Why Representation Matters
Structural variants can overlap in genomic coordinates, one event can be nested inside another, and several distinct variant states can occur at the same locus. These features complicate association analysis because a person may carry more than one event, while a locus-level question may count that person only once. The data representation therefore determines what the analysis compares: a particular event, a mutually exclusive variant state, or any qualifying event in a region.
A useful starting distinction is that overlap in the genome is not the same as overlap among carriers. Two deletions can span overlapping intervals but occur in different people. Conversely, two events with different coordinates can be carried by the same person. The event calls and the sample-level genotype or carrier indicators must both be considered.
The carrier definition must be stated in advance. It may refer to a genotype or copy-number state meeting a specified threshold, rather than merely a variant record appearing in a call file. A person with a non-reference copy-number state need not be represented by exactly one event record, and an event-level encoding does not by itself resolve whether multiple calls lie on the same chromosome or on different chromosomes.
Event Indicators and the No-Double-Counting Rule
For two events, the locus-level indicator is the indicator of their union. If a person carries both, adding the two event indicators counts that person twice; the union indicator does not.
The product term subtracts one for a dual carrier, correcting the extra count. For \(K\) events, a safe individual-level implementation is \(L_i=1\) if \(\sum_{j=1}^{K} A_{ij}>0\), and zero otherwise. Summing each event's carrier count and calling that the number of locus carriers is incorrect whenever individuals can carry multiple events. More generally, the event count \(\sum_j A_{ij}\) and the locus indicator \(L_i\) answer different questions: the former is the number of event indicators present, while the latter records whether any qualifying event is present.
Worked Example: Overlapping Events and Carrier Counts
In a hypothetical case-control sample, events A and B overlap in genomic coordinates. Their sample-level carrier patterns are summarized below. The four rows are mutually exclusive carrier categories, so every person appears exactly once.
| Carrier category | Cases | Controls |
|---|---|---|
| Neither A nor B | 70 | 65 |
| A only | 12 | 6 |
| B only | 5 | 7 |
| Both A and B | 3 | 2 |
There are \(70+12+5+3=90\) cases and \(65+6+7+2=80\) controls. Among cases, the number carrying at least one event is \(12+5+3=20\); among controls it is \(6+7+2=15\). These are locus-level carrier counts.
The event-specific counts differ. Event A has \(12+3=15\) case carriers and \(6+2=8\) control carriers. Event B has \(5+3=8\) case carriers and \(7+2=9\) control carriers. Adding event-specific carrier counts would give \(15+8=23\) event-carrier records among cases and \(8+9=17\) among controls, because dual carriers are counted twice. Those totals are not counts of distinct people.
For an unadjusted carrier-versus-noncarrier odds ratio, the locus-level estimate is \(\mathrm{OR}_{L}=(20/70)/(15/65)=(20\times65)/(70\times15)=1300/1050=1.238\). This describes the odds of case status among carriers of either event relative to the odds among people carrying neither event. It does not establish a causal effect.
The event-specific odds ratios, each comparing carriers of that event with everyone else, are different. For A, \(\mathrm{OR}_{A}=(15/75)/(8/72)=(15\times72)/(75\times8)=1080/600=1.800\). For B, \(\mathrm{OR}_{B}=(8/82)/(9/71)=(8\times71)/(82\times9)=568/738=0.7696\). Each comparison uses the indicated event carrier group and its complement. The two tests are not independent: some individuals contribute to both carrier groups.
Three Common Association Encodings
The encoding should follow the scientific contrast, not merely the layout of a variant file. Three useful choices are event-specific indicators, mutually exclusive state categories, and a locus-level any-event indicator.
| Encoding | What it represents | Typical interpretation |
|---|---|---|
| One indicator per event | Carrier status for each named event | Association of that event, conditional on other included predictors if modeled jointly |
| Mutually exclusive state categories | One assigned state per person at a defined locus | Each state compared with a chosen reference state |
| Any-event locus indicator | At least one qualifying event in a defined set | Carriers of any included event compared with noncarriers of all included events |
For a binary outcome \(Y_i\), a logistic regression with event indicators can be written as \(\operatorname{logit}\{P(Y_i=1\mid A_{i1},A_{i2},C_i)\}=\beta_0+\beta_A A_{i1}+\beta_B A_{i2}+\gamma^\mathsf{T}C_i\), where \(C_i\) denotes prespecified covariates. Under this model, \(\exp(\beta_A)\) is the conditional odds ratio for event A when event B and the covariates are held fixed. This additive model on the log-odds scale assumes no A-by-B interaction; an interaction term is needed if the association for one event may differ according to carriage of the other. It also requires independent individuals, a correctly specified conditional mean model, and adequate information to estimate its parameters. A model using \(L_i\) instead estimates one contrast for any event in the set, not separate effects for A and B.
When variant states are genuinely alternatives at a locus, a categorical predictor can make the intended comparison explicit. With reference state \(R\) and alternative states \(D\), \(U\), and \(C\), a logistic model can include indicators for \(D\), \(U\), and \(C\), leaving \(R\) as the reference. Each exponentiated coefficient is then an odds ratio against \(R\), conditional on included covariates. This representation is inappropriate if people can occupy multiple states under the definition being used; in that situation, forcing the data into one category discards information or misclassifies co-carriers.
