Population Structure as Predictive Signal Contamination

Why a Pooled Prediction Can Mislead

A polygenic score may predict a phenotype in a sample that contains multiple ancestry groups. That pooled association can arise in two distinct ways: people with higher scores may tend to have higher phenotypes within ancestry groups, or groups with higher average scores may also have higher average phenotypes. The second pattern can make a score look predictive even when it has little or no association with the phenotype within any group.

This distinction matters when interpreting prediction. A pooled association can be useful for describing prediction in the particular mixture of groups studied, but it does not by itself show that the score has the same predictive relationship within groups or in another population. Here, \(A\) denotes an ancestry group, \(S\) a polygenic score, and \(Y\) a phenotype. These are statistical variables; group membership is not a biological cause established by the decomposition.

The Covariance Decomposition

Suppose \(S\) and \(Y\) have finite variances, and \(A\) identifies the group for each observation. Write \(\mu_{S,a}=E[S\mid A=a]\) and \(\mu_{Y,a}=E[Y\mid A=a]\) for the group-specific means. The law of total covariance gives:

The first term averages the within-group covariances, weighting each group by its proportion in the pooled population. The second term is the covariance, across groups, of their mean scores and mean phenotypes. It is positive when groups with higher mean scores also tend to have higher mean phenotypes, negative when the mean differences tend to run in opposite directions, and zero when there is no such covariance.

The decomposition has a direct counterpart for the pooled linear regression slope. Provided \(\operatorname{Var}(S)>0\), the population slope from the linear regression of \(Y\) on \(S\) with an intercept is \(\operatorname{Cov}(S,Y)/\operatorname{Var}(S)\). Total variance decomposes in the same way:

Consequently, the pooled slope is

This expression is not generally an average of group-specific slopes: its numerator and denominator each contain a between-group term. If the within-group slope is a common value \(\beta_W\) in every group, then \(\operatorname{Cov}(S,Y\mid A=a)=\beta_W\operatorname{Var}(S\mid A=a)\). The pooled slope is therefore

Even under this common-within-slope condition, the pooled slope can differ from \(\beta_W\). The between-group covariance adds to or subtracts from the pooled numerator, while between-group score differences also add to the denominator.

Why the Decomposition Holds

Add and subtract the conditional means inside each deviation from the overall mean:

Multiply these expressions and take expectations. The product of the two within-group deviations contributes \(E[\operatorname{Cov}(S,Y\mid A)]\). Each cross-product has expectation zero: conditional on \(A\), the within-group deviation has mean zero, while the group-mean deviation is fixed. The product of the two group-mean deviations contributes \(\operatorname{Cov}(E[S\mid A],E[Y\mid A])\). Adding the nonzero terms yields the stated covariance decomposition. This argument requires finite second moments, but does not require a linear model or a causal interpretation.

How Allele Frequencies Can Shift Mean Scores

For an additive score \(S=\sum_{j=1}^{m}w_jG_j\), let \(G_j\) be the allele dosage at variant \(j\), coded as 0, 1, or 2 copies of the allele used in the score, and let \(w_j\) be its fixed score weight. If the frequency of that allele in group \(a\) is \(p_{ja}\), then \(E[G_j\mid A=a]=2p_{ja}\). Linearity of expectation gives the group mean score:

This identity does not require variants to be independent. It shows that differences in allele frequencies can shift the expected score between groups. For a score to be predictively associated with the phenotype in a pooled sample, however, a score difference alone is not enough: the group mean phenotypes must also differ in a way that produces nonzero between-group covariance, or there must be within-group association.

When Within- and Between-Group Signals Coexist

The pooled slope is especially easy to misread when a score has a genuine within-group association and the group means are also aligned. The next example calculates both components, rather than treating all pooled signal as either valid or spurious.

Interpretation and a Common Pitfall

“Signal contamination” describes the risk of interpreting pooled prediction as though it were entirely a within-ancestry genetic relationship. It does not mean that every between-group component is meaningless or that a pooled score cannot predict in a specified mixed population. If the target population has the same group composition and group-specific means as the evaluation sample, between-group differences may contribute to prediction in that target. The concern is that the contribution can change when the group mixture, group means, or within-group relationships change.

A common mistake is to reason: “The pooled score predicts the phenotype, so higher scores must predict higher phenotypes within each ancestry group.” The conclusion does not follow. As the first example showed, pooled covariance can be nonzero when every within-group covariance is zero. The correction is to separate the within-group and between-group components before interpreting the pooled result. A related mistake is to treat a frequency-derived difference in mean scores as proof of a causal genetic effect on the phenotype. The mean-score identity describes score distributions; it does not identify the cause of group phenotype differences.

The decomposition is a statement about covariance in a defined population, not a procedure for deciding which ancestry groups should be compared or for establishing causal effects. It is useful because it makes explicit what a pooled association combines. Any claim about performance in a particular setting must be tied to that setting and its population composition.