Global Ancestry Adjustment in Admixed Cohorts

Why Adjust for Global Ancestry?

In an admixed cohort, individuals can differ in their genome-wide proportions of ancestry from several reference populations. Those proportions may be associated with both a SNP genotype and an outcome. If they are omitted from an association model, a genotype-outcome association can partly reflect differences in ancestry rather than the conditional SNP effect of interest. Including global ancestry proportions as covariates is one way to account for this broad pattern of population structure.

The adjustment is not a claim that ancestry is a simple biological cause, nor does it guarantee that all confounding has been removed. It defines an association conditional on the ancestry variables included in the model. Its usefulness depends on how well those variables represent relevant structure, whether the model is correctly specified, and whether there is enough genotype variation across ancestry values to estimate the adjusted effect.

Definition: A person’s global ancestry proportions are estimated fractions of genome-wide ancestry assigned to specified ancestral sources. For \(K\) sources, write these proportions as \(A_{i1},\ldots,A_{iK}\) for individual \(i\), where \(A_{ik}\geq 0\) and \(\sum_{k=1}^{K}A_{ik}=1\). They summarize ancestry across the genome; they are not the ancestry assigned to a particular genomic region.

Association Regression With Ancestry Covariates

For a quantitative outcome \(Y_i\), let \(G_i\) be the SNP dosage for individual \(i\), usually the number of copies of a specified allele, coded from 0 to 2. Let \(C_{ij}\) denote other measured covariates, such as age or sex. A linear model for one SNP is

Y_i=\beta_0+\beta_G G_i+\sum_{k=1}^{K-1}\gamma_k A_{ik} +\sum_{j=1}^{q}\delta_j C_{ij}+\varepsilon_i.

One ancestry proportion is omitted because the \(K\) proportions sum to one. Including an intercept and all \(K\) proportions would create exact linear dependence: \(1=\sum_{k=1}^{K}A_{ik}\). The omitted source is the reference for interpreting the included ancestry coefficients; it is not an ancestry that has been removed from the individuals.

The model assumes that the conditional mean of the outcome is represented by the stated linear predictor and that the residuals have mean zero conditional on the included predictors. For usual model-based standard errors and tests, one also needs an appropriate variance model and independence assumptions, or a method that accounts for dependence such as relatedness. Approximate normality of residuals is chiefly relevant to small-sample inference; it is not a requirement for the coefficient to be defined. The genotype effect is assumed constant across the modeled ancestry range unless interactions or other varying-effect terms are included.

Interpretation: In the linear model, \(\beta_G\) is the expected outcome difference for a one-copy increase in the coded allele, comparing individuals with the same included global ancestry proportions and other covariates. The units are outcome units per allele copy. It is a conditional association, not automatically a causal effect.

For a binary outcome, the same covariate strategy can be used in a logistic regression. In that case, \(\beta_G\) is a conditional change in log odds per allele copy, and \(\exp(\beta_G)\) is the corresponding conditional odds ratio. The outcome model and its assumptions must be appropriate for the data; adding ancestry covariates does not by itself make a model suitable.

Worked Example: Interpreting an Adjusted SNP Coefficient

Suppose a hypothetical linear regression of a biomarker, measured in milligrams per liter, on a SNP dosage and two ancestry proportions reports \(\hat{\beta}_G=0.42\) with \(SE=0.15\). Assume the model includes an intercept, the two proportions for a three-source ancestry model, and relevant measured covariates. The third proportion is omitted because the three sum to one.

The estimated conditional association is \(0.42\) milligrams per liter per additional copy of the coded allele. It compares individuals at the same modeled global ancestry proportions and covariate values. A Wald statistic for \(H_0:\beta_G=0\) is \(z=0.42/0.15=2.80\). Under a large-sample standard normal approximation, the two-sided p-value is about \(0.0051\).

An approximate 95% confidence interval is \(0.42\pm1.96(0.15)=0.42\pm0.294\), or \((0.126,0.714)\) milligrams per liter per allele copy. Under the model and its standard-error assumptions, this procedure has 95% long-run coverage for the conditional coefficient. The interval does not mean there is a 95% probability that this fixed coefficient lies inside this particular interval.

