From Variant-Level Genotypes to Individual-Level Scores
A defined rare-variant set can contain many loci, while a downstream analysis may need one or a few values per individual. Carrier collapsing reduces the genotypes across the set to a yes-or-no indicator: did the individual carry at least one alternate allele at any included variant? An unweighted burden score instead adds contributions across variants, treating each included variant equally under the chosen scoring rule. These summaries are related, but they do not retain the same information.
The earlier material on genotype dosage coding established how to represent the alternate-allele count at a locus. Here, let \(g_{ij}\) be the genotype dosage for individual \(j\) at variant \(i\), coded as the number of copies of the specified alternate allele. For a fully called, diploid, biallelic genotype, \(g_{ij}\) is 0, 1, or 2. Let the set contain \(K\) variants, with the same set membership and allele coding used for every individual.
Thus \(C_j\) answers whether the person carries any included alternate allele; it is binary. \(B_j\) counts alternate-allele copies across the set, and \(V_j\) counts distinct included variants at which the person has at least one alternate copy. With complete diploid calls, \(C_j\) is 1 exactly when \(B_j>0\), and \(V_j\) is 1 exactly when \(C_j=1\) only if the individual carries a variant at just one locus. In general, \(C_j\), \(V_j\), and \(B_j\) are different summaries.
Conditions for a Meaningful Score
The arithmetic definitions are simple, but meaningful scores depend on careful inputs. First, the variant set must be specified: which loci are included, and which allele at each locus is counted? Second, allele coding must be harmonized so that \(g_{ij}\) consistently counts the intended alternate allele. Third, the genotype calls used in the sums must be valid and complete, or missingness must be handled explicitly. A missing call is not a reference genotype and cannot be silently entered as zero.
For ordinary diploid autosomal, biallelic genotypes, the range of \(B_j\) is 0 to \(2K\), the range of \(V_j\) is 0 to \(K\), and \(C_j\) is either 0 or 1. These bounds change for different ploidy or genotype representations. If variants have different ploidy, or if the dosage is an estimated expected allele count rather than a hard genotype count, the score should be described according to that coding; one should not automatically interpret a fractional dosage as an observed allele copy.
No association model or distributional assumption is needed merely to calculate these summaries. But a score is not itself evidence of association, and its construction does not establish that all variants have the same biological effect. It only defines a compact representation of the observed genotypes.
Worked Example: Collapsing Carriers Across Three Variants
Consider four hypothetical individuals genotyped at three included diploid variants. Each entry is alternate-allele dosage; all calls are complete.
| Individual | Variant 1 | Variant 2 | Variant 3 |
|---|---|---|---|
| A | 0 | 0 | 0 |
| B | 1 | 0 | 0 |
| C | 0 | 2 | 0 |
| D | 1 | 0 | 1 |
For individual A, all dosages are zero, so the per-variant indicators are \(0,0,0\). Therefore \(C_A=I(0+0+0>0)=0\), \(V_A=0\), and \(B_A=0+0+0=0\).
For B, the indicators are \(1,0,0\). Hence \(C_B=I(1>0)=1\), \(V_B=1\), and \(B_B=1+0+0=1\). B carries one alternate copy at one included variant.
For C, dosage 2 at Variant 2 gives indicator 1, not 2. Thus the indicators are \(0,1,0\), \(C_C=1\), \(V_C=1\), and \(B_C=0+2+0=2\). C is a carrier at one locus, with two alternate copies there.
For D, the indicators are \(1,0,1\), so \(C_D=1\), \(V_D=2\), and \(B_D=1+0+1=2\). C and D have the same allele burden, but C has two copies at one locus while D has one copy at each of two loci. The collapsed indicator records neither distinction.
Three Summaries, Three Questions
Carrier collapsing is useful when the target summary is simply whether any qualifying variant is present. It treats a heterozygous and a homozygous alternate genotype alike at that locus, and it treats one carried variant and several carried variants alike once the indicator is collapsed. This can simplify a genotype table substantially, but it discards dosage and across-locus count information.
The distinct-variant count \(V_j\) retains how many different loci are carried, but not the number of alternate copies at any one locus. The allele burden \(B_j\) retains the total number of alternate copies, but not how those copies are distributed among loci. These differences matter whenever scores are described, compared, or used as an individual-level predictor.
Worked Example: Distinguishing Allele Burden From Variant Count
Suppose three hypothetical individuals have complete dosages at four included variants:
| Individual | Dosages across four variants |
|---|---|
| E | 1, 1, 0, 0 |
| F | 2, 0, 0, 0 |
| G | 1, 1, 1, 0 |
For E, \(B_E=1+1+0+0=2\), while \(V_E=1+1+0+0=2\) and \(C_E=1\). E has two alternate copies across two distinct variants.
