Pearson Tests for Hardy-Weinberg Equilibrium

What the Pearson Hardy-Weinberg Test Assesses

For a biallelic autosomal variant, the Hardy-Weinberg model specifies genotype proportions \(p^2\), \(2pq\), and \(q^2\), where \(p\) and \(q=1-p\) are the two allele frequencies. The earlier material on expected genotype counts established how to estimate these proportions and turn them into expected counts. Here the question is whether the observed genotype counts depart more than expected from those model-based counts.

The Pearson goodness-of-fit test summarizes the discrepancies across the three genotype categories. Its reference distribution is approximate: under the null model and suitable regularity conditions, the statistic is compared with a chi-square distribution with one degree of freedom. A small upper-tail p-value is evidence against the Hardy-Weinberg proportions for the sampled population; it does not identify the reason for a departure.

Assumptions and the One-Degree-of-Freedom Reference

The test requires genotype counts from a clearly defined group of diploid individuals at a biallelic autosomal locus. It assumes individuals contribute independent observations, genotype calls are sufficiently accurate for the counts to be meaningful, and the null model of Hardy-Weinberg proportions is appropriate for the population being assessed. The model is generally motivated by conditions such as random mating with respect to the locus and no consequential population mixture or other process disrupting the proportions. These are modeling conditions, not guarantees supplied by the calculation.

The chi-square approximation also requires sufficiently large expected counts. A common practical rule is that each expected count should be at least about 5; this is a diagnostic guideline rather than a universal theorem. When expected counts are sparse, the Pearson statistic can still be computed, but its chi-square p-value may be poorly calibrated. A small expected count is not repaired by a large total sample size if one genotype category remains rare.

The degrees of freedom reflect both the three genotype categories and the fitted allele frequency. The three observed counts sum to \(n\), so only two count contrasts can vary independently. Estimating \(p\) from those same counts uses one of these dimensions: changes in the fitted frequency account for one direction of discrepancy. One independent direction of departure remains, giving \(3-1-1=1\) degree of freedom. This is why the test is not compared with a two-degree-of-freedom distribution when the allele frequency is estimated from the sample. If the allele frequency were fixed in advance rather than estimated, the usual goodness-of-fit count would instead be \(3-1=2\) degrees of freedom.

The justification is asymptotic. Under the null, the three genotype counts have a multinomial distribution with probabilities determined by \(p\). For large samples with probabilities away from zero, their centered, scaled count vector is approximately normal. Its possible fluctuations lie in the two-dimensional plane imposed by the fixed total. Estimating \(p\) fits one direction of that fluctuation, leaving a one-dimensional residual component. The Pearson quadratic form in that remaining component is asymptotically distributed as \(\chi^2_1\). This approximation depends on the null model and adequate expected counts; it is not an exact finite-sample identity.

Using the Statistic and Its P-Value

The null hypothesis is that the genotype proportions follow Hardy-Weinberg proportions at the allele frequency estimated from the sample. The alternative is that they do not. Because the Pearson statistic adds squared discrepancies, departures in either direction increase \(X^2\); the p-value is therefore an upper-tail probability. A large statistic means the observed counts are relatively far from the fitted expected counts, after scaling each discrepancy by its expected count.

A p-value is calculated under the null model and the approximation conditions: it is the probability of obtaining a statistic at least as large as the observed one if those conditions hold. It is not the probability that the null is true, nor is it the probability that a particular biological or technical explanation is correct. The significance threshold \(\alpha\) is chosen before interpreting the result; the p-value is not a measure of the size or importance of the departure.

Diagnostics, Alternatives, and Limits

Before interpreting a Pearson test, inspect the observed and expected counts together. Check that the genotype categories are defined consistently, the three counts total the number of included samples, and the allele frequency was computed from allele copies rather than from the proportion of individuals with a genotype. Confirm that the expected counts support the approximation and that the sampled individuals can reasonably be treated as independent. Unexpectedly large discrepancies should prompt scrutiny of the data and model, not just a report of a p-value.

For small expected counts, an exact Hardy-Weinberg test is a principal alternative because it avoids relying on the same large-sample chi-square approximation. The Pearson method is simple and interpretable when counts are adequate; an exact method is more suitable when that approximation is doubtful. Neither method identifies why counts depart from the model. Detailed evaluation of possible causes requires additional context and diagnostics.

A test can also flag a tiny departure in a very large sample, so statistical significance should not be confused with practical importance. Conversely, a nonsignificant result is not proof of equilibrium, especially with limited sample size. Treat the result as one quality-control signal to interpret alongside data quality, population definition, and other evidence. A test outcome by itself is not a justification for deleting a variant or a sample.

Common Mistakes

• Using two degrees of freedom automatically. With allele frequency estimated from these genotype counts, one parameter is fitted; the Pearson test uses \(df=1\).
• Using allele frequency as the genotype proportion. Estimate \(\hat{p}\) from the total number of \(A\) allele copies, then calculate all three expected counts under the model.
• Ignoring small expected counts. A computed statistic does not guarantee a trustworthy chi-square p-value. Inspect all expected counts first.
• Calling a nonsignificant result proof of equilibrium. Failure to reject means the test did not find sufficient evidence of departure; it does not establish that all assumptions hold.
• Treating a significant result as an explanation. It signals mismatch with the fitted proportions, not a specific biological mechanism or genotyping error.