Worked Rare Structural-Variant Burden Analysis

Why a Worked Burden Analysis Needs More Than a Test

A rare structural variant (SV) may be represented by several calls, and one person may carry multiple qualifying events in the same gene. A gene-level burden analysis therefore requires decisions before testing: which events count, how events are combined within each person, and which comparison is made between cases and controls. If any of these steps is unclear, a small p-value—or a large one—can be difficult to interpret.

This tutorial follows one hypothetical case-control analysis from event eligibility to sparse-data inference. The primary analysis uses an unweighted carrier definition: a participant is a carrier if they have at least one event that passes a prespecified inclusion mask. The test compares the proportions of carriers in cases and controls. This simple design makes the analysis transparent, but it does not adjust for covariates or establish a causal effect.

Definition: In a gene-level carrier burden analysis, an eligible event is an SV call that satisfies the study's prespecified event-level inclusion rules. A participant is a carrier for the tested gene if they have at least one eligible event assigned to that gene. The burden is the number or proportion of participants classified as carriers, not the raw number of calls.

Specify the Analysis Before Counting

Suppose a hypothetical study includes 120 cases and 120 controls. The gene and the primary mask were specified before looking at the case-control carrier counts. For this example, the primary mask includes deletion and duplication calls that passed the study's stated variant-level quality filters, met its breakpoint-confidence rule, were assigned to the gene using its prespecified overlap rule, and had an estimated frequency below 1% in the reference used by the analysis. These are example-specific choices, not universal thresholds. A real analysis must document the precise quality metrics, frequency source, gene definition, and overlap rule.

The previous material on variant-level filtering and breakpoint harmonization motivates applying these rules consistently. Here the key requirement is that case and control calls are processed using the same mask. A call that fails a required criterion is not rescued merely because it appears in a case. Conversely, a call is not eligible just because it overlaps the gene: it must meet every specified inclusion condition.

Worked Example: Deciding Whether Events Enter the Primary Mask

Consider three hypothetical calls assigned to the tested gene. Call A is a deletion that passes the quality filters, meets the breakpoint-confidence rule, satisfies the overlap definition, and has estimated frequency 0.4%. Call B is a duplication that passes the quality and overlap rules but fails the breakpoint-confidence rule. Call C passes the quality, breakpoint, and overlap rules, but its estimated frequency is 1.3%.

Call A: It passes every stated criterion, including $0.4\% < 1\%$, so it is eligible.

Call B: It fails the required breakpoint-confidence rule, so it is not eligible even though its other listed properties pass.

Call C: It fails the frequency criterion because $1.3\% \not< 1\%$, so it is not eligible under this primary mask.

The result is one eligible event out of these three calls. This decision concerns event inclusion; it does not yet determine how many distinct people are carriers.

Aggregate Events to People

For an unweighted carrier burden, event calls must be collapsed to one binary status per participant for the gene. A participant with one qualifying event and a participant with three qualifying events each contribute one carrier. This avoids treating multiple events in one person as independent people and prevents the carrier count from exceeding the number of participants.

The event-to-person mapping also needs consistent handling of overlapping and multiallelic calls. If calls represent the same underlying event, the harmonized representation should not create multiple carriers. If a person has distinct eligible events in the gene, they still count once in the binary burden. The unit of the case-control comparison is the participant, not the call.

Worked Example: Counting Distinct Carriers

Suppose the eligible calls for a hypothetical gene have been assigned to two event groups, E1 and E2. E1 is observed in 2 cases and 5 controls; E2 is observed in 2 cases and 6 controls. One case and one control appear in both groups.

For cases, add the group counts and subtract the overlap: $2 + 2 - 1 = 3$ distinct carriers. For controls, $5 + 6 - 1 = 10$ distinct carriers. Thus there are 4 case event-person records across the two groups but only 3 case carriers; there are 11 control event-person records but only 10 control carriers.

The corresponding noncarrier counts are $120 - 3 = 117$ cases and $120 - 10 = 110$ controls. The binary carrier table is therefore based on 3, 117, 10, and 110—not on the sums of event counts. Each person contributes to exactly one cell.

Build the Case-Control Comparison

For the primary mask, the hypothetical study has 3 carriers among 120 cases and 10 carriers among 120 controls. The carrier proportions are $3/120 = 0.0250$, or 2.50%, in cases and $10/120 = 0.0833$, or 8.33%, in controls. Their difference, cases minus controls, is $0.0250 - 0.0833 = -0.0583$, or about 5.83 percentage points lower in cases in this sample.

A descriptive odds ratio can also summarize the table. The odds of carrier status among cases are $3/117$; among controls they are $10/110$. Their ratio is $(3/117)/(10/110) = (3 \times 110)/(117 \times 10) = 0.2821$. This is a sample estimate of the carrier odds ratio, not a risk ratio and not evidence by itself that the SV burden changes disease risk. In a case-control study, the sampled numbers of cases and controls are set by design, so the proportions above describe carrier prevalence within each sampled group.

Group	Carrier	Noncarrier	Total
Cases	3	117	120
Controls	10	110	120
Total	13	227	240

Exact Inference for Sparse Carrier Counts

With only 13 carriers in the entire sample, a large-sample approximation to a test statistic may be unreliable. Fisher's exact test provides a conditional test for association in a $2 \times 2$ table. Under the null hypothesis of no association, and conditional on the observed row and column totals, the number $X$ of carriers among the 120 cases follows a hypergeometric distribution. This test is appropriate for sparse counts, but it does not correct confounding, population structure, relatedness, or differential call quality.

