One Set, Two Different Effect Targets
A gene or genomic region may contain several rare variants, each with its own association effect. Testing variants one at a time can be uninformative when each is observed in few people. Set tests combine information across variants, but different combinations answer different questions about the pattern of effects. A burden test is most sensitive when effects align with a prespecified direction, often meaning that included variants affect the phenotype in the same direction after coding and weighting. A variance-component test can detect a pattern of effects that varies across variants, including effects with opposing signs.
These are different effect targets, not different definitions of what it means for a set to be associated. The usual gene-level null is that every variant effect in the set is zero. A burden statistic may have little sensitivity to some alternatives under that null, while a variance-component statistic may be less sensitive to a well-aligned alternative. Neither test, by itself, establishes that a variant is causal.
Model and Assumptions for Comparing Targets
To make the distinction precise, consider a score vector \(U=(U_1,\ldots,U_K)'\) for the variant effects, calculated under the joint null after accounting for covariates. Let \(V\) be its covariance matrix under that null. In a large-sample score-test approximation, \(U\) has mean zero and covariance \(V\) under \(H_0\); under a local alternative its mean is approximately \(V\beta\). The approximation relies on a correctly specified phenotype model, appropriate handling of covariates and relatedness where relevant, and adequate score-test calibration. Sparse variant counts or model misspecification can make the approximation poor.
A burden direction can be represented by a fixed vector \(a\), chosen before testing. Its score is \(a'U\), with null variance \(a'Va\). The standardized one-direction statistic is \(T_B=(a'U)^2/(a'Va)\), provided \(a'Va>0\). Its local noncentrality depends on the projection of the alternative score mean onto \(a\): \((a'V\beta)^2/(a'Va)\). Thus the test is most responsive when the effects, after the score information and weights are accounted for, align with the chosen direction. A two-sided squared statistic can detect either sign of that aggregate, but it does not separately test every possible direction.
A simple variance-component statistic is \(Q=U'U\) when the score coordinates are standardized and independent under the null. More generally, a quadratic statistic has form \(U'AU\) for a specified positive semidefinite matrix \(A\). Under the null its distribution depends on the covariance structure: in general it is a weighted sum of chi-square variables, not automatically a standard chi-square distribution. The familiar \(Q=\sum_j Z_j^2\) has a chi-square distribution with \(K\) degrees of freedom only when the standardized scores \(Z_j\) are independent standard normal under the null.
Worked Example: Aligned Effects and a Burden Direction
Suppose two hypothetical variants have standardized score coordinates with \(V=I\), and consider a burden direction \(a=(1,1)'/\sqrt{2}\). For illustration, let the local alternative effects be \(\beta=(0.30,0.30)'\). The effects are aligned with the chosen direction.
The burden projection is \(a'\beta=(0.30+0.30)/\sqrt{2}=0.4243\). Because \(V=I\), the burden noncentrality is \((a'\beta)^2=0.4243^2=0.1800\). The squared length of the effect vector is also \(0.30^2+0.30^2=0.1800\). Both targets therefore register a signal in this simple illustration; the burden test concentrates on the specified direction, whereas the quadratic target retains information in both coordinates.
These numbers describe local signal strength, not an observed p-value or a guarantee of significance. They also depend on the stated identity-covariance setup. With unequal information or correlated scores, the relevant burden projection and variance must use \(V\), not an unadjusted sum of effects.
Why Opposing Effects Separate the Targets
A burden statistic can lose signal when positive and negative effects cancel in its chosen direction. The joint null is still false in that situation. A quadratic statistic avoids signed cancellation by squaring score components, though it pays for this flexibility and may have less power than a well-chosen burden test when effects truly align.
Worked Example: Opposing Effects Can Cancel in a Burden
Use two independent, standardized score coordinates and the same direction \(a=(1,1)'/\sqrt{2}\). Suppose the hypothetical local alternative is \(\beta=(0.40,-0.40)'\), with \(Z\sim N(\beta,I)\). This explicitly assumes unit score variances under the alternative as well as independent coordinates. The joint null is false because both effects are nonzero.
The burden projection is \((0.40-0.40)/\sqrt{2}=0\), so this particular directional burden has zero local noncentrality. In contrast, the quadratic signal magnitude is \(0.40^2+(-0.40)^2=0.3200\).
For the quadratic statistic \(Q=Z_1^2+Z_2^2\), each coordinate has \(E(Z_j^2)=\operatorname{Var}(Z_j)+\{E(Z_j)\}^2=1+\beta_j^2\) under the stated model. Therefore \(E(Q)=2+0.40^2+(-0.40)^2=2.320\). The null expectation is 2, so the increase is 0.3200. The expectation is a description of the statistic under this model, not a p-value or a direct measure of power. If alternative score variances were not one, the correct expression would be \(E(Q)=\sum_j\operatorname{Var}(Z_j)+\sum_j\{E(Z_j)\}^2\); the increase cannot be attributed to the means alone without the variance assumption.
Correlation Changes the Calibration
Variant scores are often correlated because genotypes are correlated through linkage disequilibrium. A test that treats correlated scores as independent can have an incorrect null distribution. The burden variance \(a'Va\) directly incorporates the covariance. For a joint quadratic test, covariance-aware calibration is also essential; one option in a multivariate normal score model is the Wald quadratic \(U'V^{-1}U\), when \(V\) is nonsingular. Its null distribution is approximately chi-square with \(K\) degrees of freedom under that model. This covariance-adjusted joint test is not identical to every variance-component procedure, but it illustrates why the covariance cannot be ignored.
