Matching Shrinkage to Sample Size and Polygenicity

Why Shrinkage Depends on More Than Sample Size

A GWAS weight is estimated with uncertainty. Shrinking that estimate toward zero can reduce the noise in a polygenic score, but it also pulls genuine effects toward zero. The appropriate balance depends on how precisely effects were estimated and on what kinds of effect sizes the model considers plausible. A procedure suited to a few relatively large effects need not be well suited to many tiny effects.

Polygenicity describes how widely genetic effects are spread across variants. In a sparse architecture, only a relatively small fraction of candidate variants have nonzero effects. In a highly polygenic architecture, effects occur at many variants, often with individually small magnitudes. These are descriptions of an effect-size distribution, not guarantees about which individual variants are causal.

The central question is not whether shrinkage is desirable in general. It is whether a particular shrinkage rule is calibrated to the discovery sample’s precision and to a defensible model of the effects. The calculations below first use an idealized independent-variant model to make this relationship explicit. Real score construction must account for linkage disequilibrium as appropriate; the independent model here is a teaching approximation, not a claim that GWAS variants are generally independent.

A Basic Model for Sampling Error and Shrinkage

This likelihood approximation assumes that the model for the association is appropriate, the estimate is approximately normal, and the stated standard error reflects its sampling uncertainty. In actual analyses, standard errors can differ by variant and the GWAS may estimate marginal effects in the presence of LD. The single value \(v\) is used here to isolate how sample size and effect-size assumptions affect shrinkage.

Under a zero-centered normal prior, \(\beta_j\sim N(0,\tau^2)\), the posterior mean has a simple form. The prior says effects are distributed around zero with variance \(\tau^2\); it does not say that every effect is exactly zero. The posterior mean is a linear shrinkage estimate:

When \(v\) is large relative to \(\tau^2\), the estimate is pulled strongly toward zero. When \(v\) is small relative to \(\tau^2\), the data receive more weight and \(\lambda\) is closer to one. Under the stated normal prior and likelihood, the posterior variance is \(\tau^2v/(\tau^2+v)\). It is also the expected squared error of the posterior mean when averaged over effects drawn from that prior and observations drawn from the likelihood.

Why Polygenicity Changes the Shrinkage Rule

A single normal prior is a natural description when many effects are expected to be small and distributed around zero. A sparse model instead allows a point mass at zero together with a distribution for nonzero effects. This changes how evidence is translated into weights: a sparse model can shrink weak-looking estimates heavily while retaining more of an estimate that is sufficiently compelling given the low prior probability of a nonzero effect.

For a normal slab, the assumptions are \(\beta_j=0\) with probability \(1-\pi\), and \(\beta_j\sim N(0,\tau^2)\) with probability \(\pi\). Under the normal likelihood above, the posterior mean is the posterior inclusion probability times the posterior mean conditional on inclusion. Conditional on inclusion, that mean is \(\tau^2\hat{\beta}_j/(\tau^2+v)\). The inclusion probability depends on how likely the observed estimate is under the slab relative to the point mass at zero.

The following comparison makes the role of the assumed distribution concrete. Both models assign the same marginal prior variance to an effect, but one places effects across all variants and the other concentrates them in a small fraction.

Estimation Error Across Sparse and Dense Architectures

The bias-variance trade-off can also be examined directly for a fixed set of true effects. Suppose an estimator uses \(\tilde{\beta}_j=\lambda_j\hat{\beta}_j\), with the same unbiased normal sampling model and variance \(v\). Conditional on a true effect \(\beta_j\), its expected squared error is

The first term is squared shrinkage bias: the estimate’s expected value is \(\lambda_j\beta_j\). The second is sampling variance, because multiplying the estimate by \(\lambda_j\) multiplies its variance by \(\lambda_j^2\). Differentiating the expression with respect to \(\lambda_j\), setting the derivative to zero, and solving gives the fixed-effect oracle choice:

This choice depends on the unknown true effect, so it is a benchmark rather than a practical recipe. It shows why shrinking every estimate by the same amount can be inefficient when the architecture is heterogeneous: strong effects can tolerate less shrinkage, while true null effects benefit from being set to zero. A sparse model attempts to learn which pattern is plausible from noisy observations; a highly polygenic model instead expects many small effects and typically distributes shrinkage across many variants.

Choosing Assumptions Without Treating Them as Facts

Discovery sample size affects the evidence available about each effect. With greater precision, a model can often allow less shrinkage for estimates supported by the data. But sample size alone does not determine the right architecture. If effects are in fact widespread and small, a prior that expects only a few large effects may discard useful weak signals. If effects are sparse, a model that spreads weight across many noisy estimates may retain unnecessary noise.

The prior’s scale and shape matter as well as its sparsity. A normal prior specifies a common scale for effects. A spike-and-slab prior adds a nonzero fraction and a slab scale. Continuous-shrinkage approaches use a different way to represent varying degrees of shrinkage across variants. Each model encodes assumptions; none discovers the true architecture without uncertainty. When the assumed distribution is poorly matched to the data-generating effects, the resulting weights can have more estimation error than weights from a better-matched model.

The posterior mean minimizes posterior expected squared error under the model used to compute it. To see why, for any proposed estimate \(a\),

The second term does not depend on \(a\), and the first is minimized when \(a=E[\beta_j\mid\hat{\beta}_j]\). This result is conditional on the model and the squared-error loss. It does not establish that a model’s prior assumptions are correct, nor does low estimation error for individual weights automatically imply a particular score’s predictive performance in another population.

Common Pitfall: “Larger Samples Always Need Less Shrinkage”

A common mistake is to conclude that a sufficiently large discovery sample always justifies weak shrinkage for every variant. The correction is that precision is only one part of the calculation: the effect-size distribution also matters. Under the normal model, the shrinkage factor is \(\tau^2/(\tau^2+v)\), so it depends on the prior scale as well as the sampling variance. Under a sparse model, posterior inclusion depends on the prior fraction of nonzero effects and the slab distribution as well as the observed estimate and its standard error.

A second mistake is to treat a low posterior inclusion probability as proof that a variant has no effect, or to treat a posterior mean as a directly observed biological truth. Both are model-based summaries that depend on the assumptions and data. In practice, shrinkage choices should be considered in relation to plausible architecture and discovery precision, and their consequences should be assessed using appropriate evidence rather than by declaring one architecture universally correct.