Effect-Size Heterogeneity Across Ancestries

What Does an Ancestry Difference in Effect Mean?

A polygenic score can perform differently across ancestry groups, but that observation alone does not show that genetic effects differ. A group difference in a regression coefficient may reflect a genuine difference in the effect of a genetic variant on the phenotype. It may also arise because the groups differ in allele frequency, linkage disequilibrium (LD), phenotype measurement, or the precision of the estimates. These explanations have different implications and should not be conflated.

The key is to specify which effect is being compared and on what scale. For a quantitative phenotype, a per-allele regression coefficient describes the expected change in the phenotype, in its stated units, associated with one additional copy of an allele, conditional on the model's covariates. It is not automatically a causal effect. A difference in estimated coefficients is evidence about a difference in the corresponding association parameters only to the extent that the variants, phenotype, model, and comparison are comparable.

Define the Regression Effect Before Comparing It

Consider a biallelic variant coded as dosage \(G\), the number of copies of a specified effect allele, with values 0, 1, or 2. In ancestry group \(k\), a linear regression for a quantitative phenotype can be written as

where \(Y\) is the phenotype, \(C\) denotes included covariates, \(\alpha_k\) is a group-specific intercept, and \(\beta_k\) is the per-allele association in phenotype units per allele copy. Under this model, \(\beta_k\) is the change in the conditional mean of \(Y\) associated with one additional effect-allele copy, holding the included covariates fixed.

The comparison of interest is often \(\Delta=\beta_A-\beta_B\). Interpreting it requires the same allele coding, phenotype definition and units, and a suitably comparable covariate model in both groups. For instance, a coefficient in kilograms cannot be directly compared with a coefficient in pounds without conversion. Even when these conditions hold, the coefficient is an association unless the study design and assumptions justify a causal interpretation.

A regression coefficient for a score is similarly specific to how the score is constructed and scaled. A coefficient per one score point is not directly comparable if a score point represents different amounts of genetic variation in the groups. Standardizing a score can aid some comparisons, but then the coefficient is expressed per group-specific standard deviation and answers a different question from a per-allele effect.

Testing a Difference and Its Assumptions

The contrast can be tested directly with a regression interaction. Let \(I_B\) equal 1 for group B and 0 for group A. A combined model can include group-specific intercepts and a genotype-by-group term:

In this parameterization, the genotype slope in group A is \(\beta\), the slope in group B is \(\beta+\delta\), and the difference (B minus A) is \(\delta\). The null hypothesis of equal slopes is \(H_0:\delta=0\). This test addresses a difference in the specified regression associations; it does not by itself identify why a difference exists.

The interaction approach and the difference-of-estimates approach express the same comparison when they use equivalent models and data. Valid standard errors require an appropriate variance model. In a simple linear model, normal-error assumptions support exact small-sample inference; with sufficiently large samples, suitable robust standard errors can provide asymptotic inference under weaker distributional assumptions. Participants should be independent, or the standard errors must account for dependence such as relatedness or clustering. The model must also adequately handle relevant covariates and population structure. If the two group estimates are correlated, the variance of their difference is not the sum of their variances.

Why Allele Frequency Can Change Without Effect Heterogeneity

A per-allele effect and the amount of genetic variation contributed by a variant are different quantities. Under Hardy–Weinberg equilibrium, a dosage \(G\) for an allele with frequency \(p\) has variance \(2p(1-p)\). If the same per-allele effect \(\beta\) applies in both groups, the variance of the variant's additive genetic contribution \(\beta G\) is \(\beta^2 2p(1-p)\). Thus, different allele frequencies can change the variance explained, and can affect the precision of estimates, without changing \(\beta\).

LD and the Difference Between a Tag and a Causal Variant

A regression association at a marker that tags a causal variant can differ across groups even if the causal variant has the same effect. The marker's association depends partly on how strongly it covaries with the causal genotype. As LD differs, that covariance can differ. This is distinct from a change in the causal variant's effect.

For a simple illustration, let \(Z_C\) and \(Z_T\) be standardized genotypes at a causal variant and a tag variant. Suppose \(Y=\beta Z_C+\varepsilon\), where \(\varepsilon\) has mean zero and is uncorrelated with both genotypes. The population regression slope of \(Y\) on \(Z_T\) alone is

where \(r_{TC}\) is the correlation between the standardized genotypes. The slope at the tag therefore depends on both the causal effect and the LD correlation.

Phenotype Scale, Measurement, and Estimation Error

Effects depend on the phenotype scale. If a group measures the same underlying phenotype in different units, raw regression coefficients will change with the unit conversion. If one group records height in centimeters and another in meters, the numerical coefficient differs by a factor of 100 even though the underlying association is unchanged. Differences in instruments, protocols, or phenotype definitions can be more complicated: they may change the construct being measured, not merely its units.

Independent random error in a quantitative outcome generally increases residual variation and reduces precision; under a correctly specified linear model, it does not necessarily attenuate the expected unstandardized slope. But group-specific scaling changes the slope's units, and outcome noise can weaken standardized associations or prediction metrics. Error in genotype measurement, such as imperfect dosage measurement, can also distort regression slopes. The direction and size of these effects depend on the measurement process, so comparability should be investigated rather than presumed.

Finally, estimates vary by chance. A larger standard error means that a given observed difference is less informative about the underlying contrast. A non-significant heterogeneity test is not proof of identical effects; a significant test is not a measure of the contrast's practical importance, nor proof of a biological mechanism. Report the estimated difference, its uncertainty, and the scale and model used.

A Short Justification for the Contrast Standard Error

For estimates from independent samples, the variance of a difference follows from the variance rule for independent random variables. If each estimate is approximately unbiased for its group parameter, then \(\hat{\Delta}=\hat{\beta}_A-\hat{\beta}_B\) estimates \(\Delta=\beta_A-\beta_B\), and

Taking the square root gives the standard error used in the first worked example. If the estimates are dependent, the covariance term must be included:

With approximate normality, dividing the estimated contrast by its standard error gives an approximately standard normal statistic under \(H_0:\Delta=0\). The approximation and variance calculation are only as suitable as the model and sampling assumptions behind the estimates.

Common Mistake: Calling Every Coefficient Difference Biological Heterogeneity

• Wrong reasoning: “The score slope is smaller in group B, so genetic effects must be weaker in that group.”
• Correction: First establish that the two slopes refer to comparable score or variant definitions, phenotype units, and models. Then consider whether allele-frequency differences, LD between markers and causal variants, measurement differences, or sampling uncertainty could explain the contrast.
• Wrong reasoning: “The heterogeneity test was not significant, so the effects are the same.”
• Correction: A non-significant result means the data did not provide sufficient evidence against the tested equality under the model. Examine the confidence interval to see which differences remain compatible with the data.