Continuous Shrinkage With PRS-CS

Why continuous shrinkage?

A marginal GWAS effect is not generally an estimate of only that variant’s joint effect: correlations among variants can make marginal estimates reflect nearby effects as well. Clumping discards some correlated variants, while a genome-wide shrinkage model retains them and uses linkage disequilibrium (LD) to estimate their effects jointly. PRS-CS does this using a continuous shrinkage prior: most effects are pulled toward zero, while the prior’s local scales allow some effects to remain comparatively large.

The result is a vector of posterior effect estimates that can be used as weights in a polygenic risk score. Those estimates depend on both the GWAS summary statistics and an LD reference. The method therefore relies on correctly aligned variants and a reference panel that represents the discovery GWAS population’s LD well enough for the model to be informative.

Summary-statistic model and prior

For a basic derivation, suppose there are \(p\) variants, the discovery-sample genotype columns have been centered and scaled to variance one, and the phenotype is centered. Let \(X\) be the \(n \times p\) genotype matrix, \(y\) the phenotype vector, and \(\beta\) the vector of joint effects. Assume the linear model \(y=X\beta+\epsilon\), with independent errors \(\epsilon\sim N(0,\sigma^2 I)\). Define the LD matrix \(R=X^\mathsf{T}X/n\) and the vector of marginal regression effects \(\hat b=X^\mathsf{T}y/n\). Conditional on \(X\), this model gives \(\hat b\mid\beta,\sigma^2\sim N(R\beta,\sigma^2 R/n)\). In applications, the LD matrix is estimated from a reference panel, and the summary-statistic likelihood is adapted to the available GWAS scales and sample sizes.

This is a continuous prior, not a rule that assigns some variants exactly zero effects. Its scale mixture gives substantial concentration near zero while retaining heavier tails than a single normal prior. Consequently, weak signals tend to be strongly shrunk, but a variant with evidence for a larger effect can receive less shrinkage through its local scale. The prior does not itself establish that any variant has a nonzero causal effect.

In the basic model, the prior scales are independent across variants before seeing data. The posterior effects are not independent: the likelihood connects them through \(R\). This distinction is central to PRS-CS. Local scales determine how readily each effect can move away from zero under the prior, while LD in the likelihood determines how the variants’ evidence is jointly interpreted.

Deriving the LD-aware posterior update

Condition on \(\phi\), all local scales, and \(\sigma^2\). For this derivation assume \(R\) is positive definite. The likelihood density for \(\hat b\) is proportional to \(\exp\{-n(\hat b-R\beta)^\mathsf{T}R^{-1}(\hat b-R\beta)/(2\sigma^2)\}\). Expanding the quadratic in \(\beta\) gives a term \(\beta^\mathsf{T}R\beta-2\beta^\mathsf{T}\hat b\). The prior contributes the quadratic term \(\beta^\mathsf{T}D^{-1}\beta\), where \(D\) is diagonal with \(D_{jj}=\phi\psi_j\). Combining them yields a multivariate normal posterior:

The derivation follows by collecting the quadratic and linear terms: the posterior precision, after factoring out \(1/\sigma^2\), is \(nR+D^{-1}\), and the corresponding linear term is \(n\hat b\). Completing the square gives the stated mean and covariance. The off-diagonal entries of \(R\) make the posterior update LD-aware: the estimate at one variant depends on the other variants’ effects as well as its own marginal association.

This is a conditional posterior distribution, given the scales. PRS-CS also infers the local scales, so the final posterior effect estimate reflects uncertainty over those scales rather than merely substituting one chosen scale into this formula. In practice, the method uses posterior computation to obtain effect estimates. The conditional update makes clear why a common global scale does not force every variant to have the same final weight.

How the global parameter changes weights

Holding a local scale \(\psi_j\) fixed, increasing \(\phi\) increases the prior variance \(\sigma^2\phi\psi_j\) and reduces prior precision. The likelihood then has more influence, so posterior effects generally move less toward zero. Decreasing \(\phi\) increases prior precision and generally shrinks effects more strongly. This is an overall tendency, not a guarantee that every coordinate’s posterior mean changes monotonically in a correlated multivariate model.

The local scale is what makes the shrinkage adaptive across variants. Even under a small global scale, a sufficiently large \(\psi_j\) can give one variant a less restrictive prior variance than its neighbours. Conversely, small local scales keep some effects strongly concentrated near zero. PRS-CS therefore differs from applying one identical fixed normal prior variance to every effect, even though \(\phi\) is shared.

Assumptions, diagnostics, and alternatives

The derivation depends on a suitable summary-statistic likelihood, aligned alleles, and an LD matrix that describes correlations in the discovery sample. The idealized version also assumes standardized genotypes, a linear phenotype model with independent Gaussian errors, and a well-conditioned positive-definite \(R\). Real GWAS may involve different phenotype scales, sample sizes, covariates, or imprecise LD estimates; their summary statistics and LD reference must be put on compatible scales for the implemented method.

Before interpreting weights, check that variant identifiers, effect alleles, genome build, and genotype scaling agree between the GWAS and LD reference. Confirm that the reference contains the variants used in the analysis and that the LD blocks are plausible. Marked discrepancies between expected and reference allele frequencies or LD patterns can indicate mismatched populations or variant alignment errors. Unstable computations or highly sensitive effects can also arise when the LD matrix is nearly singular or poorly estimated. These checks flag potential problems; a plausible-looking score alone does not validate the reference or model.

The earlier material on infinitesimal-prior shrinkage uses a common Gaussian prior scale, which gives a simple form of shrinkage. PRS-CS adds variant-specific continuous local scales, allowing a heavier-tailed prior and adaptive shrinkage. LDpred2, covered in the earlier material on genome-wide effect shrinkage, provides a different Bayesian modelling approach to joint effect estimation. These methods differ in their prior constructions and computational procedures; none removes the need for compatible summary statistics and LD information. PRS-CS also does not guarantee accurate prediction when the discovery and target populations differ or when the phenotype model is inadequate.