Hardy-Weinberg Deviations in Cases

Why Case Genotypes Need Not Follow Hardy–Weinberg Proportions

A Hardy–Weinberg check asks whether genotype proportions are consistent with random union of alleles at a locus, given the allele frequency in the group being examined. Among controls, a substantial departure can be a useful quality-control signal. Among cases, however, disease status itself may change the genotype distribution: if genotype affects disease risk, selecting people because they have the disease also selects genotypes unequally.

This distinction matters in a case-control study. A case-only deviation is not automatically evidence that genotype calls are wrong. It may reflect the biological association being studied. Conversely, the fact that a disease association can affect cases does not mean every case-only deviation is genuine or that every disease model must produce one. The relationship depends on how risk varies across genotypes.

For this tutorial, assume a biallelic autosomal locus, correctly classified case and control status, and a source population in HWE before selection into cases or controls. Initially, also assume no population mixture or genotype-dependent participation. These assumptions let us isolate the effect of conditioning on disease status. The same logic applies if the effect allele is coded differently, provided the genotype labels and frequency are used consistently.

Conditioning on Disease Status

Let \(f_g=P(\text{case}\mid g)\) be the probability of being a case for genotype \(g\), where \(g=0,1,2\) is the number of effect alleles. In the source population, the genotype proportions are \(q^2,2pq,p^2\). Among cases, each source proportion is multiplied by its case probability and then normalized to make the proportions sum to one:

This is Bayes’ rule applied to each genotype. If the \(f_g\) values differ, the case distribution is reweighted relative to the source population. Its allele frequency must then be recalculated from the case genotypes; it is not generally the source-population frequency \(p\). HWE expectations for cases are based on that recalculated frequency.

The key comparison is between the reweighted case proportions and the HWE proportions calculated using the case allele frequency. A shift in allele frequency alone is not an HWE departure: the genotype proportions could shift while continuing to satisfy HWE at the new frequency.

Comparing Cases and Controls in a Case-Control Study

In a case-control sample, cases and controls are often sampled in numbers chosen by the study design, rather than in proportion to disease prevalence. That design choice does not change genotype proportions within the case group or within the control group, assuming sampling within each disease-status group does not depend on genotype. It does mean that the combined sample should not be treated as a source-population sample for this calculation.

For controls, the analogous conditional probability is \(P(g\mid\text{control})=P(g)(1-f_g)/\sum_{j=0}^{2}P(j)(1-f_j)\). The control genotype proportions can therefore differ from the source proportions too, although for a rare disease the difference may be small. The case and control tests should be carried out separately, using each group’s own estimated allele frequency.

For observed genotype counts \(O_0,O_1,O_2\) among \(n\) individuals, estimate the effect-allele frequency as \(\hat p=(O_1+2O_2)/(2n)\), and set \(\hat q=1-\hat p\). The HWE expected counts are \(n\hat q^2,2n\hat p\hat q,n\hat p^2\). The Pearson statistic is the sum of \((O-E)^2/E\) over the three genotypes; with the allele frequency estimated from the same counts, the usual large-sample reference has \(df=1\). The Pearson approximation requires expected counts large enough for that approximation to be reliable; an exact HWE test is an alternative when counts are sparse.

A Useful Exception: Multiplicative Genotype Weights

A disease association need not create case disequilibrium. Suppose the probabilities of being a case are proportional to \(1,r,r^2\) for genotypes \(0,1,2\), respectively, for some positive \(r\). These weights act multiplicatively per effect allele. Applying them to source HWE proportions gives unnormalized weights \(q^2,2pqr,p^2r^2\). These are the HWE proportions formed from the allele weights \(q\) and \(pr\), before normalization:

After division by \((q+pr)^2\), the case proportions are HWE proportions with effect-allele frequency \(p'=pr/(q+pr)\). Thus this particular pattern of case weighting changes the allele frequency but preserves HWE exactly. This is a mathematical exception, not a general claim that common disease-association models always preserve HWE: actual probabilities of being affected, and the sampling scheme, determine the relevant weights.

Checks, Failure Modes, and Interpretation

A case-control comparison is most informative when the groups are examined separately and the sampling and risk assumptions are made explicit. Before interpreting a departure, check that genotype counts use the same allele and genotype coding in both groups, that allele frequency was estimated separately in each group, and that the expected counts support the chosen test. A case-control imbalance in sample size does not justify pooling the groups for a single HWE test.

Other features can complicate the clean mechanism above. Population substructure can produce non-HWE genotype proportions within a combined group even if subgroups are each in HWE. Genotyping error, relatedness, selection into the study, and missingness can also affect observed counts. These possibilities mean that a case-only departure is compatible with a genuine association, not diagnostic of one. Likewise, a control group close to HWE is reassuring for this particular check but does not rule out all data-quality problems or establish the absence of association.

A common mistake is to say, “The cases fail HWE, so the variant must be a bad call.” That reasoning ignores disease-dependent selection: if genotype affects case probability, case frequencies are reweighted, and the resulting case proportions need not satisfy HWE. The correction is to compare cases and controls separately, assess whether the observed pattern is plausible under genotype-dependent risk, and consider technical and sampling explanations before making a quality-control decision. A case-only HWE test is not itself a test of association, because both a genuine association and other mechanisms can produce departures.