Empirical Bayes Shrinkage of Selected GWAS Effects

Why Shrink Selected Effects?

A genome-wide association study can report an estimated effect for each tested variant, but the variants that pass a significance threshold are not a representative sample of all variants. Even when estimates are unbiased before selection, those that happen to be unusually large in magnitude are more likely to be reported. This is the winner's curse: selected discovery estimates tend to overstate the underlying effects.

Empirical Bayes shrinkage addresses a related question: given a collection of estimated effects, what effect estimate is sensible for each locus if effects across loci are drawn from a shared distribution? The method combines each locus's discovery estimate with a prior distribution learned across loci. Because the data available for learning that distribution may themselves be selected, the selection mechanism must also be represented in the prior-fitting step.

This tutorial treats effect estimates on a common scale, such as log odds per allele. Shrinkage moves an estimate toward the prior mean, often zero, but is not a correction that guarantees an unbiased estimate. Its quality depends on the likelihood, prior, and selection model being reasonable.

Posterior Shrinkage for a Known Prior

For a moment, treat \(\tau^2\) as known. The likelihood for \(\beta_j\) is proportional to \(\exp\{-(Y_j-\beta_j)^2/(2s_j^2)\}\), and the prior is proportional to \(\exp\{-\beta_j^2/(2\tau^2)\}\). Multiplying them gives the posterior density. Completing the square gives its mean and variance:

The posterior mean is a weighted average of the raw estimate and the prior mean of zero. Its weight on the raw estimate is \(w_j=\tau^2/(\tau^2+s_j^2)\), which lies between zero and one. Less precise estimates, with larger \(s_j\), receive more shrinkage; estimates with high precision receive less. The posterior standard deviation is the square root of the posterior variance. In this model it is smaller than the raw standard error \(s_j\), because the prior contributes information.

Learning the Prior From Selected Loci

If estimates from all tested loci are available, a marginal likelihood can be used to estimate \(\tau^2\). Under the normal model, integrating out \(\beta_j\) gives \(Y_j\sim N(0,\tau^2+s_j^2)\). But if the estimation data contain only loci passing a threshold, fitting this untruncated marginal distribution to those estimates will generally overstate the spread. The selected estimates are disproportionately large in magnitude.

Suppose selection retains a locus when \(|Y_j/s_j|>c_j\), with known threshold \(c_j\). Given \(\tau\), the marginal density of \(Y_j\) is \(f_j(y;\tau)=\phi(y;0,\tau^2+s_j^2)\), and the probability of selection is \(P_j(S_j\mid\tau)=2\{1-\Phi(c_js_j/\sqrt{\tau^2+s_j^2})\}\). Thus the density for a selected estimate is \(f_j(y;\tau)/P_j(S_j\mid\tau)\), for values inside the selected region. Multiplying these conditional densities across independent selected loci, and maximizing over \(\tau\), gives a selection-adjusted empirical Bayes estimate. If loci have different standard errors or thresholds, each denominator must reflect its own selection rule.

For example, a normal density calculation at \(y=0.120\) with predictive variance \(0.0020\) is \(\exp\{-0.120^2/(2(0.0020))\}/\sqrt{2\pi(0.0020)}\approx0.2437\) per log-odds unit. If the selection probability under that model is \(0.0148\), the conditional density at this selected value is \(0.2437/0.0148\approx16.47\) per log-odds unit. The numerator is the unselected marginal density; dividing by the probability of selection normalizes it over the selected region.

Putting the Learned Prior Into a Locus Estimate

After estimating the prior variance, substitute it into the posterior formulas to obtain empirical Bayes summaries. Conditional on the observed estimate and a fixed fitted prior, the posterior formula is unchanged by the fact that the estimate passed a deterministic threshold: once \(Y_j\) is observed, knowing that it passed a rule based on that same \(Y_j\) supplies no additional information about \(\beta_j\). Selection adjustment is essential when learning the prior from the selected collection. This distinction does not remove uncertainty about the fitted prior or fix an incorrect selection model.

Assumptions, Diagnostics, and Alternatives

The normal likelihood is an approximation for an estimated effect and standard error; it is more credible when the underlying association analysis has adequate information and the standard error is reliable. The prior assumes effects are exchangeable around zero with a common variance. A collection containing distinct biological classes, substantial directional effects, or very different effect distributions may not be well represented by one zero-centered normal prior. The selection model must match the actual reporting rule, including the threshold and whether selection was one- or two-sided.

Diagnostics should examine whether the fitted prior reproduces the observed distribution after applying the selection rule. Compare observed and model-predicted selected effect magnitudes, check whether residual patterns vary with standard error or allele frequency, and verify that the selected-locus set and thresholds were recorded correctly. A few extreme loci can dominate a variance estimate; sensitivity analyses that omit influential loci can reveal this instability. Correlated variants also violate an independent-locus likelihood and can make the effective number of observations much smaller than the count of reported variants.

A raw discovery estimate is simple and preserves the original likelihood-based analysis, but does not use cross-locus information to temper extreme estimates. Empirical Bayes shrinkage borrows strength through a learned prior and is computationally convenient, but depends on the prior family and plug-in hyperparameter estimate. Selection-adjusted likelihood methods explicitly model the distribution among selected estimates when learning the prior. Bootstrap and resampling corrections, covered in the earlier material on selection, instead use repeated resampling to assess or correct selection effects and can be sensitive to how the resampling scheme represents discovery and selection. No approach makes an unrecorded selection rule or a misspecified effect distribution harmless.