Ancestry-Stratified Association Analysis

Why Analyze Association Within Ancestry Strata?

A genotype–phenotype association calculated across a mixed-ancestry sample can reflect differences between ancestry groups, differences within those groups, or both. An ancestry-stratified analysis asks a narrower question: within each defined ancestry stratum, how does the phenotype vary with genotype? Estimating the association separately makes those within-stratum comparisons visible rather than allowing the sample-wide pattern to stand in for them.

This approach is useful only when the genotype varies within a stratum. If every person in a stratum has the same genotype dosage, that stratum provides no comparison of phenotype across genotype values. Even when variation exists, a very small number of observations at one genotype value can make the fitted effect sensitive to individual observations. The analysis therefore requires both a model for the within-stratum association and a careful check of the data supporting each estimate.

A Separate Linear Model in Each Stratum

Consider a quantitative phenotype \(Y\) and an additive genotype dosage \(G\), coded as 0, 1, or 2 copies of a specified allele. For stratum \(s\), fit the simple linear model

where \(i\) indexes individuals, \(\alpha_s\) is the stratum-specific intercept, and \(\beta_s\) is the stratum-specific genotype effect. In this model, \(\beta_s\) is measured in phenotype units per additional allele copy. It describes a linear change in the conditional mean phenotype across dosage values within stratum \(s\); it does not by itself establish a causal effect.

The usual least-squares calculation assumes that the mean phenotype is linear in dosage within each stratum, that observations are independent, and that the error has mean zero conditional on dosage. The usual standard error and small-sample \(t\)-based inference additionally rely on constant error variance across dosage values; exact \(t\)-based inference assumes normally distributed errors. These conditions should be considered in context. In particular, related individuals can violate the independence assumption if their dependence is not accounted for.

The genotype dosage should use the same allele definition and coding in every stratum. An estimate of \(2\) phenotype units per copy means that the fitted mean differs by \(2\) units for each one-copy increase in dosage. It does not require that all three dosage categories occur in a stratum, but missing categories limit which dosage contrasts the data directly support.

How the Within-Stratum Slope Is Calculated

For a stratum with \(n_s\) individuals, let \(\bar{G}_s\) and \(\bar{Y}_s\) denote the mean genotype dosage and mean phenotype. Define the within-stratum sums of squares and cross-products as

If \(S_{GG,s}>0\), the least-squares slope and intercept are

These formulas center genotype and phenotype on their own stratum means. Thus the slope is obtained from within-stratum variation, not from differences between stratum means.

Worked Examples

Check Whether Each Stratum Supports Its Estimate

Before interpreting a slope, inspect genotype counts within every stratum. The most basic diagnostic is whether there is any genotype variation at all. If \(S_{GG,s}=0\), every dosage equals \(\bar{G}_s\), and the slope formula has a zero denominator. No within-stratum genotype effect is estimable from those data. A slope calculated from other strata cannot fill that gap without changing the analysis question.

When \(S_{GG,s}>0\), the slope is mathematically estimable, but that fact alone does not mean the data support a dependable estimate. A useful review includes:

• Counts by dosage: Record how many individuals have dosage 0, 1, and 2 in each stratum. Empty or nearly empty categories limit direct comparisons.
• Range of observed dosage: Check whether the fitted slope relies on extrapolation beyond observed values, as it would for dosage 2 in the second example.
• Influential observations: A rare dosage category may contain a single person whose phenotype has a large effect on the estimated slope.
• Phenotype information: A genotype category with several people can still be uninformative if phenotype measurements are highly variable or missing.
• Model assumptions: Consider whether a linear change per allele copy and the required error assumptions are plausible within the stratum.

For the simple linear model, if residuals are \(e_{is}=Y_{is}-\hat{\alpha}_s-\hat{\beta}_sG_{is}\), the residual sum of squares is \(SSE_s=\sum_i e_{is}^2\). When \(n_s>2\), the estimated residual variance is \(MSE_s=SSE_s/(n_s-2)\), and the usual estimated standard error of the slope is

This formula shows two ways precision can suffer: residual variability can be large, or genotype variation \(S_{GG,s}\) can be small. It does not remove the need to inspect dosage counts. In particular, a small standard error under a model does not make an unobserved genotype category observed, nor does it guarantee that the model describes the data well.

Why the Stratum-Specific View Matters

Ancestry strata can differ in genotype frequencies, phenotype distributions, or both. A separate slope makes the genotype–phenotype comparison within each stratum explicit and prevents between-stratum mean differences from being treated as if they were within-stratum genotype variation. The interpretation remains conditional on how strata were defined and on the adequacy of the within-stratum model; stratification does not automatically remove every source of confounding.

Strata should be defined using ancestry information rather than chosen after inspecting which grouping produces a preferred association. Also report strata that are too sparse to support an informative estimate. Omitting such a limitation can make a set of separate slopes appear more comprehensive than the genotype data allow.

Common Mistakes

• Wrong: “The slope can be estimated because the full sample contains all three genotype dosages.” Correction: Check variation within each stratum. A stratum with one observed dosage has \(S_{GG,s}=0\) and no estimable within-stratum slope.
• Wrong: “A computed slope is necessarily well supported.” Correction: A slope can exist with very sparse dosage counts and be highly sensitive to one observation. Report the counts and assess the range of observed genotypes.
• Wrong: “A slope based on dosages 0 and 1 demonstrates the observed phenotype pattern at dosage 2.” Correction: The dosage-2 pattern is not observed in that stratum; any fitted value there extends the model beyond the data.
• Wrong: “A stratum-specific association proves the allele causes the phenotype.” Correction: The slope describes an association under the specified model. It does not establish causation.