Winner’s Curse in Polygenic Score Weights

Why Selected GWAS Effects Matter for a Score

A polygenic score combines genetic variants using estimated effects as weights. When variants are selected because their GWAS results pass a significance threshold, the selected estimates tend to be unusually large in magnitude: some reflect genuine effects, and some reflect favorable sampling error. This winner’s curse matters beyond the individual effect estimates. It affects the score built from them and can make the score look more predictive when it is evaluated in the same data that selected and weighted its variants.

The distinction between discovery and evaluation is central. The discovery sample determines which variants enter the score and supplies their estimated weights. If the score is then associated with the phenotype in that same sample, the association reuses the data that produced those weights. Independent evaluation avoids that direct reuse, although the weights can still be noisy and selection can still affect which variants and effects the score carries forward.

Selection Changes the Distribution of the Weights

For a simple illustration, suppose the estimate has a normal sampling distribution, \(\hat{\beta}\sim N(\beta,SE^2)\), and a variant is selected if its signed test statistic exceeds a positive threshold \(c\): \(\hat{\beta}/SE>c\). Write \(\delta=\beta/SE\), and let \(\phi\) and \(\Phi\) be the standard normal density and cumulative distribution functions. Under this model,

The added term is positive. It arises because the estimate must be high enough to cross the threshold; estimates with sufficiently negative sampling error cannot be selected by this positive-direction rule. With a two-sided rule, the signed conditional mean depends on the true effect and the direction of selection. In particular, across null variants the selected signs can be balanced even though the selected absolute estimates are large. Winner’s curse concerns the magnitude of estimates conditional on selection, not a claim that every selected signed estimate is biased upward.

A score for an individual with genotypes \(G_1,\ldots,G_m\) can be written as \(S=\sum_{j=1}^{m}\hat{\beta}_jG_j\). Inflated or noisy estimates affect both the relative weighting of variants and the variation of the resulting score. Thus a large score association in discovery data cannot, by itself, show that these weights will capture the same signal in new individuals.

How the Discovery Sample Can Make a Score Look Better

The following derivation uses a deliberately simple setting so the mechanism is visible. Assume the selected genotype columns are centered and scaled to have sample variance one, and are mutually orthogonal in the discovery sample. Suppose the weights are the joint ordinary least-squares estimates for those columns. Let \(Y\) be the centered outcome, with sample variance one, and let \(X\) be the matrix of selected genotype columns. Then \(X'X/n=I\), where \(I\) is the identity matrix, and the estimated weight vector is \(\hat{\boldsymbol{\beta}}=X'Y/n\). The discovery score vector is \(S=X\hat{\boldsymbol{\beta}}\).

Here the subscript \(D\) denotes the discovery sample. Since the outcome variance is one, the squared sample correlation between the score and outcome is

The same-sample fit is therefore directly tied to the squared size of the estimated weights. Selection favors variants with large observed estimates, and those estimates are also what construct the score. This reuse produces an optimistic discovery-sample association. The identity is exact under the stated orthogonal, joint-fit setup; it is not a universal formula for every GWAS score. In particular, marginal GWAS estimates and correlated genotypes do not generally satisfy these equalities.

What Changes in an Independent Sample

For a fixed score evaluated in a new population, let \(\boldsymbol{\beta}\) be the vector of true effects, \(\mathbf{w}\) the weights estimated in discovery, and \(\Sigma_G\) the genotype covariance matrix in the evaluation population. Assume a linear outcome model \(Y=\boldsymbol{\beta}'\mathbf{G}+\varepsilon\), with residual \(\varepsilon\) uncorrelated with the genotypes. Then

These expressions describe an association for the chosen weights in the evaluation population. If genotypes are uncorrelated and standardized there, \(\Sigma_G=I\), so the covariance is \(\sum_j w_j\beta_j\) and the score variance is \(\sum_j w_j^2\). Under that simplifying assumption, a selected weight can be large because of its sampling error without contributing a correspondingly large true effect to the covariance. Its squared size still contributes to score variance.

In an independent sample, the discovery noise that selected a weight is not reused to calculate the evaluation phenotype. Consequently, selection alone does not create a positive expected association with a wholly null outcome when the evaluation data and genotypes are independent of discovery and there is no genuine genetic association. For nonzero effects, however, noisy selected weights can reduce how well the score aligns with the true effects. Which way a particular realized evaluation estimate moves also depends on its sampling variability.

Linkage Disequilibrium and the Scope of the Simple Calculation

The orthogonal-variant model isolates the selection-and-reuse mechanism, but genetic variants can be correlated through linkage disequilibrium. With correlated genotypes, \(\Sigma_G\) need not be the identity matrix: a weight at one variant can contribute to the score’s covariance through its correlation with another variant. The independent-sample formulas using \(\Sigma_G\) accommodate that covariance when the stated linear model and residual assumption hold.

In discovery data, the exact score-outcome covariance is determined by the observed genotype-outcome cross-products and the chosen weights. If the weights are a joint least-squares fit to the included variants, the fitted-score identities follow from the least-squares equations, with the genotype covariance matrix retained rather than replaced by \(I\). If weights are marginal GWAS estimates, or selection and fitting use different specifications, the orthogonal joint-fit identity need not hold. The basic warning remains: performance measured in data that helped select variants or estimate weights is not an independent assessment of generalization.

Why the Distinction Matters

A discovery-sample score association combines biological signal with the consequences of selecting and weighting variants using the same outcome data. The score can still be useful, and selection does not imply that every selected variant is false. Rather, the observed discovery estimates are not a neutral description of the true effects once selection has occurred. Reporting an in-sample association as though it were expected performance in new individuals confuses fit to the discovery data with evidence of transportable prediction.

Independent evaluation should apply the already specified score weights to genotypes in data not used to select variants or estimate those weights. The distinction also applies when interpreting a score’s apparent explained variance: it is a sample-specific summary, and its interpretation depends on how the score was constructed and evaluated. For binary outcomes, the relevant performance measure and its interpretation require additional care; the linear \(R^2\) calculations here are for a quantitative outcome with the stated variance model.

Common Mistakes

• “Every selected effect is biased upward.” Selection tends to inflate magnitude; signed estimates can be positive or negative, and null variants selected by a two-sided rule can have balanced signs.
• “A strong discovery association proves the score predicts well elsewhere.” The discovery phenotype helped choose or weight the score. Independent evaluation is needed to assess performance without that direct reuse.
• “The discovery \(R^2\) identity applies to every PRS.” The simple identity here assumes orthogonal standardized variants and joint least-squares weights. Correlated variants and marginal GWAS weights require attention to genotype covariance and the actual estimation procedure.
• “Winner’s curse means there is no real signal.” Some selected effects may be genuine. The issue is that selected estimates can overstate their magnitudes, which can distort score weights and apparent performance.