Definition: For a table with $N$ total participants, $K$ total carriers, and $n$ cases, the conditional carrier count under the null has probability $$ P(X=x)=\frac{\binom{K}{x}\binom{N-K}{n-x}}{\binom{N}{n}}. $$ A conventional two-sided Fisher exact p-value sums the probabilities of all tables with the same margins whose probability is no greater than the probability of the observed table.

Worked Example: Two-Sided Fisher Exact Test for the Primary Mask

For the table above, $N=240$, $K=13$, $n=120$, and the observed case-carrier count is $x=3$. Under $H_0$ of no association, the probability of a table with $x$ case carriers is

P(X=x)=\frac{\binom{13}{x}\binom{227}{120-x}}{\binom{240}{120}}.

The observed table has probability $P(X=3)$. The two-sided probability-based Fisher test includes tables with probability no greater than this observed probability. For these margins, those outcomes are $x=0,1,2,3$ and $x=10,11,12,13$. Therefore,

p=\sum_{x=0}^{3}\frac{\binom{13}{x}\binom{227}{120-x}}{\binom{240}{120}} +\sum_{x=10}^{13}\frac{\binom{13}{x}\binom{227}{120-x}}{\binom{240}{120}} \approx 0.08354.

At a prespecified $\alpha=0.05$, this p-value is greater than the significance level, so the analysis does not reject $H_0$ at that level. The data do not provide statistically significant evidence of an association under this test and mask. This does not demonstrate that the carrier burdens are equal: the observed sample proportions differ, and the small number of carriers limits the information available.

Why the Inclusion Mask Matters

A burden result is conditional on the definition of an eligible event. A stricter mask might exclude lower-confidence events; a more permissive mask might include some calls excluded from the primary analysis. Because changing the mask changes who is classified as a carrier, it changes the table and the inferential result. The mask should be scientifically justified and reported, rather than selected after trying many definitions and highlighting whichever one gives the preferred conclusion.

The following hypothetical sensitivity counts illustrate this dependence. They are alternative classifications of the same 120 cases and 120 controls, not additional participants. The strict mask removes some primary-mask carriers, while the permissive mask adds one case carrier. The proportions and odds ratios can be recalculated directly; any exact test must also be recomputed using the new table.

Worked Example: Comparing Alternative Inclusion Masks

Under the primary mask, the carrier counts are 3 cases and 10 controls. Under a stricter mask, suppose they are 2 cases and 7 controls. The case and control proportions are $2/120=0.0167$ (1.67%) and $7/120=0.0583$ (5.83%). The odds ratio is $(2 \times 113)/(118 \times 7)=226/826=0.2736$.

Under a more permissive mask, suppose the counts are 4 cases and 10 controls. The proportions are $4/120=0.0333$ (3.33%) and $10/120=0.0833$ (8.33%). The odds ratio is $(4 \times 110)/(116 \times 10)=440/1160=0.3793$.

All three hypothetical masks show lower carrier proportions in cases than in controls, but the magnitude changes. These descriptive comparisons do not establish that the association is robust, and the primary-mask p-value of 0.08354 cannot be carried over to either alternative table. Each alternative test requires its own exact calculation, and mask sensitivity should be considered alongside the prespecified primary result.

Assumptions, Limits, and Interpretation

The exact calculation conditions on the table margins and assumes that, under the null, the case labels are exchangeable with respect to carrier status given those margins. In a simple analysis this requires independent participants and comparable ascertainment of eligible SVs across groups. If participants are related, ancestry or other covariates differ systematically, or call quality differs between cases and controls, the unadjusted table may not support a valid association interpretation. Fisher's exact test addresses sparse cell counts; it does not repair these design or measurement problems.

A gene-level unweighted burden also treats all eligible events as equivalent for the carrier indicator. That is a modeling choice: it discards differences in event size, location, dosage, and predicted functional effect. The burden test asks whether the defined carrier status is associated with case-control group under the chosen mask. It does not identify which event drives a result, prove a molecular mechanism, or imply that carrying an event causes disease.

Common Mistake

Wrong reasoning: “There are 15 event-person records, so there are 15 carriers.” This double-counts the people who carry events in both groups.
Correction: Collapse eligible events to one carrier indicator per participant for the gene. In the aggregation example, the union is 3 case carriers and 10 control carriers.
Wrong reasoning: “The Fisher p-value is above 0.05, so the null is true.”
Correction: A p-value of 0.08354 means that, under the null and the conditional model, the probability of a table at least as extreme by the specified probability ordering is about 8.354%. It is not the probability that the null is true, and failure to reject is not proof of no association.
Wrong reasoning: “Changing the mask gives another chance to find significance, so report only the most favorable mask.”
Correction: State the primary mask in advance, show justified sensitivity analyses, and interpret all results as conditional on their event definitions.

Key takeaway: A rare-SV burden analysis is a linked sequence of decisions: define eligible events, aggregate them into distinct participant-level carriers, construct the case-control table, and use an inference procedure suited to the sparse counts. The result applies to the stated mask and study design, not automatically to every SV in the gene.

Check Your Understanding

Use the definitions and hypothetical analysis above to answer these questions.

Why does a participant with two eligible events in the tested gene count once in an unweighted carrier burden?
Under the primary mask, how many noncarriers are there among cases and among controls?
What quantities are held fixed in the conditional hypergeometric calculation for the Fisher exact test?
At $\alpha=0.05$, what conclusion follows from the reported p-value, and what conclusion does not follow?
Why must the exact test be recalculated after changing the inclusion mask?
Name one source of confounding or measurement difference that Fisher's exact test does not address.