The Hypothesis Tested By A Rare-Variant Burden

Why Test a Gene-Level Burden?

Rare variants can be too uncommon for a separate test of each variant to be informative. A burden test instead summarizes genotype information across a defined set of variants in a gene or genomic region and asks whether that aggregate is associated with a trait. Its central hypothesis is therefore about a regional summary, not about the evidence for each variant considered separately.

To state that hypothesis precisely, suppose a study has \(n\) individuals and a fixed set of \(m\) included variants. Let \(G_{ij}\) denote the genotype quantity for individual \(i\) at variant \(j\), as supplied to the analysis, and let \(w_j\) be the prespecified weight for that variant. The weighted burden for individual \(i\) is

B_i=\sum_{j=1}^{m} w_jG_{ij}.

This definition describes the aggregate as a mathematical object; it does not prescribe how genotype quantities, variant inclusion, or weights must be chosen. Those choices determine exactly which variants contribute and how they contribute, and they must be specified for the test to have a clear interpretation.

Definition: A gene-level burden hypothesis concerns whether an individual's weighted aggregate \(B_i\), formed from a defined set of variants in a gene or region, is associated with the trait after any specified adjustment. In a model with a burden effect parameter \(\beta_B\), the usual two-sided hypotheses are \(H_0:\beta_B=0\) versus \(H_a:\beta_B\ne 0\). The null says there is no association with this particular burden under the model; it does not, by itself, say that every included variant has no effect.

The Burden Null Is Not a Set of Separate Variant Tests

For a quantitative trait, a simple working model for the aggregate association is

Y_i=\beta_0+\beta_BB_i+\gamma^\mathsf{T}C_i+\varepsilon_i,

where \(Y_i\) is the trait, \(C_i\) is a vector of specified covariates, \(\gamma\) is its coefficient vector, and \(\varepsilon_i\) is the residual. The burden test asks whether \(\beta_B=0\) in this model. For the ordinary linear-model test, the relevant assumptions include independent observations, a correctly specified conditional mean, and residuals with mean zero and constant variance; exact small-sample \(t\)-inference additionally assumes normally distributed residuals. Other trait types require an appropriate model and test, but the target remains an association with the defined aggregate.

A different question is whether each variant has its own effect. For example, a variant-level mean model might include separate terms \(\beta_jG_{ij}\). Its joint null is \(H_0:\beta_1=\cdots=\beta_m=0\); separate single-variant tests instead ask one question for each \(\beta_j\). Neither is automatically identical to the burden null. A burden collapses several genotype quantities to one number, so the burden test targets a one-dimensional pattern of association.

Key distinction: A significant burden test is evidence against no association with the specified weighted aggregate, under the model and test used. It is not a declaration that every included variant is individually associated, nor is it a general test that can identify which variant is responsible.

The relationship between the hypotheses can be seen in a simple derivation. Suppose the conditional mean includes variant-specific effects:

E(Y_i\mid G_i,C_i)=\beta_0+\gamma^\mathsf{T}C_i+\sum_{j=1}^{m}\beta_jG_{ij}.

If the variant effects follow the particular pattern \(\beta_j=\theta w_j\) for every included variant, then

\sum_{j=1}^{m}\beta_jG_{ij} =\sum_{j=1}^{m}\theta w_jG_{ij} =\theta\sum_{j=1}^{m}w_jG_{ij} =\theta B_i.

Under this constrained pattern, the genetic contribution is represented by a single burden coefficient, \(\theta\). The burden null tests \(\theta=0\) within that model. Without the constraint, a single aggregate coefficient does not describe every possible pattern of variant-specific effects. In particular, an aggregate result cannot recover all the individual \(\beta_j\) values.

Worked Example: Testing an Aggregate in a Small Hypothetical Dataset

Consider six hypothetical individuals with a quantitative trait \(Y\) and a burden score \(B\) already calculated from a fixed variant set and fixed weights. The scores are illustrative burden units, not a claim about any real study.

Individual	\(B_i\) (burden units)	\(Y_i\) (trait units)
1	0	2
2	0	3
3	1	3
4	1	4
5	2	5
6	2	5

Use the simple linear model \(Y_i=\beta_0+\beta_BB_i+\varepsilon_i\), assuming independent, normal errors with mean zero and common variance. The observed values show no covariate adjustment is being made in this example. The sample means are \(\bar{B}=6/6=1\) burden unit and \(\bar{Y}=22/6=3.6667\) trait units. The slope estimate is

\hat{\beta}_B =\frac{\sum_i(B_i-\bar{B})(Y_i-\bar{Y})}{\sum_i(B_i-\bar{B})^2} =\frac{5}{4} =1.25\text{ trait units per burden unit}.

The fitted intercept is \(3.6667-(1.25)(1)=2.4167\) trait units. The residual sum of squares from these fitted values is \(1.0833\), so with \(df=n-2=4\), the residual variance estimate is \(1.0833/4=0.2708\). Thus the standard error of the slope is

SE(\hat{\beta}_B) =\sqrt{\frac{0.2708}{4}} =0.2602\text{ trait units per burden unit}.