Why Omitting Ancestry Can Change the SNP Slope

A short omitted-variable calculation explains the purpose of adjustment. Consider a simplified model with one ancestry covariate \(A\), a SNP genotype \(G\), an intercept, and an error: \(Y=\beta_0+\beta_GG+\gamma A+\varepsilon\), where \(E(\varepsilon\mid G,A)=0\). If \(Y\) is instead regressed on an intercept and \(G\) alone, the population slope is

\frac{\operatorname{Cov}(G,Y)}{\operatorname{Var}(G)} = \beta_G+ \gamma\frac{\operatorname{Cov}(G,A)}{\operatorname{Var}(G)}.

To see this, take covariance with \(G\) on both sides of the outcome model. The intercept has zero covariance with \(G\), and the conditional mean-zero error gives \(\operatorname{Cov}(G,\varepsilon)=0\). Thus \(\operatorname{Cov}(G,Y)=\beta_G\operatorname{Var}(G)+\gamma\operatorname{Cov}(G,A)\); division by \(\operatorname{Var}(G)\) gives the expression. The unadjusted slope differs from \(\beta_G\) when ancestry is associated with the outcome (\(\gamma\ne0\)) and with genotype (\(\operatorname{Cov}(G,A)\ne0\)).

This identity clarifies both the benefit and the limitation. If the included ancestry variables adequately capture the relevant structure and the regression model is appropriate, adjustment can remove this particular source of slope distortion. It cannot remove distortion from omitted structure, incorrect functional form, measurement error, or other unmodeled associations.

Worked Example: Quantifying Omitted-Ancestry Distortion

In a hypothetical cohort, suppose the correctly specified conditional model is \(Y=3+1.5G+4A+\varepsilon\). Here \(Y\) is in arbitrary outcome units, \(G\) is allele dosage, and \(A\) is one ancestry proportion. Suppose the cohort’s genotype and ancestry summaries are \(\operatorname{Var}(G)=0.50\) and \(\operatorname{Cov}(G,A)=0.08\), with \(E(\varepsilon\mid G,A)=0\).

The slope from regressing \(Y\) on \(G\) alone is \(\beta_G+\gamma\operatorname{Cov}(G,A)/\operatorname{Var}(G)\). Substituting the values gives \(1.5+4(0.08/0.50)=1.5+4(0.16)=2.14\) outcome units per allele copy. In this hypothetical setting, the unadjusted slope is \(2.14\), whereas the conditional slope is \(1.5\). The difference, \(0.64\), is attributable to the omitted ancestry term under the stated model and moments; this calculation is not a general correction formula when the model assumptions fail.

Interpreting Multiple Ancestry Proportions

With three ancestry sources, for example, \(A_{i1}+A_{i2}+A_{i3}=1\). If source 3 is omitted, a coefficient \(\gamma_1\) describes the modeled outcome change associated with a one-unit increase in \(A_{i1}\) while \(A_{i2}\) and the other covariates are held fixed. Since the proportions must still sum to one, this change corresponds to replacing source 3 proportion with source 1. It is not a comparison that increases one proportion while leaving every other proportion unchanged.

The same constraint matters when interpreting fitted values. A difference between two valid ancestry profiles can be calculated by multiplying each included proportion difference by its coefficient. This describes the model’s conditional mean contrast at fixed genotype and other covariates. It does not establish that changing ancestry would change the outcome.

Worked Example: A Contrast Between Valid Ancestry Profiles

Consider a hypothetical fitted model with three ancestry sources, source 3 omitted, and ancestry coefficients \(\hat{\gamma}_1=0.8\) and \(\hat{\gamma}_2=0.3\). The outcome is measured in units of a laboratory score. Compare profiles \((A_1,A_2,A_3)=(0.6,0.3,0.1)\) and \((0.2,0.3,0.5)\), holding genotype and all other covariates fixed.