For F, \(B_F=2+0+0+0=2\), but \(V_F=1+0+0+0=1\) and \(C_F=1\). F has two alternate copies at one locus. The equal burdens \(B_E=B_F=2\) do not imply the same distribution of genotypes.
For G, \(B_G=1+1+1+0=3\), \(V_G=3\), and \(C_G=1\). All three individuals are carriers, but their distinct-variant counts and allele burdens differ. Reporting “carrier” alone would not communicate those differences; reporting “burden” without defining the score would leave it unclear whether the number means allele copies or distinct loci.
Construction as a Reproducible Sequence
A reliable calculation follows the same order for every individual. The sequence below makes the coding choices visible and helps catch errors before a score is used.
Record the included loci and the alternate allele counted at each one. Confirm allele harmonization and the applicable ploidy.
Distinguish a valid dosage of zero from an unobserved or invalid genotype. Decide how incomplete individuals will be treated before calculating a complete-set score.
For each valid dosage, set \(I_{ij}=1\) if \(g_{ij}>0\), and otherwise set \(I_{ij}=0\).
Use the indicator OR for \(C_j\), sum the indicators for \(V_j\), or sum dosages for \(B_j\). Keep the definitions with the resulting data.
Worked Example: A Missing Call Is Not a Zero
Consider a hypothetical individual with observed dosages \(1,0,\text{missing}\) at three included variants. The first two calls establish one alternate copy, but the third genotype is unknown.
If the third call were incorrectly replaced by zero, the apparent allele burden would be \(1+0+0=1\), the distinct-variant count would be \(1+0+0=1\), and the collapsed indicator would be 1. Those values describe the observed calls with an assumed reference genotype at the missing locus, not the complete three-variant genotype.
The complete-set \(B_j\) and \(V_j\) cannot be determined from these data: the missing dosage could add 0, 1, or 2 allele copies, and could add zero or one distinct variant. The carrier indicator is known to equal 1 because the observed dosage of 1 already establishes carrier status; however, that does not make the complete allele burden known. A sound analysis must preserve the missingness distinction and apply a prespecified rule, such as excluding incomplete records for a complete-set score or using a justified genotype-imputation procedure. It must not silently recode missing as reference.
Diagnostics, Alternatives, and Limitations
Several simple checks can reveal construction errors. For complete diploid calls, verify that \(0\leq V_j\leq K\), \(0\leq B_j\leq 2K\), and that \(C_j=0\) if and only if \(B_j=0\). Also verify that \(V_j\leq B_j\), because every locus with at least one alternate copy contributes at least one copy to the allele sum. A violation usually indicates inconsistent coding, an invalid dosage, a missing call mishandled as a number, or a mistaken calculation.
The main alternatives here are different summaries of the same variant-level data, not competing significance tests. Use \(C_j\) when the question is presence versus absence of any included alternate allele. Use \(V_j\) when the number of distinct carried loci is relevant. Use \(B_j\) when total alternate-allele copies are the intended unweighted score. Retaining the full genotype vector preserves more information than any single summary, but does not provide the same compact one-number representation.
Each summary loses information. Carrier collapsing treats all positive genotypes alike and may group people carrying different variants. The distinct-variant count ignores homozygous versus heterozygous dosage. The allele burden can equate different configurations, as in the example where one person had two copies at one locus and another had one copy at each of two loci. An unweighted score also assigns the same numerical contribution to each included variant by construction; that is a scoring choice, not proof that their functional or clinical effects are equal. These scores alone do not establish causation or determine the direction or magnitude of an association.
Check Your Understanding
For each question, calculate or interpret the requested score using the definitions above. Solutions follow the questions.
- An individual has dosages \(0,1,2,0\) at four fully called diploid variants. Find \(C_j\), \(V_j\), and \(B_j\).
- Two individuals have genotypes \(1,1,0\) and \(2,0,0\). Which scores are equal, and which differ?
- Why is an individual with observed dosages \(0,1,\text{missing}\) not necessarily a noncarrier, even though the missing genotype cannot be treated as zero?
- For complete diploid data at \(K=5\) variants, an individual has \(V_j=3\). Give the possible bounds for \(B_j\) based on this information alone.
Solutions. (1) The indicators are \(0,1,1,0\), so \(C_j=1\) and \(V_j=2\). The allele burden is \(B_j=0+1+2+0=3\). (2) Both have \(C_j=1\) and \(B_j=2\). The first has \(V_j=2\), while the second has \(V_j=1\); their genotype configurations differ. (3) The observed dosage of 1 establishes that the individual carries at least one included alternate allele, so the collapsed carrier indicator is known to be 1. The missing call prevents knowing the complete-set allele burden and distinct-variant count, because it could contribute additional copies or a carried locus. (4) Three distinct loci each contribute at least one alternate copy, so \(B_j\geq3\). Those three loci can contribute at most two copies each, and the other two loci contribute at most two each, so \(B_j\leq 10\). Thus the bounds are \(3\leq B_j\leq10\).