Identifiability of Ancestry-Specific Effects

Why Identifiability Comes Before Estimation

A model can be fitted by software and still fail to answer the scientific question it was intended to address. In an admixed cohort, a pooled genotype association may be estimable even when the data cannot distinguish an effect associated with ancestry A from one associated with ancestry B. The key question is not merely whether a regression returns coefficients, but whether the observed ancestry-specific genotype components provide distinct information about those coefficients.

This tutorial considers a single variant and two ancestral sources, labelled A and B. The effects are regression associations per additional alternate allele assigned to the corresponding ancestry component. Identifiability is a property of the model together with the design: it does not establish that either effect is causal, and it is not the same as having a small standard error.

Model and Rank Condition

For person \(i\), let \(G_{iA}\) and \(G_{iB}\) be the numbers of alternate alleles on ancestry A and ancestry B chromosome segments at the variant. For a diploid individual, each is a nonnegative integer and \(G_{iA}+G_{iB}\) is the total alternate-allele dosage. Let \(L_{iA}\) denote the number of the person's two chromosome copies assigned to ancestry A at that locus; then \(L_{iB}=2-L_{iA}\). A local-ancestry covariate may be included among the nuisance predictors.

Consider the linear model

Here \(\boldsymbol{x}_i\) contains the intercept and any other specified covariates, \(\boldsymbol{\gamma}\) is their coefficient vector, and \(\beta_A,\beta_B\) are the ancestry-specific associations in outcome units per alternate allele. For ordinary least squares inference, assume the conditional mean follows this model and the errors have mean zero, constant variance \(\sigma^2\), and are independent across individuals. Normal errors are additionally needed for exact finite-sample \(t\)- and \(F\)-inference. These assumptions concern estimation and inference; the rank condition below determines whether the coefficients can be separated by the design.

Write \(X\) for the matrix of nuisance columns and \(Z=[G_A\;G_B]\) for the two genotype-component columns. When \(X\) has full column rank, define the residual-maker matrix \(M_X=I-X(X^\mathsf{T}X)^{-1}X^\mathsf{T}\). It removes the linear parts of the genotype components explained by the nuisance predictors. The effects are separately identifiable if and only if \(Z^\mathsf{T}M_XZ\) has rank two. Equivalently, the two columns of \(M_XZ\) must be nonzero and not exact scalar multiples of one another.

This condition makes the role of adjustment explicit: two genotype columns can vary in the raw data but become indistinguishable after their shared variation with ancestry or other covariates is removed. A zero determinant means exact non-identifiability. A small positive determinant means the effects are technically identifiable but the data provide little information for distinguishing them.

What Allele Frequencies and Local Ancestry Contribute

Let \(p_A\) and \(p_B\) denote the alternate-allele frequencies on ancestry A and B chromosome segments in the population represented by the model. Under a simple allele-sampling description, the conditional expectations are \(E(G_{iA}\mid L_{iA})=L_{iA}p_A\) and \(E(G_{iB}\mid L_{iB})=L_{iB}p_B\). A contrast between \(p_A\) and \(p_B\) can therefore make the ancestry-specific components behave differently across people, especially when local ancestry counts vary. Local-ancestry variation can also give individuals different numbers of opportunities to carry alleles from each ancestry.

Neither feature guarantees identifiability. If a particular ancestry contributes no chromosome copies at the locus in the observed cohort, its genotype component is zero for everyone. Conversely, a frequency contrast does not rescue a design in which the resulting genotype columns are exactly redundant after adjustment. Frequency contrasts and local-ancestry variation are clues about where independent variation may come from; the rank condition is the definitive design check.

A useful distinction is between structural and practical identifiability. Structural non-identifiability occurs when the model and design cannot distinguish parameter values at all. Practical weakness occurs when the columns are not exactly redundant but are so strongly correlated that small changes in outcomes can produce large changes in the separate coefficient estimates.