The test statistic for \(H_0:\beta_B=0\) is \(t=1.25/0.2602=4.804\), with 4 degrees of freedom. Its two-sided \(p\)-value is approximately \(0.0086\). Under the stated model assumptions, these hypothetical data provide evidence of a positive association between this burden and the trait. The result concerns the aggregate's association; it does not identify which variant contributed to it.

Interpreting a Significant Aggregate Result

A weighted burden test is tied to the exact aggregate supplied to it. The test result depends on the included variants and on their weights, as well as on the trait, covariates, and statistical model. A change to the aggregate changes the hypothesis. Consequently, a result should be reported with enough information to identify the tested region, variant set, weighting scheme, model, and direction and scale of the association.

A small \(p\)-value means that, if the burden null and the assumptions used for the test were correct, a test statistic at least as extreme as the observed one would be unlikely. It is not the probability that the null is true. Statistical significance also does not establish that the gene causes the trait: population structure, technical artifacts, or other sources of bias can produce or distort an association. Sequencing quality and data validity are therefore part of the interpretation, not details that a low \(p\)-value can replace.

Worked Example: A Significant Burden Does Not Identify a Variant

Suppose a hypothetical analysis tests one prespecified weighted burden for a gene and reports an estimated burden coefficient of \(0.42\) trait units per burden unit, with \(SE=0.155\) in the same units. Assume the test uses a standard-normal reference distribution and the model and calibration assumptions are adequate. The test statistic is

z=\frac{0.42}{0.155}=2.710.

The two-sided standard-normal \(p\)-value is approximately \(0.0067\), so this analysis provides evidence against the null of no association with the tested aggregate. Now suppose the separate estimates for three included variants have individual \(p\)-values of \(0.18\), \(0.31\), and \(0.44\). Those values do not contradict the burden result: the aggregate test asks a different question and can have evidence that is not present in any one single-variant test.

The correct conclusion is limited: the specified weighted burden is associated with the trait under the analysis model. The result does not establish that all three variants are associated, that any particular one is associated, or that changing the included variants or weights would yield the same result. It also does not show that the gene is causal.

What a Non-Significant Burden Result Means

A failure to reject \(H_0:\beta_B=0\) means the analysis did not find sufficient evidence of association with the tested aggregate at the chosen significance level. It is not proof that the aggregate has exactly zero effect, and it is not proof that every included variant is harmless. Limited sample size, sparse genotype information, imperfect weights, or a pattern of effects not well represented by the aggregate can all limit the test's ability to detect association.

This distinction matters because the burden is one summary of a region, not a complete description of all possible variant effects. A non-significant burden result leaves open the possibility of variant-specific effects that the summary does not capture. Conversely, a significant burden result supports an aggregate association but does not supply individual-variant conclusions. Those questions require analyses that target them directly.

Worked Example: A Non-Significant Test Is Not Proof of No Association

In a second hypothetical quantitative-trait analysis, suppose the estimated coefficient for a prespecified burden is \(0.12\) trait units per burden unit with \(SE=0.10\) in the same units. Under a standard-normal reference approximation,

z=\frac{0.12}{0.10}=1.20, \qquad p=2P(Z\geq 1.20)\approx 0.230,

where \(Z\) has a standard normal distribution. At \(\alpha=0.05\), the test does not reject the burden null. The result is not evidence that the coefficient is exactly zero: it says this analysis did not provide strong evidence against zero for this aggregate. It also cannot rule out associations with individual variants or a different prespecified aggregate. No conclusion about those other hypotheses follows from this \(p\)-value alone.

Common Pitfalls and Practical Reporting

Wrong reasoning: “The gene-level burden is significant, so every included rare variant must be associated.” Correction: The test concerns one weighted regional summary; it does not test every variant separately.
Wrong reasoning: “The burden is significant, so this gene causes the trait.” Correction: The result is an association under a specified analysis, not proof of causation.
Wrong reasoning: “The burden is not significant, so none of the variants has an effect.” Correction: Non-significance is a lack of sufficient evidence for this aggregate hypothesis, not a proof of no effect.
Wrong reasoning: “The \(p\)-value is the probability that the burden null is true.” Correction: It is a probability calculated assuming the null and test assumptions, not a posterior probability of the null.
Wrong reasoning: “Any significant score for this gene tests the same hypothesis.” Correction: Different included variant sets or weights define different burden variables and therefore different hypotheses.

A clear report names the region and the burden definition, states the null being tested, describes the trait model and covariate adjustment, and gives the estimated association and its uncertainty in context where available. If many regions or burden definitions were examined, the reported significance must also be interpreted in light of that broader testing. Keeping the target narrow prevents a regional association from being mistaken for a variant-level or causal conclusion.

Key takeaway: A rare-variant burden test evaluates association with a defined weighted aggregate for a gene or region. Statistical significance supports that aggregate-level claim under the stated model; it does not establish which variants are associated, that all included variants contribute, or that the gene causes the trait.

Check Your Understanding

Answer each question in terms of the hypothesis and the scope of the conclusion.

In a model with burden coefficient \(\beta_B\), write the usual two-sided null and alternative hypotheses.
How does a burden null differ from the joint null that all variant-specific coefficients equal zero?
A burden test is significant, but no single-variant test is significant. Is that necessarily a contradiction? Explain.
What does a non-significant burden test allow you to conclude, and what does it not establish?
Name two features of a burden analysis that must be specified to identify the aggregate hypothesis being tested.