Winner’s Curse After GWAS Significance Selection

Why Significance Selection Inflates Estimates

A genetic association study tests many variants, then commonly highlights those whose test statistics pass a significance threshold. Even when each effect estimate is unbiased before selection, estimates that happen to be unusually far from zero are more likely to pass. Among selected variants, sampling errors therefore tend to push estimates away from zero. This selection-induced difference between the average reported estimate and the underlying effect is called the winner’s curse.

The effect is about the estimate, not a change in the true genetic effect. A particular estimate may be close to, above, or below its true value. The key result is an expectation: across repeated studies of the same underlying effect, what is the mean estimate among studies in which the variant is significant?

Sampling Model and Selection Rule

Here \(c\) is the critical value for the chosen significance level. For a two-sided test with level \(\alpha\), \(c\) satisfies \(P(|Z|>c\mid\beta=0)=\alpha\). The standard normal density and cumulative distribution function are denoted by \(\phi\) and \(\Phi\), respectively. The model assumes an unbiased estimate before selection, a normal sampling distribution, and a standard error treated as fixed. These are useful approximations for deriving the effect; departures from them can change the amount of inflation.

The quantity \(\delta\) is the true effect in standard-error units. It determines how likely the variant is to pass the threshold: a larger positive \(\delta\) makes positive-significance results more likely. Since \(\hat{\beta}=SE\,Z\), the conditional expectation of the estimate follows directly from the conditional expectation of \(Z\).

Deriving the Conditional Expectation

For a two-sided test, selection includes both tails: unusually large positive and unusually large negative test statistics. Under \(Z\sim N(\delta,1)\), the selection probability is

To find the mean among selected results, use the truncated-normal tail identities. If \(Z\sim N(\delta,1)\), then \(E[Z\,\mathbf{1}(Z>c)]=\delta\Phi(\delta-c)+\phi(c-\delta)\), and \(E[Z\,\mathbf{1}(Z<-c)]=\delta\Phi(-c-\delta)-\phi(c+\delta)\). These follow by writing \(Z=\delta+U\), where \(U\sim N(0,1)\), and integrating the standard normal density; in particular, \(\int_a^\infty u\phi(u)\,du=\phi(a)\).

Adding the contributions from the two selected tails gives the total selected first moment. Dividing by the probability of selection gives the conditional mean:

Multiplying by \(SE\) and using \(\beta=SE\,\delta\) yields the result for the estimated effect:

For a one-sided rule \(Z>c\), only the upper tail contributes. The corresponding result is

The extra term is positive because selection requires an unusually large positive estimate. This directional formula is not interchangeable with the two-sided formula: a two-sided rule can also select a negative estimate.

Genome-Wide Thresholds and Effect-Size Inflation

For a conventional two-sided genome-wide significance level near \(5\times10^{-8}\), the standard normal critical value is approximately \(c=5.45\). The formula shows that inflation depends not only on this threshold but also on the true signal-to-noise ratio \(\delta\). A variant with a large \(\delta\) is likely to pass even without an unusually favorable sampling error, so the conditional shift can be modest. A weaker variant passes mainly in studies with unusually large estimates, so its selected estimate can be much more inflated relative to its true effect.

Signed Estimates and Reported Magnitudes

The two-sided formula above concerns the signed estimate in the allele coding used for the analysis. When the true effect is positive but small, some selected studies can still have significant negative estimates. If results are instead summarized by the magnitude of the estimated effect, the relevant quantity is \(E[|\hat{\beta}|\mid |Z|>c,\beta]\), which is not generally the absolute value of the conditional signed mean.

The two selected tails contribute positively to the absolute first moment. For \(Z\sim N(\delta,1)\), their combined contribution is \(\delta\{\Phi(\delta-c)-\Phi(-c-\delta)\}+\phi(c-\delta)+\phi(c+\delta)\). Dividing by the selection probability gives

This expression is useful when discussing effect sizes by absolute magnitude or when the estimated sign is used to orient the reported effect. In particular, even at a null effect, the selected signed estimates average to zero under symmetric two-sided selection, while their selected absolute magnitudes are positive.

Interpretation and Common Pitfall

The winner’s curse matters because significance selection affects how discovery estimates should be interpreted: the estimate reported for a selected variant is not an unbiased summary of its true effect when conditioning on having passed the threshold. The formulas quantify this conditional expectation under a specified model and threshold. They do not imply that every selected estimate is too large, nor do they provide a correction for an individual variant without additional information.

• Wrong reasoning: “The estimate was unbiased before testing, so it remains unbiased among significant variants.” Correction: Unbiasedness before selection means \(E[\hat{\beta}\mid\beta]=\beta\) across all studies. Selection changes the distribution being averaged, giving \(E[\hat{\beta}\mid |Z|>c,\beta]\), which generally differs from \(\beta\).
• Wrong reasoning: “A significant estimate must be inflated.” Correction: Inflation describes a conditional average over repeated studies. A particular selected estimate can be below or above that average, and it can even be closer to the true effect than the conditional mean suggests.
• Wrong reasoning: “The two-sided conditional signed mean is the mean reported effect magnitude.” Correction: Signed and absolute estimates differ when selected results occur in both tails. Use the absolute-value expectation when the question concerns magnitudes.

The calculation is conditional on a fixed true effect and standard error. Across variants with different true effects or precisions, the average among all selected variants also depends on how those effects and standard errors are distributed. Thus, the result for a single variant should not be mistaken for a universal inflation factor across a collection of GWAS findings.