Why Selected Genetic Effects Tend to Look Too Large
A genetic association estimate combines an underlying effect with sampling noise. Even when an estimate is unbiased before looking at the data, selecting associations because their estimates are large changes the distribution of the estimates being reported. Among estimates that pass a sufficiently demanding threshold, positive noise values are more likely to have helped an association pass than negative noise values are. The selected estimates therefore tend, on average, to be farther from zero than the corresponding true effects.
This conditional inflation is called the winner’s curse. It matters when interpreting discovery results: the reported effect for a selected locus is not automatically a reliable estimate of its true magnitude. The claim is about the average among estimates selected by a rule, not a guarantee that every selected estimate is too large.
Sampling Model and Threshold Selection
Consider one genetic association with true effect \(\beta\) and discovery estimate \(\hat{\beta}\). For this tutorial, assume the estimate has a normal sampling distribution with known standard error \(SE\):
This model says that, before selection, the estimate is centered at the true effect and has sampling variance \(SE^2\). It is an approximation commonly useful for association estimates with adequate information. The calculation below treats \(SE\) as fixed, assumes the model is appropriate, and treats the threshold \(c\) as specified in advance. It does not account for uncertainty in the standard error or for other reasons an estimate might be selected.
For positive, direction-specific selection, retain the estimate when \(\hat{\beta}>c\). Standardize the threshold by defining \(a=(c-\beta)/SE\). Let \(\phi\) and \(\Phi\) denote the standard normal density and cumulative distribution functions. The probability of passing the threshold is \(1-\Phi(a)\). Conditional on passing it, the mean estimate is
The additional term is positive for any finite threshold, so the conditional mean exceeds \(\beta\). Its size depends on the threshold relative to the true effect and on the standard error. A higher threshold relative to the true effect generally makes passing more dependent on a favorable noise realization.
Why the Conditional Mean Is Inflated
Write the estimate as \(\hat{\beta}=\beta+SEZ\), where \(Z\sim N(0,1)\). The event \(\hat{\beta}>c\) is equivalent to \(Z>a\), with \(a=(c-\beta)/SE\). Therefore,
The conditional expectation of \(Z\) follows by integrating its density over the retained tail. Since the standard normal density satisfies \(\phi'(z)=-z\phi(z)\), integrating \(z\phi(z)\) from \(a\) to infinity gives \(\phi(a)\). The conditional probability in the denominator is \(P(Z>a)=1-\Phi(a)\). Thus,
Substituting this result gives the conditional-mean formula above. The derivation also explains the direction of the effect: the retained values of \(Z\) are all above \(a\), and their conditional mean is positive. Before selection, \(E(Z)=0\); after selection, the noise component has a positive mean.
If selection is based on absolute magnitude and the true effect is zero, positive and negative estimates are retained symmetrically. In that special case, the expected magnitude among selected estimates has a simple expression:
The factor of two from the two tails appears in both the expectation numerator and the selection probability denominator and cancels. For nonzero \(\beta\), the two tails need not contribute equally, so the one-sided formula should not be applied to absolute-value selection without accounting for both tails.
Worked Example: Absolute-Value Selection Under a Null Effect
Suppose a hypothetical variant has true effect \(\beta=0\), and its discovery estimate has \(SE=0.030\) effect units. Researchers report it only if \(|\hat{\beta}|>0.075\) effect units. What is the expected absolute estimate among reported variants under this model?
The standardized cutoff is \(0.075/0.030=2.5\). Under the null model, the probability of selection is \(2[1-\Phi(2.5)]\), approximately \(2(0.0062097)=0.01242\). The conditional expected magnitude is
Thus, although the true effect is zero, the expected absolute estimate among those passing this threshold is about 0.08468 effect units. This does not mean every selected estimate will have that magnitude; it is the conditional average under the stated model.
Worked Example: A Positive Effect and a One-Sided Threshold
Suppose a hypothetical association has true effect \(\beta=0.080\) effect units and discovery \(SE=0.020\) effect units. A positive estimate is selected when it exceeds \(c=0.100\) effect units. Find the probability of selection and the expected estimate among selected results.
Here \(a=(0.100-0.080)/0.020=1\). The selection probability is \(1-\Phi(1)\approx0.1587\), and the conditional mean is
The expected selected estimate is about 0.1105 effect units, approximately 0.03050 effect units above the true effect. This is an average conditional on passing the threshold, not the expected estimate across all discovery studies. Without selection, the model gives \(E(\hat{\beta})=0.080\).
Worked Example: A More Stringent Threshold
Consider a hypothetical association with true effect \(\beta=0.020\) effect units and \(SE=0.015\) effect units. Only positive estimates above \(0.050\) effect units are reported. Calculate the selected mean and interpret it.
The standardized cutoff is \(a=(0.050-0.020)/0.015=2\). The probability of selection is \(1-\Phi(2)\approx0.02275\). Using \(\phi(2)\approx0.0539910\),
Among results that pass this threshold, the expected discovery estimate is about 0.05560 effect units, about 0.03560 above the true effect. The selection probability is only about 2.275%, so the selected group consists of unusually high estimates relative to the full sampling distribution. The calculation describes that conditional group, not the behavior of every study or every locus.