The included-proportion differences are \(0.6-0.2=0.4\) for source 1 and \(0.3-0.3=0\) for source 2. The fitted conditional mean contrast is \(0.8(0.4)+0.3(0)=0.32\) laboratory-score units. This is equivalently a 0.4 increase in source 1 proportion accompanied by a 0.4 decrease in the omitted source 3 proportion. It is a model-based contrast, not a causal effect of ancestry.

Diagnostics, Alternatives, and Limitations

Before interpreting an adjusted coefficient, check whether the data support the comparison the model implies. If genotype dosage is nearly determined by ancestry, there may be little within-ancestry genotype variation. The model may then have a large standard error, unstable coefficients, or strong sensitivity to a few observations. A coefficient can be algebraically estimable yet depend heavily on extrapolation where the data have little overlap.

Useful checks include inspecting genotype distributions across ancestry ranges, assessing correlations among ancestry covariates and other predictors, and examining whether outcome residual patterns vary systematically with ancestry. For continuous ancestry proportions, plots or flexible checks can reveal curvature that a linear term misses. Compare the stated model with reasonable alternatives and report sensitivity when conclusions change materially. A small p-value is not a diagnostic that the ancestry model is adequate.

Global ancestry proportions also average over the genome. They may not represent all relevant population structure at a particular SNP, and they may be measured with error. Relevant structure can remain if the outcome depends nonlinearly on ancestry, if effects vary with ancestry, or if environmental and social factors associated with ancestry are inadequately modeled. Adding ancestry covariates does not ensure causal identification, eliminate unmeasured confounding, or make an association transportable to a different population.

Principal components of genetic variation offer another way to summarize broad genetic structure; their coordinates need not correspond directly to interpretable ancestry proportions. Mixed models can account for genetic similarity among related or structured individuals through a covariance model. These approaches address related but not identical modeling goals. Depending on the study, ancestry proportions, principal components, mixed-model terms, and measured covariates may be considered together, but each addition should be justified and checked for collinearity and adequate data support.

Global ancestry adjustment is distinct from modeling ancestry at a specific genomic region. It should not be interpreted as a test of regional ancestry, nor should it be assumed to capture every locus-specific pattern. The adjusted SNP coefficient answers the conditional association question defined by the regression; other scientific questions require different models.

Common mistake: “I included global ancestry proportions, so population structure is fully controlled and the SNP coefficient is causal.” The correction is that adjustment controls only the modeled relationships represented by those covariates under the model assumptions. Check overlap, fit, measurement quality, and remaining sources of confounding, and describe the coefficient as a conditional association unless causal assumptions are separately defended.

A Practical Analysis Sequence

Specify the target association.
State the outcome, allele dosage coding, effect scale, and covariates needed for the scientific question.

Encode ancestry without redundancy.
Include \(K-1\) proportions with an intercept, or use another nonredundant parameterization, and identify the reference source.

Fit and check the model.
Assess genotype-ancestry overlap, collinearity, functional form, residual patterns, and whether dependence among observations is handled.

Report the conditional result.
Give the effect estimate, units or effect scale, standard error or interval, p-value when relevant, covariates, and key limitations.

Check Your Understanding

Answer each question, then compare with the solutions.

In a model with four ancestry proportions summing to one and an intercept, why should one proportion be omitted?
A linear model estimates a SNP coefficient of \(-0.25\) for an outcome measured in seconds. State its conditional interpretation.
Suppose \(\beta_G=0.8\), \(\gamma=3\), \(\operatorname{Cov}(G,A)=0.10\), and \(\operatorname{Var}(G)=0.50\). What slope is expected if ancestry is omitted under the simplified model?
Name two reasons global ancestry adjustment might not remove all relevant structure.

Solutions. (1) The proportions sum to one, so including all four alongside an intercept creates exact linear dependence; omit one as the reference. (2) At the same included ancestry proportions and other covariates, one additional copy of the coded allele is associated with an estimated 0.25-second lower outcome. This is conditional association, not necessarily causation. (3) The omitted-variable expression gives \(0.8+3(0.10/0.50)=0.8+0.6=1.4\) outcome units per allele copy. (4) Examples include nonlinear outcome relationships with ancestry, ancestry measurement error, unmodeled effects that vary across ancestry, and structure not captured by genome-wide proportions.