Worked Example: Correlated Scores Affect the Test
Suppose a hypothetical two-variant score vector is \(U=(2,1)'\), and its null covariance is \(V=\begin{pmatrix}1&0.5\\0.5&1\end{pmatrix}\). Use the unscaled burden direction \(a=(1,1)'\). Its score is \(a'U=2+1=3\), and its null variance is \(a'Va=1+0.5+0.5+1=3\). Thus \(T_B=3^2/3=3.000\), which has a one-degree-of-freedom chi-square reference distribution under the score-test approximation; its p-value is about 0.0833.
For a covariance-adjusted joint quadratic, \(V^{-1}=\begin{pmatrix}1.333&-0.6667\\-0.6667&1.333\end{pmatrix}\), so \(U'V^{-1}U=4.000\). Under the stated nonsingular, correctly calibrated two-dimensional null model, this has an approximate chi-square distribution with 2 degrees of freedom, giving \(p=e^{-4/2}=0.1353\). At \(\alpha=0.05\), neither statistic rejects. These different p-values arise because the statistics test different directions and use different reference distributions; they are not contradictory evidence about a single identical statistic.
Short Justification: What the Quadratic Measures
The expectation of a squared score separates its mean contribution from its variance contribution. For a random variable \(Z_j\) with finite second moment, \(E(Z_j^2)=\operatorname{Var}(Z_j)+\{E(Z_j)\}^2\). Summing this identity gives the expected value of the independent-coordinate quadratic, without requiring the coordinates themselves to be independent for this expectation calculation:
Under the null, if each \(Z_j\) has mean zero and variance one, \(E(Q)=K\). Under an alternative, the expectation increases by the sum of squared means only if the variances remain one. If the variances change, their changes also affect \(E(Q)\). This calculation explains how a quadratic can retain signal from effects of either sign, but it does not by itself establish a null distribution or compute power.
Choosing, Diagnosing, and Interpreting a Set Test
The choice between targets is driven by plausible effect patterns and the intended interpretation. If included variants are expected to have effects in a common direction after allele coding and weighting, a burden target can be efficient. If effects may differ in magnitude or sign, a variance-component target is more robust to cancellation. When the pattern is uncertain, an approach combining directional and quadratic evidence may be considered, but its calibration must account for the combination procedure; it is not valid to select whichever unadjusted p-value is smaller.
Before interpreting either result, check that variant alleles and effect directions are harmonized, the variant set and weights were specified appropriately, and the score covariance and phenotype-model assumptions are credible. Look for extreme or unstable single-variant contributions, sparse carrier counts, population structure, relatedness, and calibration problems. A nominally significant set result may reflect model failure, an unsuitable set definition, or an unaccounted testing burden across many sets. Conversely, a non-significant result is not evidence that all effects are zero.
| Feature | Burden target | Variance-component target |
|---|---|---|
| Main pattern detected | Effects aligned with a prespecified weighted direction | Effects spread across variants, including opposing directions |
| Cancellation risk | Can be substantial when effects oppose | Squaring or quadratic aggregation avoids signed cancellation |
| Potential advantage | Can concentrate power when the direction is well specified | Can detect heterogeneous effects without a common sign |
| Important limitation | May miss alternatives poorly aligned with the chosen weights | May be less powerful for a strongly aligned pattern and requires appropriate covariance calibration |
Neither result identifies which variant is responsible, establishes a causal effect, or guarantees portability to another ancestry group. Effects are defined on the model’s phenotype and covariate scale; differences in allele frequency, linkage disequilibrium, measurement, or effect heterogeneity can alter the evidence across populations. A set-level association is therefore evidence about the tested set under the specified model, not a causal conclusion about every included variant.
Common Mistakes
- Equating a burden null with the gene-level null. A burden score near zero may result from cancellation even when some \(\beta_j\) are nonzero. The joint null concerns all variant effects, not only one aggregate.
- Assuming a quadratic statistic is automatically chi-square. This is true for a sum of squares of independent standard normal scores with the corresponding degrees of freedom, not for arbitrary correlated or weighted scores. Use a null distribution justified by the covariance and procedure.
- Interpreting \(E(Q)\) as evidence of significance. An expectation under an alternative is not an observed statistic, a p-value, or a power calculation. Variances as well as means determine it.
- Calling a detected set causal. Association can arise through linkage, confounding, or model misspecification. The test supports association under its assumptions, not causation.
Check Your Understanding
Answer each question, then compare with the worked solutions below.
- For three variants, what is the joint no-association null, and how does it differ from a null about one burden direction?
- With independent standardized scores and effects \((0.2,-0.2)\), what is the burden projection for \(a=(1,1)'/\sqrt{2}\), and what is the squared effect length?
- If a quadratic statistic is the sum of four independent standard-normal score squares under the null, what is its null distribution?
- Why is it unsafe to calculate a burden statistic’s null variance as the sum of component variances when scores are correlated?
Solutions. (1) The joint null is \(\beta_1=\beta_2=\beta_3=0\); a burden-direction null concerns only the specified aggregate and can hold despite nonzero effects that cancel. (2) The projection is \((0.2-0.2)/\sqrt{2}=0\); the squared effect length is \(0.2^2+(-0.2)^2=0.0800\). (3) It is chi-square with 4 degrees of freedom, by the sum-of-squares result for independent standard normal variables. (4) The variance includes covariance terms: \(\operatorname{Var}(a'U)=a'Va\). Omitting off-diagonal entries of \(V\) generally gives the wrong standardization and can miscalibrate the test.