample and population being tested, and the quality evidence available for that locus. The earlier material on Hardy-Weinberg deviations in controls and cases explains why the sample context matters: a departure in cases can have a different interpretation from one in controls.

For observed counts \(O_{AA}, O_{AB}, O_{BB}\), estimate the allele frequency from the called genotypes, for example \(\hat{p}=(2O_{AA}+O_{AB})/(2n)\), where \(n=O_{AA}+O_{AB}+O_{BB}\). The corresponding expected counts are \(n\hat{p}^{\,2}\), \(2n\hat{p}(1-\hat{p})\), and \(n(1-\hat{p})^2\). The Pearson statistic compares each observed count with its expected count. With a biallelic variant and an estimated allele frequency, its usual large-sample reference distribution has \(df=1\). The earlier material on Pearson and exact Hardy-Weinberg tests covers the test procedures; here the emphasis is on reading their results responsibly.

Assumptions and the Meaning of the Test

The HWE proportions describe a model, not a universal requirement for every sample. For the usual interpretation, consider a diploid autosomal locus in a population with random mating, no relevant selection, mutation, or migration affecting the locus, and a sufficiently large population. The sample should represent one population in a way that does not pool groups with substantially different allele frequencies. Statistical testing also relies on appropriately called genotypes and independent sampled individuals; close relatedness can undermine the usual reference distribution.

In a QC setting, these conditions are not guaranteed. A test may flag a departure because one or more model assumptions fail, because genotypes are miscalled, or because random sampling produced a difference. It may also flag a departure for a biologically meaningful reason. The null hypothesis is compatibility with the HWE model in the tested sample, not “the data are error-free.” A p-value is the probability, under the null model and test assumptions, of obtaining a statistic at least as extreme as the one observed. It is not the probability that the null is true, or the probability that a genotype is wrong.

A short derivation clarifies why the usual Pearson test has one degree of freedom. There are three genotype counts, but their sum is fixed at \(n\), leaving two independent count contrasts. Estimating the allele frequency from the same genotype counts uses one of those degrees of freedom. Thus the usual test has \(3-1-1=1\) degree of freedom. This reference distribution is an approximation whose adequacy depends on the sample and expected counts; exact methods are useful when counts are sparse.

A p-value addresses the strength of evidence against the specified model, not the practical size or cause of a departure. To describe the data, report the observed and expected counts and identify whether the departure is mainly a deficit or an excess of heterozygotes relative to expectation. Then assess plausible explanations and follow-up evidence. The same p-value can accompany different effect sizes, sample sizes, and QC implications.

How Different Causes Can Produce Departures

Some departures are compatible with technical problems. For example, allele dropout or systematic misclassification can convert heterozygotes into homozygote calls, producing a heterozygote deficit. Other calling problems can create different patterns, including excess apparent heterozygotes. These are hypotheses to check against the assay and calling evidence, not diagnoses inferred from the direction of the departure alone.

Population mixture is another important possibility. If groups with different allele frequencies are pooled, the pooled genotypes need not follow HWE proportions calculated from the pooled allele frequency—even if each group is individually in equilibrium. This is why examining ancestry or other meaningful strata can change interpretation.

Sampling Variation, Sample Size, and Test Choice

A modest departure can occur by chance, especially in a small sample. Conversely, a small proportional difference can yield a small p-value in a very large sample. The counts, the departure's size, and the study context should therefore accompany any p-value. Failure to reject HWE is not proof that the population meets every HWE assumption or that calls are correct; a small sample may simply provide little evidence either way.

For an individual biallelic locus with sufficiently large expected counts, the Pearson test offers a convenient approximation. An exact HWE test avoids the chi-square approximation and is often preferred when expected counts are small; it still tests compatibility with an HWE model and does not diagnose the cause. If a sample contains meaningful subgroups, examining justified strata can help assess mixture. These approaches answer different questions and are not substitutes for examining genotype-call evidence.

In genome-wide QC, many loci are tested. Even when each null model is correct, some small p-values will occur by chance across a large set of tests. Use an appropriate multiple-testing strategy when making scan-wide claims, and do not let a threshold replace context. A result may warrant review without warranting automatic exclusion; conversely, absence of a small p-value does not certify a locus as reliable.

Practical Interpretation and Common Pitfalls

A useful review follows a sequence: verify the sample being tested and the genotype counts; compare observed and expected counts; describe the direction and size of the departure; assess whether the test approximation is suitable; then evaluate plausible causes using independent evidence. Check whether the pattern varies across batches or relevant population strata, and inspect technical call evidence before deciding whether to flag, re-call, or filter a variant. When the data are from cases, interpret departures in light of the case-selection context rather than treating them as identical to control-sample QC signals.

• Mistake: “The p-value is tiny, so the genotypes are wrong.” A small p-value is evidence against the tested HWE model under its assumptions. It does not identify genotyping error; mixture or other model departures can also explain it. Seek supporting technical evidence.
• Mistake: “The test was not significant, so the locus passed QC.” Non-significance is not proof of equilibrium or accuracy. Consider sample size, power, call quality, and other QC checks.
• Mistake: “A heterozygote deficit has one unique cause.” Several mechanisms can produce similar count patterns. Treat the direction as a clue for investigation, not a causal conclusion.
• Mistake: “A pooled test describes every subgroup.” A pooled departure may arise because allele frequencies differ among groups. Review scientifically meaningful strata before attributing the signal to the assay.

The limitations are fundamental: HWE testing is a one-locus summary, depends on assumptions that may not hold, and does not reveal which individuals are miscalled. It should be combined with other QC evidence and interpreted alongside the research design. A QC action should be proportionate to the strength of the departure and the corroborating evidence, not determined by a p-value cutoff alone.