Cancellation From Opposing Variant Effects

Why Opposing Effects Matter

A burden test compresses several variants into one score for each person. This can be powerful when included variants tend to shift the trait in the same direction: their contributions reinforce one another. But if some variants increase the trait and others decrease it, the signed contributions may offset. A small burden-test signal can then occur even when individual variants have meaningful effects.

The direction of an effect depends on how the outcome is coded. For a quantitative trait, a positive effect means that carrying an additional effect allele is associated with a higher value of the trait, conditional on the model. A negative effect means a lower value. In a case-control analysis, the direction instead depends on which outcome is coded as the event. Neither direction by itself establishes causation.

How Cancellation Enters the Burden Score

Consider \(m\) variants and let \(G_{ij}\) denote the dosage or carrier indicator for variant \(j\) in person \(i\), according to the chosen burden definition. With prespecified nonnegative weights \(w_j\), the burden is \(B_i=\sum_{j=1}^{m}w_jG_{ij}\). A standard linear-model description of a quantitative trait is

where \(\beta_j\) is the conditional change in the trait per unit of \(G_{ij}\), and \(\varepsilon_i\) is an error term with mean zero given the genotypes. For the simple calculations below, assume the genotypes are independent across variants and the errors are independent of the genotypes, with constant variance. These assumptions make the source of cancellation transparent; real analyses may use covariate-adjusted models and correlated variants.

If the burden is fitted as the single genetic predictor, its population slope is the covariance of \(B_i\) with \(Y_i\), divided by the variance of \(B_i\). Under the stated independence assumptions, this gives

For a carrier indicator with carrier probability \(p_j\), \(\operatorname{Var}(G_{ij})=p_j(1-p_j)\). The numerator is a signed sum: a positive \(\beta_j\) contributes in one direction and a negative \(\beta_j\) in the other. The denominator is nonnegative, so it does not undo the numerator’s cancellation. This formula also shows why the burden result is not a simple unweighted average of variant effects: variants with different weights and genotype variation contribute differently.

A Derivation of the Cancellation Formula

The key calculation follows from covariance linearity. Write the burden and genetic component of the trait as \(B_i=\sum_jw_jG_{ij}\) and \(Y_i-\mu=\sum_k\beta_kG_{ik}+\varepsilon_i\). Then

Because the errors have mean zero given the genotypes, \(\operatorname{Cov}(B_i,\varepsilon_i)=0\). If variants are independent, \(\operatorname{Cov}(G_{ij},G_{ik})=0\) whenever \(j\ne k\). Only terms with \(j=k\) remain, so

Similarly, independence gives \(\operatorname{Var}(B_i)=\sum_jw_j^2\operatorname{Var}(G_{ij})\). Dividing the covariance by this variance yields the stated burden slope. If variants are correlated, the double sum contains cross-variant covariance terms too; cancellation can then depend on linkage disequilibrium as well as on the signs of the \(\beta_j\).

Cancellation in a Score Test

The same idea appears in a burden score test. For a simple linear model with an intercept, independent variants, and centered genotype contributions, the expected score contribution from variant \(j\) under the alternative is proportional to \(n\,w_j\beta_j\operatorname{Var}(G_{ij})\), where \(n\) is the sample size. A burden score adds those contributions. Its null standard error reflects the total genotype information and the residual variation; the opposing signs affect the score numerator, not merely its standard error.

This distinction matters: a weak result can reflect low information because few people carry the included variants, cancellation because their effects point in opposing directions, or both. A burden p-value alone does not tell which explanation applies. The score calculation below is an illustration under a simple model, not a substitute for the appropriate covariate-adjusted test or its calibration.

Diagnosing Opposing Effects

Variant-level effect estimates can help identify possible cancellation, but they must be read with their uncertainty. Rare alleles often have few carriers, so separate estimates can be unstable and their confidence intervals wide. A positive estimate for one variant and a negative estimate for another is therefore a clue to investigate, not by itself proof that their true effects differ in direction.

A useful diagnostic asks whether the pattern is coherent with the burden construction and the study design. Check the outcome coding, effect-allele coding, dosage definition, and weight signs first; inconsistent coding can create an apparent direction conflict. Then inspect variant-level estimates with standard errors or confidence intervals, and consider whether the pattern follows a prespecified functional distinction, such as two annotation classes.

Why Prespecification Matters

A mask is a rule for deciding which variants contribute to a burden. If biology or annotation motivates distinct groups before the association results are examined, testing those masks can provide a useful way to investigate whether a broad mask combines variants with different likely roles. For example, separate masks might be defined by functional consequence or another established annotation rule. Each mask answers a different, explicitly defined question.

By contrast, repeatedly changing masks after seeing which variants have positive or negative estimates can make a seemingly compelling result partly a product of selection. The chosen mask may be unusually favorable in the same data, and the reported p-value may not account for that search. A mask motivated by observed directions should therefore be described as exploratory, not as independent confirmation. Prespecification does not guarantee a valid biological grouping, but it makes the analysis question clear and limits data-driven selection.

Common Mistake

Wrong reasoning: “The burden test was not significant, so none of the variants affects the trait.” This conclusion does not follow. A burden test tests an aggregate direction under its chosen mask and weights. Opposite contributions can make that aggregate small, and rare carriers can also make estimates imprecise.

Correction: Treat the burden result as evidence about the specified aggregate, not as a separate verdict on every variant. Examine variant-level estimates and uncertainty to assess whether opposing effects are plausible, check whether the burden assumptions are appropriate, and use alternative masks only when their definitions are justified. A nonsignificant result is not proof of no variant effects.