Worked Example: A Multiallelic Locus
Suppose a hypothetical locus is represented by four mutually exclusive states: reference \(R\), deletion \(D\), duplication \(U\), and a defined complex state \(C\). In 200 cases, the counts are \(R=170\), \(D=14\), \(U=12\), and \(C=4\). In 200 controls, they are \(R=180\), \(D=8\), \(U=9\), and \(C=3\). The totals in each group are 200.
For a categorical logistic model using \(R\) as the reference, the unadjusted deletion-versus-reference odds ratio is \(\mathrm{OR}_{D:R}=(14/170)/(8/180)=(14\times180)/(170\times8)=2520/1360=1.853\). It compares the odds of case status for the deletion state with those for the reference state, excluding the other two states from this particular contrast.
The any-nonreference locus encoding instead combines \(D\), \(U\), and \(C\). It gives 30 nonreference cases and 20 nonreference controls, with 170 reference cases and 180 reference controls. Thus \(\mathrm{OR}_{L}=(30/170)/(20/180)=(30\times180)/(170\times20)=5400/3400=1.588\). This is a different estimand from the deletion-versus-reference odds ratio: it combines three alternative states, and its value depends on their frequencies and associations. Calling the first estimate “the locus effect” would therefore be misleading.
The categorical model can also retain separate contrasts for the other states. For example, \(\mathrm{OR}_{U:R}=(12/170)/(9/180)=(12\times180)/(170\times9)=2160/1530=1.412\). These calculations are descriptive unadjusted odds ratios. A regression analysis with covariates would estimate conditional contrasts and would require the model assumptions stated for that analysis.
Choosing an Encoding and Checking Its Meaning
A practical analysis begins by defining the event set and the target contrast. If the question is whether one well-defined event is associated with an outcome, use that event's carrier or genotype coding. If the question is whether any event in a prespecified region is associated, construct a single locus-level indicator from the union of the carriers. If the question distinguishes deletion, duplication, and complex states, preserve those states as separate categories when their definitions are mutually exclusive and meaningful.
A locus-level union can increase the number of people contributing to a carrier group, but it can also combine events with different biological effects. Conversely, event-level analysis preserves distinctions but can produce sparse groups and correlated estimates when carriers overlap. Neither encoding is universally superior. The choice defines the scientific question and should be reported alongside the event set, carrier rule, reference group, and handling of people carrying multiple events.
A short derivation explains why the union indicator is the correct quantity for distinct carriers. Let \(N_A=\sum_i A_{i1}\), \(N_B=\sum_i A_{i2}\), and \(N_{AB}=\sum_i A_{i1}A_{i2}\). The number of distinct people carrying at least one event is \(\sum_i L_i=\sum_i(A_{i1}+A_{i2}-A_{i1}A_{i2})=N_A+N_B-N_{AB}\). Thus, adding the event-specific counts alone overstates the distinct-carrier count by exactly the number of dual carriers. For more than two events, directly forming \(L_i=I(\sum_j A_{ij}>0)\) avoids needing to enumerate every overlap pattern.
Worked Example: Auditing an Event and Locus Design Matrix
Consider six hypothetical people with event indicators and a binary outcome as shown. Events A and B overlap in coordinates, but the table, rather than their coordinates, specifies who carries each event.
| Person | Outcome \(Y\) | \(A_i\) | \(B_i\) | \(L_i\) | \(A_i+B_i\) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 1 | 1 |
| 2 | 0 | 0 | 1 | 1 | 1 |
| 3 | 1 | 1 | 1 | 1 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 6 | 0 | 0 | 1 | 1 | 1 |
There are two A carriers (people 1 and 3) and three B carriers (people 2, 3, and 6), giving \(2+3=5\) event-carrier records. There are only four distinct locus carriers (people 1, 2, 3, and 6), because person 3 carries both. Applying the identity gives \(\sum_i L_i=\sum_i A_i+\sum_i B_i-\sum_i A_iB_i=2+3-1=4\).
The final column is an event count, not a locus carrier indicator: its value is 2 for person 3. In a model with separate event indicators, that person contributes to both event predictors. In a locus-carrier model, the person contributes once with \(L_i=1\). The two columns therefore cannot be interchanged without changing the association being estimated. This small table is an encoding audit, not enough data for reliable association inference.
Common Mistakes
- Counting records as people. Adding event-specific carrier counts includes a dual carrier once for each event. Use the union of carrier sets, or subtract overlaps, to count distinct people.
- Assuming coordinate overlap implies shared carriers. Genomic intervals describe event locations; sample-level indicators describe who carries them. Inspect both.
- Treating a multiallelic locus as one undifferentiated event. An any-event contrast can hide differences among states. Use separate state contrasts when that is the scientific question.
- Calling an event coefficient a marginal effect. In a joint model, an event coefficient is conditional on the other included predictors and covariates, with interpretation depending on the model specification.
- Assuming overlapping event tests are independent. Shared carriers create dependence among event-level estimates and tests; separate event counts do not make the underlying samples independent.
Check Your Understanding
Use the definitions and encodings in this tutorial to answer the following questions.
- If 18 people carry event A, 11 carry event B, and 4 carry both, how many distinct people carry at least one event?
- Explain why overlapping genomic coordinates do not, by themselves, tell you how many people carry both events.
- In a logistic model with indicators for A and B, what contrast does the coefficient for A represent when both indicators are included and there is no interaction?
- Why can an any-event locus odds ratio differ from each event-versus-reference odds ratio at a multiallelic locus?
- What information is lost if a person carrying two qualifying events is represented only by an event count rather than separate event indicators or a locus-level carrier indicator?