Winner's Curse In Instrument Selection

Why Selection Changes the Estimates We See

Genetic variants are often selected as instruments because their estimated associations with an exposure pass a significance threshold. But selection uses noisy estimates: among variants whose true exposure effects are modest, those that happen to receive unusually large estimates are more likely to be retained. Consequently, the exposure associations reported for selected variants tend to be shifted in the direction used for selection. This is the winner’s curse.

The issue is not that every selected estimate must be too large, nor that the true effect itself changes. It is a statement about a conditional distribution: after conditioning on a selection event, the estimates have a different distribution from estimates considered before selection. This matters in Mendelian randomization because the exposure-association estimate is used in the denominator of a Wald ratio, and because strength summaries such as an F statistic are calculated from the same selected association.

That qualification is important. If a variant is selected by a one-sided rule \(\hat{\gamma}/SE > c\), then the conditional signed mean exceeds \(\gamma\), even when \(\gamma\) is negative. But an estimate shifted upward from a very negative true effect can be closer to zero, not farther from it. In practice, variants may be selected by a two-sided test and then oriented to the allele associated with higher exposure. The simple one-sided model below describes selection in a prespecified direction; it gives a clear calculation when the true effect is oriented in that direction. With uncertain or incorrectly estimated direction, signed and absolute-magnitude claims must not be conflated.

A Normal-Selection Calculation

Suppose the discovery-sample estimate follows the standard sampling model \(\hat{\gamma} \sim N(\gamma, SE^2)\), with a known standard error \(SE\). For a one-sided selection rule, retain the variant when \(\hat{\gamma}/SE > c\), where \(c\) is a fixed positive threshold. This is equivalent to \(\hat{\gamma} > cSE\). The model assumes the sampling distribution is approximately normal, the standard error is treated as fixed, and selection is based on this estimate and threshold.

Write \(Z=(\hat{\gamma}-\gamma)/SE\), so \(Z\sim N(0,1)\). Define \(a=c-\gamma/SE\). Selection is the event \(Z>a\). For a standard normal variable, the conditional mean above a threshold is \(E[Z\mid Z>a]=\phi(a)/[1-\Phi(a)]\), where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution functions. It follows that:

The fraction is positive, so the conditional mean signed estimate exceeds \(\gamma\). If the true effect is oriented in the selected direction (\(\gamma\geq 0\)), this is an upward shift in that direction. If \(\gamma\) is not oriented that way, it does not by itself establish inflation in absolute magnitude. The shift is generally larger when selection is more stringent relative to the true signal.

Consequences for Wald Ratios and Strength

For a single valid variant, let \(\beta\) be its true outcome association and \(\theta\) the causal effect per exposure unit. Under the instrumental-variable model with no direct effect of the variant on the outcome, \(\beta=\theta\gamma\). The Wald estimate is \(\hat{\theta}=\hat{\beta}/\hat{\gamma}\). If selection inflates a positive exposure estimate while the outcome estimate is obtained independently, the denominator used in the ratio tends to be too large relative to the true exposure association. This can pull the ratio toward zero.

This is a direction-of-effect explanation, not an exact identity for the expected ratio. Ratios are nonlinear, and the expectation of a ratio is not generally the ratio of expectations. A first-order approximation, when the selected denominator is sufficiently far from zero and its variability is not dominant, is to compare the ratio of the outcome mean to the conditional exposure mean with the true ratio. If the denominator is unstable, this approximation can be poor. Under an independent outcome sample, selection on the exposure estimate does not itself shift the outcome estimate’s conditional mean; sample overlap or correlated estimation errors require a different analysis.

Selection also affects apparent strength. For one variant, a common association-strength statistic is \(F=Z^2\) in the corresponding one-degree-of-freedom setting. If selection requires \(Z>2.5\), then every retained estimate has \(F>6.25\) under that definition. This truncation is built into the selection rule: the observed strength among retained variants cannot be read as an unbiased measure of their underlying strength. A large selected F statistic does not undo the fact that the same data were used both to select the variant and to estimate its exposure association.

Winner’s curse and weak-instrument bias are related but distinct. Winner’s curse concerns the conditional behavior of exposure-association estimates after selection. Weak-instrument bias concerns the behavior of causal estimators when instruments provide limited information; in one-sample settings, its direction can depend on the design and sources of confounding. Correcting selection does not automatically remove weak-instrument bias, and a strength diagnostic alone does not correct winner’s curse.

Practical Mitigation and Comparison

Methods differ in whether they prevent reuse of the selected estimate, model the selection event, or reduce noisy estimates toward plausible values. The right choice depends on data availability and on how selection was conducted.

Independent replication is often the most transparent remedy. If discovery and replication samples are independent, then conditional on selection in discovery, the replication estimate has its usual sampling distribution around the true association, assuming the replication model and sample are appropriate. Selection-adjusted likelihood instead retains discovery information but explicitly uses its conditional distribution. For the one-sided normal example, the density used after selection is the normal density divided by \(P(\hat{\gamma}>cSE\mid\gamma)\), for values above the threshold. Ignoring that normalizing probability treats a truncated sample as if it were unselected.

No method is universally best. External estimates can be less precise or drawn from populations with different allele frequencies or exposure measurement. Sample splitting trades precision for independence. Selection-adjusted analysis can be sensitive to an incorrectly recorded threshold, multiple filtering stages, or dependence among variants. Shrinkage can stabilize estimates but is not a substitute for checking its assumptions. These approaches should not be described as interchangeable, and a correction for exposure selection alone does not address other violations of the instrumental-variable assumptions.

Diagnostics, Assumptions, and Failure Modes

A practical assessment begins by documenting which exposure data were used to select variants, the selection threshold, and which data supplied the association estimates used in the causal ratios. Compare discovery and independent replication estimates when available. Systematic shrinkage from discovery to replication, especially among variants near the selection threshold, is consistent with winner’s curse, although differences can also reflect sampling variation or differences between study populations. Report whether strength statistics were calculated from the same data used for selection.

The normal calculation is a model, not a universal correction formula. Its assumptions include an approximately normal estimate with a suitable standard error, selection based on a known threshold, and a clearly defined direction of selection. Genome-wide selection can involve two-sided tests, multiple screening stages, allele orientation, and correlated variants. In those settings, the exact conditional distribution depends on the full selection procedure. If selection is two-sided, a useful approximation is to orient the estimate by its discovery sign and analyze positive magnitudes when the selected sign is likely to match the true sign. When the sign is uncertain, wrong-direction selections matter and the one-sided formula should not be treated as an exact magnitude correction.

A common mistake is to say, “Every significant estimate is farther from zero than its true effect, so dividing by it must bias the causal estimate toward zero.” The first claim is not valid without a direction condition: under a positive one-sided threshold, the signed conditional mean exceeds the true effect even if the true effect is negative, in which case it may move closer to zero. The second claim is also too absolute: random outcome error, denominator variability, sample overlap, and the exact selection mechanism affect ratio behavior. The safer conclusion is conditional and qualified: selection shifts the signed exposure estimate in the selected direction; with a correctly oriented positive true effect and an independent outcome estimate, an inflated denominator can attenuate the typical Wald ratio.