Variant Selection And Winner’s Curse In MR

Why Variant Selection Matters

In Mendelian randomization (MR), variants are often selected because their estimated associations with an exposure are sufficiently strong in a discovery genome-wide association study (GWAS). This is a practical way to identify candidate instruments, but it changes how the exposure estimates should be interpreted. Among variants that pass a threshold, some estimates will be large partly because their sampling errors happened to point away from zero.

This selection-related exaggeration is called the winner’s curse. It matters even when the variants are valid instruments and the exposure and outcome association estimates come from independent samples. A selected exposure estimate may overstate instrument strength; because it appears in the denominator of a ratio estimate, the corresponding MR estimate can also be distorted. Power calculations that use selected estimates can be too optimistic.

A Model for Selection on the Exposure Association

For variant \(j\), let \(\beta_{Xj}\) denote its true association with the exposure, measured in exposure units per allele, and let \(\hat{\beta}_{Xj}\) be its discovery estimate. Suppose, before selection, the estimate is unbiased and approximately normal:

Here \(SE_{Xj}\) is the standard error in the discovery data. A common selection rule retains a variant if its association passes a threshold, for example if \(|\hat{\beta}_{Xj}/SE_{Xj}|>c\), where \(c\) is a chosen critical value. The reported estimate is then not an unselected draw from the model above: it is a draw conditional on passing the rule.

To see the basic direction of the effect, suppose the allele has been oriented so that the true association is positive, and selection retains it when \(\hat{\beta}_{Xj}/SE_{Xj}>c\). Estimates just below the threshold are excluded; selected estimates are drawn from the upper part of the sampling distribution. Consequently, their conditional mean exceeds the true association. The same principle applies to two-sided selection after the allele is oriented so that the estimated association is positive.

Why Conditioning Inflates the Estimate

The truncation calculation makes the selection mechanism explicit. Write \(Z=\mu+U\), where \(U\sim N(0,1)\), and select when \(Z>c\). The event is then \(U>c-\mu\). The standard normal identity \(\int_a^\infty u\phi(u)\,du=\phi(a)\) follows from \(\phi'(u)=-u\phi(u)\). Therefore:

The final term is positive for every finite threshold, so the conditional mean exceeds \(\mu\). When the true effect is close to the selection boundary, the retained estimates can be especially sensitive to this mechanism: a larger share of variants near that boundary need favorable sampling errors to pass. If the true effect is zero, variants that pass a one-sided threshold do so because their estimates are positive, on average; this does not make their true effects positive.

The calculation describes selection on one estimate under a normal approximation. Real discovery analyses may test many variants, use stringent thresholds, and account for linkage disequilibrium. Those details affect which variants are selected and the magnitude of the bias, but they do not change the central point: an estimate chosen for being extreme is not an ordinary unbiased estimate conditional on that choice.

Consequences for Ratio Estimates

For a single valid instrument, the Wald ratio is the estimated variant–outcome association divided by the estimated variant–exposure association. Let \(\beta_{Yj}\) be the true outcome association and \(\hat{\beta}_{Yj}\) its estimate. Under the simple valid-instrument model, \(\beta_{Yj}=\theta\beta_{Xj}\), where \(\theta\) is the causal effect on the outcome per unit increase in the exposure. The estimated ratio is:

If selection makes the discovery denominator too large and the outcome estimate comes from independent data, the numerator is not pushed upward by that exposure-based selection. The ratio will therefore tend to be smaller in magnitude than the corresponding ratio using the true exposure association. With a positive causal effect and a positively oriented allele, this is commonly described as attenuation toward zero.

That description is about the direction of the ratio distortion under the stated setup; it is not an exact identity for the expected ratio. Ratios are nonlinear, and the expectation of a ratio is not generally the ratio of expectations. In particular, a denominator that can be close to zero can make the ratio’s sampling behavior unstable. The illustration below compares a selected denominator with the true association rather than claiming to calculate the exact conditional expectation of a ratio.

Independence between the exposure and outcome estimation errors is important to that interpretation. If the samples overlap, the estimates may be statistically dependent. Conditioning on exposure selection can then also change the conditional behavior of the outcome estimate, depending on the overlap and the covariance of the errors. The simple “inflated denominator, otherwise unchanged numerator” explanation no longer suffices. Likewise, winner’s curse is distinct from violations of the instrumental-variable assumptions: selection on exposure association alone does not establish that a variant has a direct effect on the outcome, but it can still distort estimates and their uncertainty.

Instrument Strength and Power Calculations

The same selection effect can make an instrument look stronger than it is. For a single variant, an observed strength summary is often based on the estimated exposure association relative to its standard error, such as the squared association \(Z\)-statistic. Because variants were selected for large exposure statistics, the selected values are not representative of the unselected sampling distribution. A strength threshold does not remove this conditional bias; passing a threshold is the event that creates it.

Power calculations can inherit the same problem. Under the simple model \(\beta_{Yj}=\theta\beta_{Xj}\), the outcome signal used in a calculation is proportional to \(\theta\beta_{Xj}\). If the selected discovery estimate is substituted for \(\beta_{Xj}\), the predicted outcome signal can be too large. Holding the outcome standard error fixed, a larger predicted signal gives a larger noncentrality parameter and hence a larger calculated power. Alternatively, a study designed to reach a target power may appear to need fewer participants than it really does.

Reducing and Interpreting Selection Effects

The clearest way to avoid using a discovery estimate as though it were unselected is to estimate the exposure association in an independent replication sample. Because the replication estimate was not used to choose the variant, it does not have the same discovery-selection conditioning, though it still has ordinary sampling uncertainty. Other approaches can account for selection or shrink discovery estimates toward zero. The appropriate method depends on the selection process and the available data; simply reporting a selected estimate with more decimal places does not correct it.

When interpreting MR results, keep three quantities conceptually separate: the observed discovery association used to select a variant, the underlying exposure association that determines instrument strength, and the causal estimand targeted by the MR analysis. A strong-looking selected association is evidence about the selection event and the data, not a guarantee of a large true association or a precise causal estimate.

Common Mistakes

• “The selected exposure estimate is unbiased because it was unbiased before selection.” Unbiasedness before selection does not imply unbiasedness conditional on passing a threshold. The conditional distribution is truncated.
• “An inflated denominator makes the ratio exactly equal to the true effect times a fixed attenuation factor.” Ratios are nonlinear and both associations are estimated with error. Attenuation toward zero is a useful direction under independent outcome estimation and the simple valid-instrument model, not a universal exact formula.
• “A selected strength statistic proves the instrument is strong.” Selection makes the statistic unusually large by design. A threshold describes the selection rule; it does not remove winner’s curse or establish instrument validity.
• “Power calculated from the selected effect is the study’s actual power.” If the selected estimate overstates the underlying exposure association, a plug-in calculation can overstate the expected outcome signal and power.
• “Winner’s curse proves the causal estimate is biased in one fixed direction.” The usual attenuation explanation assumes a particular relationship between selection and estimation errors. Sample overlap, other sources of dependence, and the ratio’s instability can alter the behavior.