Decomposing Genotype Into Ancestry-Specific Components

Why Decompose an Allele Dosage?

At a biallelic locus, the usual diploid genotype dosage \(G_i\) records how many copies of a chosen allele individual \(i\) carries: 0, 1, or 2. In an admixed genome, however, those two copies may have different local ancestries. A dosage of 1 therefore does not say whether the allele copy lies on an ancestry-A chromosome, an ancestry-B chromosome, or another ancestry background. Decomposing dosage makes those contributions explicit.

This is an accounting identity, not a new association test. It clarifies how genotype variation can reflect differences in local ancestry composition, differences in allele counts among chromosomes of the same ancestry, or both. That distinction helps interpret the local-ancestry association models introduced in Regression Models for Local Ancestry Association; it does not, by itself, establish that ancestry or genotype causes a phenotype.

Define the Ancestry-Specific Counts

Consider a diploid individual \(i\) at one biallelic locus. Index the two homologous chromosomes by \(h=1,2\). Let \(X_{ih}=1\) if chromosome \(h\) carries the allele being counted and \(X_{ih}=0\) otherwise. Let \(Z_{ih}\) denote the local ancestry assigned to that chromosome at the locus. For ancestry \(a\), define \(L_{ia}\) as the number of the individual's two chromosomes assigned ancestry \(a\), and define \(G_{ia}\) as the number of counted-allele copies on those ancestry-\(a\) chromosomes.

Here \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. Thus \(L_{ia}\) is a local-ancestry copy count, while \(G_{ia}\) is an allele count restricted to chromosomes with that local ancestry. Necessarily \(0\leq G_{ia}\leq L_{ia}\). If the possible ancestries are \(1,\ldots,K\), each of the two chromosomes has one assigned ancestry, so \(\sum_{a=1}^{K} L_{ia}=2\). The ordinary allele dosage is \(G_i=\sum_{h=1}^{2}X_{ih}\).

The identity follows because every chromosome contributes its allele indicator to exactly one ancestry-specific count. It does not require independence between chromosomes, equal allele frequencies across ancestries, or a particular phenotype model. Its interpretation as separating composition from within-ancestry allele variation uses an additional reference: an allele frequency \(p_a\) for the counted allele in ancestry \(a\). These frequencies might be specified from appropriate reference data or as model parameters. They should be defined for the same allele and a relevant population; they are not supplied by the identity itself.

Separating Composition From Within-Ancestry Deviation

For each ancestry, add and subtract \(L_{ia}p_a\) from its allele count. Summing over ancestries gives a second exact identity:

The composition component is the dosage expected from the individual's local ancestry copy counts if each ancestry-\(a\) chromosome has counted-allele probability \(p_a\). The second component measures how the observed ancestry-specific counts differ from those reference-frequency expectations. It can be positive or negative. These components sum to the observed dosage; neither should be mistaken for a separately observed biological measurement.

A model-based expectation makes the interpretation precise. Suppose that, conditional on the ancestry assignments, each ancestry-\(a\) chromosome has expected counted-allele indicator \(p_a\). Linearity of expectation then gives \(E(G_{ia}\mid Z_{i1},Z_{i2})=L_{ia}p_a\), and hence \(E(G_i\mid Z_{i1},Z_{i2})=\sum_a L_{ia}p_a\). No independence assumption is needed for this expectation. It does not follow that the observed deviation is zero for an individual, or that the deviation is independent of ancestry composition in a finite sample.

Aggregating the Components Across Individuals

The identity also holds after summing over individuals. For \(n\) people, let \(G_{ia}\) and \(L_{ia}\) be defined at the same locus for each person. Then the total allele count is \(\sum_i G_i=\sum_a\sum_i G_{ia}\), and the reference-based decomposition is \(\sum_i G_i=\sum_a p_a\sum_i L_{ia}+\sum_a\sum_i(G_{ia}-L_{ia}p_a)\). This is useful for checking calculations and describing how the cohort’s observed allele count divides according to its ancestry-specific chromosome counts and deviations. It does not turn a cohort-level total into an individual-level causal explanation.

Assumptions, Diagnostics, and Alternatives

For the count identity, the essential requirements are limited: the locus is treated as diploid and biallelic; the counted allele is coded consistently; and each chromosome's allele and local ancestry are paired on that same chromosome. The reference-frequency interpretation adds the requirement that each \(p_a\) represents the allele frequency relevant to ancestry-\(a\) chromosomes in the analysis. If that interpretation is used as an expectation model, its conditional mean assumption must be appropriate for the population and locus being described. The algebraic decomposition remains true even if those reference frequencies are poorly chosen, but the composition and deviation then have a less useful interpretation.

A practical diagnostic is to verify, for every person and locus, that \(\sum_aL_{ia}=2\), that \(0\leq G_{ia}\leq L_{ia}\), and that \(\sum_aG_{ia}=G_i\). These checks can uncover inconsistent allele coding, dropped ancestry categories, or a mismatch between phased ancestry labels and genotype data. For a dosage represented as an expectation under uncertain calls, the corresponding component quantities should be constructed consistently; additivity of expected values is valid when they refer to the same underlying allele and ancestry assignments. Such a dosage does not make the ancestry assignment known with certainty.

The raw dosage \(G_i\) is the simplest summary and is often adequate when the analysis asks only about total allele copies. The counts \(G_{ia}\) retain which ancestry carries those copies. The centered decomposition goes further by expressing the dosage relative to chosen ancestry-specific frequencies. These are different representations of related data, not interchangeable claims: the centered representation depends on the \(p_a\) values, while the raw count identity does not. The decomposition is not, by itself, a regression specification, a test of association, or a test of differing genetic effects across ancestries.

Common Mistakes and Limitations

• Calling the composition component the observed genotype. It is a reference-frequency expectation based on local ancestry counts. The observed dosage includes the deviation as well.
• Assuming a zero deviation for each person. Even under a correct frequency model, individual allele counts vary. The expectation statement concerns an average conditional on ancestry, not a guarantee for every person.
• Changing allele coding midway. If one count uses the alternate allele and another uses the reference allele, the partition will not be meaningful. Use one counted allele throughout.
• Treating reference frequencies as universal constants. Their meaning depends on how ancestry groups and reference data are defined. Inaccurate or poorly matched values can distort the composition-versus-deviation interpretation, although they cannot break the basic count identity.
• Inferring causation or independence from the arithmetic. A decomposition partitions a measured dosage; it does not show that ancestry causes a phenotype or that its components are statistically independent.

The principal limitation is therefore interpretive: the exact partition is determined by counts, but the biological or statistical meaning of its centered components depends on allele definition, local-ancestry calls, and reference frequencies. A transparent analysis reports those choices rather than presenting the components as self-explanatory quantities.