A Short Justification of the Rank Test

The rank condition follows by asking whether two parameter settings can produce the same fitted mean. Suppose \((\boldsymbol{\gamma},\beta_A,\beta_B)\) and \((\boldsymbol{\gamma}',\beta_A',\beta_B')\) yield identical means. Subtract the two mean vectors, then project away the nuisance columns with \(M_X\). The nuisance terms vanish, leaving

If \(M_XZ\) has rank two, its null space contains only the zero vector. The equation then requires \(\beta_A=\beta_A'\) and \(\beta_B=\beta_B'\), so the two effects are identifiable. If its rank is below two, there is a nonzero vector \(\boldsymbol{d}\) with \(M_XZ\boldsymbol{d}=0\). Moving the effect vector by any multiple of \(\boldsymbol{d}\) changes the genotype contribution only by a vector in the nuisance-column space; the nuisance coefficients can offset that change. The same fitted means can therefore arise from distinct effect values. This proves both directions of the rank criterion.

Near-Collinearity and Weak Identification

For two residualized genotype columns, the determinant can be written as the product of their squared lengths times one minus their squared correlation. Thus the determinant is close to zero when either component has little residual variation or when the components are nearly perfectly correlated after adjustment. In the full-rank case with independent errors of variance \(\sigma^2\), the coefficient covariance matrix is \(\sigma^2(Z^\mathsf{T}M_XZ)^{-1}\). A nearly singular matrix therefore produces large standard errors and sensitivity to small modelling or data changes.

Interpreting the Design and Choosing an Alternative

A frequency contrast can help create distinct ancestry-specific genotype patterns, but it is not a requirement in itself. For example, imagine four hypothetical people who each have one A-derived and one B-derived chromosome at the locus. Let their components be \(G_A=(0,1,0,1)^\mathsf{T}\) and \(G_B=(0,0,1,1)^\mathsf{T}\). Both ancestry components have sample frequency \(2/4=0.5\) across their four chromosome copies. After removing the intercept, the centered columns are \((-0.5,0.5,-0.5,0.5)^\mathsf{T}\) and \((-0.5,-0.5,0.5,0.5)^\mathsf{T}\). Their squared lengths are both \(1\), and their cross-product is \(0\). The two effects are identifiable despite equal sample frequencies and no variation in the number of A-derived copies.

This example also shows why local-ancestry variation is informative but not logically necessary: variation in allele counts within each ancestry can itself supply two independent directions. In other designs, variation in local ancestry may be crucial. What matters is the residualized design, including any local-ancestry predictor that the model adjusts for.

The main alternative is a pooled genotype model that imposes a common effect, such as \(\beta_A=\beta_B=\beta\). It needs only the pooled genotype direction and can remain estimable when the two-effect model is not. Its gain in simplicity comes with a restriction: it does not estimate separate ancestry-specific effects. Another option is to report that the separate effects are not estimable rather than presenting arbitrary or unstable coefficients as meaningful estimates. Separate analyses within ancestry groups are not automatically a solution; they require adequate genotype variation within each group and answer a different design-based question.

For a practical assessment, examine the local-ancestry counts and ancestry-specific allele counts, then compute the rank and conditioning of the adjusted component matrix. If local ancestry is uncertain, use the uncertainty-aware approach established in the earlier material on propagating local ancestry uncertainty; treating uncertain assignments as certain can misrepresent the design information. A full-rank matrix is necessary for separate estimation, but it does not validate the model, eliminate confounding, or demonstrate a biological mechanism.

Diagnostics and Common Pitfalls

• Zero or nearly zero component variation. Tabulate ancestry-specific allele counts and local-ancestry support. A component with no residual variation cannot identify its effect.
• Exact linear dependence. Check matrix rank after including the intercept, local ancestry, and other covariates. A dropped column or singular fit is a warning, not evidence that the dropped effect is zero.
• Near-collinearity. Inspect the residualized correlation or singular values of \(M_XZ\), and compare coefficient stability under reasonable modelling choices. A nonzero determinant alone does not guarantee useful precision.
• Confusing allele-frequency contrast with identification. Different ancestry-specific frequencies may help generate distinct columns, but only the observed, adjusted design determines rank.
• Assuming a pooled result answers the separate-effect question. A common-effect estimate is identified under a restriction; it does not show that the data could estimate two unrestricted effects.

A common mistake is to reason, “The allele frequencies differ by ancestry, so the two effects must be identifiable.” The correction is to treat that contrast as a possible source of information, then verify that both ancestry-specific columns retain independent variation after nuisance adjustment. The reverse mistake is also possible: equal frequencies do not force non-identifiability, as the orthogonal-column example shows.