Interpreting the Calculation and Checking Its Fit
The key comparison is between an estimate before selection and an estimate conditional on the selection rule. Before selection, the model is unbiased: \(E(\hat{\beta})=\beta\). Conditional on exceeding a one-sided threshold, the expected estimate is larger than \(\beta\). This is a consequence of conditioning, not evidence that the sampling model was biased before results were selected.
The simple calculation is most informative when the target is a clearly defined effect and the rule is genuinely a fixed threshold on its noisy estimate. In genomic analyses, several features can complicate that description:
- Selection rule differs from a fixed threshold. Being the largest estimate among many candidates is not the same event as merely exceeding a fixed cutoff. The formula derived here conditions on the latter.
- Effect estimates have different standard errors. The conditional inflation depends on \(SE\), so one common value cannot describe loci with substantially different precision.
- Normal sampling is inadequate. Sparse counts, small samples, or poorly behaved estimates can make the assumed normal distribution inaccurate. A tail calculation based on the wrong distribution can misstate the conditional mean.
- The estimate is selected for more than its magnitude. Screening, quality filters, or a data-dependent choice of direction can alter which estimates are reported. The selection event must match the one used in the calculation.
- Effects differ across loci or cohorts. A single \(\beta\) and \(SE\) describe one model for one association. Pooling unlike effects without a suitable model can obscure what conditional average is being calculated.
A practical diagnostic is to state the selection event explicitly and compare it with the actual reporting process. Check whether the estimate and its standard error are on the stated scale, whether the cutoff is one-sided or two-sided, and whether the normal approximation is plausible in the relevant tail. If discovery estimates are larger than estimates from independent data, that pattern is compatible with selection inflation, but it is not by itself proof: sampling variation, heterogeneity, and differences in study design can also contribute.
How This Differs From Other Estimates
The conditional mean is not a replacement for every estimate of a genetic effect. It answers a particular question: under a specified sampling model, what is the average estimate among results retained by this threshold? The alternatives answer different questions.
| Quantity or approach | What it describes | How it differs from threshold conditioning |
|---|---|---|
| Unselected discovery estimate | An estimate before conditioning on passing a threshold | Has mean \(\beta\) under the stated model; does not describe only reported results |
| Threshold-conditioned mean | Expected estimate among estimates passing a specified rule | Explicitly includes the shift caused by conditioning on that rule |
| Independent replication estimate | An estimate from new data not used for discovery selection | Under an unbiased sampling model, its mean is \(\beta\), not the discovery conditional mean |
| Meta-analysis estimate | A combined estimate across studies under a specified model | Combines information across cohorts; it does not automatically remove selection in a discovery estimate that entered the analysis |
Independent replication is a useful contrast because its sampling noise is not the noise that caused the discovery estimate to pass the threshold, provided the replication data are genuinely independent and are not selected on the same result. Its estimate can still differ from the true effect by chance. The conditional expectation is also not an individual-locus correction: it describes an average under assumptions, rather than identifying the true effect from one selected estimate.
Common Mistake
A common mistake is to say, “The discovery estimator is unbiased, so its reported estimate is unbiased too.” That reasoning ignores the condition defining which estimates are reported. Unbiasedness concerns the distribution before selection: \(E(\hat{\beta})=\beta\). After selecting only estimates above \(c\), the relevant mean is \(E(\hat{\beta}\mid\hat{\beta}>c)\), which includes a positive conditional noise term. The correction is to name the selection event and interpret the conditional expectation, rather than transferring the unselected unbiasedness claim to the selected subset.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Under \(\hat{\beta}\sim N(\beta,SE^2)\), what is the standardized threshold for selecting estimates with \(\hat{\beta}>c\)?
- For \(\beta=0.040\), \(SE=0.010\), and \(c=0.050\), calculate the selection probability and conditional mean, using \(\phi(1)=0.2419707\) and \(1-\Phi(1)=0.1586553\).
- Why does an unbiased estimate before selection not imply an unbiased mean among estimates passing a positive threshold?
- For a zero true effect and a symmetric rule \(|\hat{\beta}|>c\), why is the expected signed estimate among selected estimates zero even though their expected absolute magnitude is positive?
Solutions. (1) The standardized threshold is \(a=(c-\beta)/SE\), because \(\hat{\beta}>c\) is equivalent to \(Z>a\) when \(\hat{\beta}=\beta+SEZ\). (2) Here \(a=1\), so selection probability is \(1-\Phi(1)=0.1586553\). The conditional mean is \(0.040+0.010(0.2419707/0.1586553)\approx0.040+0.015251=0.05525\) effect units. (3) The unselected mean averages positive and negative sampling errors, but threshold selection retains estimates with sufficiently positive errors more often; thus the conditional noise mean is positive. (4) Under the zero-effect normal model and symmetric selection, positive and negative tails have equal probabilities and mirror-image estimates, so their signed contributions cancel. Their magnitudes do not cancel, so the selected absolute estimates have a positive mean.