Why Relatedness Inflates Naive GWAS Evidence

Why Relatedness Matters for Association Tests

A single-variant association test asks whether genotype is associated with a phenotype after accounting for the specified covariates. Ordinary regression calculations generally treat the participants’ residual phenotype values as independent. That assumption can fail when sampled individuals are related: relatives may share genetic or environmental influences that remain in their phenotypes after adjustment. Their residuals then tend to move together.

The consequence is not simply that an analysis has “less data.” The variance of the estimated genotype effect depends on the pattern of genotype values across the correlated individuals. If genotype values tend to be similar within related pairs, positive residual covariance can increase the estimator’s variance. The usual independent-observation standard error may then be too small, making the association statistic too large in magnitude and its p-value too small. Other genotype arrangements can yield less inflation or even reduced variance.

The calculations below use a quantitative trait and a linear association model to isolate this issue. They do not imply that every relatedness pattern produces inflation, nor do they establish a particular biological cause for residual covariance.

OLS Variance with Correlated Residuals

Consider the model \(y_i=\mu+\beta x_i+\varepsilon_i\), where \(y_i\) is the quantitative phenotype, \(x_i\) is the genotype dosage for the tested variant (0, 1, or 2 copies of an allele), and \(\beta\) is the per-allele association effect in phenotype units. Assume the mean model is correct, \(E(\varepsilon\mid X)=0\), and, conditional on the genotypes, the residual vector has covariance matrix \(\sigma^2R\). The diagonal entries of \(R\) are 1; its off-diagonal entries describe residual correlations. For the calculations, take \(\sigma^2\) as known so that the effect of covariance on the standard error is separated from the additional issue of estimating residual variance.

With an intercept and no other covariates, let \(x_i\) denote the centered genotype dosage, so \(\sum_i x_i=0\), and let \(S_{xx}=\sum_i x_i^2\). The OLS slope estimate is \(\hat{\beta}=\sum_i x_i y_i/S_{xx}\). Under the null hypothesis \(H_0:\beta=0\), its conditional variance is determined by the residual covariance, not just the residual variances.

If residuals are independent, \(R=I\), so \(x^\mathsf{T}Rx=S_{xx}\) and the variance reduces to \(\sigma^2/S_{xx}\). That is the variance underlying the ordinary standard error when \(\sigma^2\) is known. When residuals are correlated, the off-diagonal terms contribute to \(x^\mathsf{T}Rx\). A convenient expression for the ratio of the covariance-aware variance to the independent-residual variance is

This ratio depends on both the covariance matrix and the genotype pattern. In a model with additional fixed covariates, the corresponding calculation uses the genotype after removing its linear association with those covariates, along with the covariance of the residuals. The same principle applies: the independent-observation formula is justified only if the relevant covariance structure supports it.

Worked Examples

Justification for the Variance Formula

The formula follows directly from the OLS slope and the covariance of a linear combination. Under the model and centering condition, the intercept contribution disappears from the centered-genotype numerator. The estimation error is therefore a weighted sum of residuals. For any fixed vector \(a\), the variance of \(a^\mathsf{T}\varepsilon\) is \(a^\mathsf{T}\operatorname{Cov}(\varepsilon\mid X)a\). Applying that fact gives the result:

If the residuals are jointly Gaussian, this linear combination is normally distributed. Under \(H_0\), the naive statistic formed using the independent-residual standard error has variance \(x^\mathsf{T}Rx/S_{xx}\), rather than 1, when \(\sigma^2\) is known. A variance ratio above 1 makes the null statistic more dispersed than the reference distribution and can increase the frequency of extreme test statistics. A ratio below 1 makes it less dispersed. When \(\sigma^2\) is estimated, the usual residual-based standard error also reflects that estimation; correlation can affect that estimate as well, so the known-variance calculation isolates rather than exhausts the potential miscalibration.

Distinguishing Covariance from Mean-Model Problems

Relatedness can raise two different concerns. This tutorial’s calculation concerns covariance: the residuals have mean zero under the specified model, but are not independent. Separately, ancestry, family environment, or other omitted factors may make the conditional mean wrong—for example, genotype and phenotype may both differ across subgroups. In that case, an apparent association can arise from mean-model misspecification, not just an underestimated standard error. Adjusting standard errors alone does not generally remove such bias.

The earlier material on calibration checks and quantile-quantile plots provides useful ways to notice that a scan’s test statistics depart from a reference distribution. Excess dispersion across many variants can be consistent with covariance-related miscalibration, but it does not identify its cause. True associations, population structure, technical artifacts, and model misspecification can also affect the distribution. A quantile-quantile plot or an inflation summary is a diagnostic clue, not proof that relatedness is responsible or a replacement for a suitable association model.

Possible responses depend on the sampling structure and the inferential target. A covariance-aware association model represents dependence directly; a robust variance estimator can adjust uncertainty under appropriate conditions. These approaches are not interchangeable in all settings, and neither repairs a misspecified mean automatically. In particular, a cluster-robust estimate generally needs enough independent clusters for its approximation to work well. A sample containing only a few large families is not made safe merely by labeling the standard error “robust.” The subsequent material on random genetic effects and mixed-model association develops one model-based approach; the point here is to understand why the independence calculation can fail.

Diagnostics, Limitations, and Common Mistakes

• Ignoring the sampling structure. Check whether participants include relatives or other groups likely to have correlated outcomes. If so, state what covariance assumption supports the reported standard errors.
• Assuming every related sample is inflated by the same amount. The variance ratio \(x^\mathsf{T}Rx/S_{xx}\) depends on genotype alignment with covariance. It can exceed, equal, or fall below 1.
• Equating a large test statistic with a causal effect. This calculation concerns uncertainty in an association estimate. It does not establish causation, and covariance correction alone does not address confounding.
• Treating a genomic inflation signal as a diagnosis. Broad test-statistic inflation can motivate investigation, but several mechanisms can produce it. Inspect study design and model assumptions as well as summary diagnostics.
• Assuming a known covariance structure in practice. The examples specify the correlations to make the arithmetic transparent. Real analyses must justify and estimate or otherwise account for dependence; a convenient assumed correlation is not automatically correct.

The central limitation of the simple pair calculations is that they use a deliberately specified covariance matrix and, for the test illustration, known residual variance. Real GWAS may involve uneven relationships, covariates, binary phenotypes, estimated variance parameters, and population structure. The simple result remains valuable because it shows the basic mechanism: the OLS coefficient is a weighted sum of outcomes, and covariance among outcomes changes the variance of that sum.