Power Loss From Opposing Rare-Variant Effects

Why Opposing Effects Matter

A burden test compresses information across variants into one score. That compression is useful when the variants in a set tend to affect the phenotype in the same direction: their contributions can reinforce one another. But if some variants increase the phenotype and others decrease it, the contributions can offset. The set may contain real effects while its combined burden score has little average association signal.

This is a power issue, not evidence that every variant is harmless. A burden test targets a particular kind of combined effect: the association between a specified weighted score and the phenotype. When the effects oppose one another, that target may be small even when individual effects are not.

For the derivation and examples below, assume independent individuals, mean-zero genotype dosages, and independent variants with genotype variances \(v_j=\operatorname{Var}(G_{ij})\). Assume the errors are independent of genotypes, have mean zero and variance \(\sigma^2\), and there is no confounding of genotype and phenotype. The effects and weights are treated as fixed. These assumptions simplify the calculation; linkage disequilibrium and other departures are discussed below. A burden score test uses a standardized association score, with its null distribution approximated by the standard normal distribution in the setting considered here.

Where Cancellation Enters the Score

Under the model, the phenotype’s genetic component is \(\sum_j\beta_jG_{ij}\). The covariance between that component and the burden score is \(\operatorname{Cov}(B_i,\sum_j\beta_jG_{ij})=\sum_jw_j\beta_jv_j\) when variants are independent. This covariance is the expected signal that the one-dimensional burden test can detect. The variance of the burden score is \(\sum_jw_j^2v_j\).

The approximate standardized mean displacement of the score statistic is therefore

The numerator is a signed sum. A variant with a positive effect contributes in one direction and a variant with a negative effect contributes in the other, assuming positive weights. If these terms nearly cancel, \(|\delta|\) is small. A small \(|\delta|\) means that the score statistic’s distribution under the alternative is not far from its null distribution, so the test often fails to reject even when some \(\beta_j\) are substantial.

For a two-sided test at significance level \(\alpha\), let \(c=z_{1-\alpha/2}\), the standard-normal critical value. Under a local-alternative approximation (or if the standardized score variance is approximately 1 under the alternative), \(Z\) is approximately \(N(\delta,1)\), and the corresponding approximate power is. For the fixed effects in the examples below, the score’s alternative variance need not equal its null variance; it depends on the genotype distribution, including higher moments. The supplied inputs do not determine fixed-effect power, so the numerical power values below are mean-shift calculations under the unit-variance approximation, not fixed-effect power estimates.

where \(\Phi\) is the standard normal cumulative distribution function. Under this unit-variance normal approximation, power depends on \(|\delta|\) for a two-sided test: a negative displacement reverses the direction of the association but does not, by itself, lower two-sided power compared with the same positive displacement.

Worked Cases: Exact and Near Cancellation

Justification and Interpretation

The score’s mean displacement follows directly from its covariance with the phenotype. In the simplified model, the burden component of an individual’s phenotype is \(\sum_j\beta_jG_{ij}\). Since each centered genotype has mean zero and the variants are independent, cross-variant covariances vanish. Hence \(\operatorname{Cov}(B_i,Y_i)=\sum_jw_j\beta_jv_j\). Under the null, the score has variance approximately \(n\sigma^2\operatorname{Var}(B_i)\). Dividing its alternative mean, approximately \(n\operatorname{Cov}(B_i,Y_i)\), by the null standard deviation, \(\sqrt{n\sigma^2\operatorname{Var}(B_i)}\), gives the stated \(\delta\). This derivation identifies the cancellation mechanism: it is the signed covariance contributions, not simply the count of positive and negative effects, that determine the mean signal.

With correlated variants, let \(\Sigma\) be the genotype covariance matrix, \(w\) the vector of weights, and \(\beta\) the vector of effects. The corresponding quantities are \(\operatorname{Var}(B_i)=w^\mathsf{T}\Sigma w\) and \(\operatorname{Cov}(B_i,\sum_j\beta_jG_{ij})=w^\mathsf{T}\Sigma\beta\). Thus the displacement becomes \(\delta=\sqrt{n}\,w^\mathsf{T}\Sigma\beta/(\sigma\sqrt{w^\mathsf{T}\Sigma w})\). Linkage disequilibrium changes the contributions through the covariance matrix; the independent-variant sum is not generally valid when variants are correlated.

The calculation is an approximation under the stated model and normal score calibration. It is most useful for understanding how the specified effects, weights, genotype variation, and sample size combine. In applications, the effects are unknown, and their estimates can be noisy—particularly for rare variants. A power calculation based on assumed effects is conditional on those assumptions; it is not a guaranteed power level for a real study.

Diagnostics, Alternatives, and Limitations

To diagnose potential cancellation, examine the expected or estimated contributions \(w_j\beta_jv_j\), not just the signs of the effects or the number of variants in each direction. Opposite signs are a warning, but cancellation depends on magnitudes, weights, genotype variances, and, with linkage disequilibrium, covariance. In real data, estimated effects may be unstable, so direction patterns should be interpreted cautiously. Choosing weights or a variant subset after inspecting the same outcome data can also inflate the false-positive rate unless that selection is accounted for.

A very small burden statistic alone cannot diagnose cancellation. It can also result from small effects, low genotype variation, a small sample, measurement error, or a mismatch between the chosen weights and the true effects. Check the genotype coding and allele harmonization, verify the intended weights, and examine whether the model assumptions are plausible. If variants are correlated, use the covariance-aware calculation rather than treating their contributions as independent.

The main strength of a burden test is its efficiency when effects align with the score’s direction: aggregating can yield a strong signal with fewer degrees of freedom than testing variants separately. Its central limitation is the same aggregation: one signed score cannot represent every mixed-direction pattern well. Variant-by-variant tests retain direction-specific information but may have low power for rare variants and require attention to multiple testing. Methods designed to detect variation in effects across a set provide another option when directions may differ; they target a different pattern of association and require their own calibration. Neither alternative makes assumptions or limitations disappear, and a significant set-level result does not identify which variants are responsible.

The practical lesson is to treat the burden test as a targeted summary, not a universal test for any genetic effect in a region. Its power is highest when the specified score aligns with the pattern of effects. When protective and risk-increasing effects coexist, a weak burden result does not rule out meaningful variant-level associations; it may show that a single-direction summary is poorly matched to them.