Sum of Single Effects Models for Fine-Mapping

Why Represent a Locus as a Sum of Effects?

A locus can contain more than one association signal. When several variants have correlated genotypes, their marginal association statistics can reflect overlapping signals, so a model that allows only one causal variant may not represent the evidence adequately. The sum of single effects (SuSiE) model instead represents the locus-wide genetic effect as a sum of components. Each component can assign its effect to at most one variant, while different components can account for different signals.

This construction links two useful ideas. A component is sparse because it is supported on no more than one variant. The overall model can nevertheless be multi-signal because it adds several such components. The number of components, \(L\), is specified for a fit; it is not itself a claim that exactly \(L\) variants are causal. A component may have substantial posterior probability on the null option, and multiple components may put probability on the same variant.

Model and Component-Level Posterior Weights

For a simple individual-level formulation, let \(y\) be the vector of a quantitative phenotype, \(X\) the matrix of genotypes for \(p\) candidate variants, and \(b\) their effect vector. After accounting for any specified nuisance covariates, write

The Gaussian residual model assumes that, conditional on the genetic effects and covariates, residuals have mean zero, common variance \(\sigma^2\), and no residual correlation. Genotype columns must use a consistent allele coding, and the phenotype and genotypes must be aligned across individuals. A summary-statistic likelihood can also be used when it accounts for linkage disequilibrium; the component idea is the same.

For component \(l\), let \(\gamma_l\) denote its assignment. It takes values \(0,1,\ldots,p\), where \(0\) is the null choice and \(j\) assigns the component to variant \(j\). Let \(\pi_{lj}\) be the prior probability of assignment to variant \(j\), and let \(\pi_{l0}\) be the prior probability of the null choice, with \(\pi_{l0}+\sum_{j=1}^{p}\pi_{lj}=1\). If \(\gamma_l=j\), the component effect is drawn from a zero-centred normal prior with variance \(V_l\); if \(\gamma_l=0\), its effect is zero. Components are assigned independently in the prior under this basic specification.

The posterior calculation for one component can be understood by holding the other component effects fixed. Define the partial residual \(r_l=y-\sum_{k\ne l}Xb_k\). The likelihood contribution for candidate assignment \(j\) is then based on \(r_l=X_j b_{lj}+\varepsilon\), where \(X_j\) is genotype column \(j\). Integrating over the normal effect prior gives a Bayes factor, \(\mathrm{BF}_{lj}\), comparing assignment to variant \(j\) with the null assignment. Its posterior assignment weight is proportional to prior weight times Bayes factor:

These are component-level posterior probabilities: for a given component, the probabilities across the null option and all variants sum to one. A large \(\alpha_{lj}\) means that, within that component’s current comparison, variant \(j\) has relatively strong posterior support. It is not yet the locus-wide inclusion probability for \(j\), because another component could also assign to \(j\).

From Components to Variant Inclusion Probabilities

In the fitted model, a variant is included if at least one component assigns to it. If the posterior distribution over component assignments were available jointly, the locus-wide inclusion probability would be the posterior probability of the union of those events. In general, component assignments can be dependent in the posterior because they explain the same data. Consequently, multiplying component non-inclusion probabilities is not an exact identity for an arbitrary joint posterior.

SuSiE uses a structured variational approximation in which the posterior over components is represented as a product of component-specific distributions. Write \(\alpha_{lj}\) for the resulting approximate posterior probability that component \(l\) assigns to variant \(j\). Under this product approximation, assignments across components are independent in the approximating distribution. The approximate probability that no component assigns to variant \(j\) is therefore the product of their non-assignment probabilities. Taking its complement gives the variant inclusion probability:

This is an inclusion probability for the union of component assignments, not a sum of component probabilities. A sum can exceed one if the same variant receives probability from several components. The product formula stays between zero and one and accounts for the possibility that more than one component assigns to the same variant.

Effect Estimates and Coordinate Updates

The model also gives component-specific effect distributions, not only assignment probabilities. In the Gaussian example, conditional on assigning component \(l\) to variant \(j\), the posterior effect is normal. Let \(S_{jj}=X_j^\mathsf{T}X_j\) and \(S_{jr}=X_j^\mathsf{T}r_l\). With prior variance \(V_l\) and residual variance \(\sigma^2\), its posterior mean and variance are

These are conditional on the component being assigned to \(j\). Averaging over the component’s assignment probabilities gives its posterior mean effect vector: the entry for variant \(j\) is \(\alpha_{lj}\mu_{lj}\), and the null assignment contributes zero. Summing these component means gives the fitted posterior mean for the total effect vector. The effect of a genotype column is measured per unit of that column; for allele dosage coded from 0 to 2, this is per additional copy of the coded allele.

In practice, component updates are coupled: changing one component changes the partial residuals used for the others. A coordinate-ascent fitting scheme repeatedly updates a component’s assignment probabilities and conditional effect distributions while holding the other component distributions fixed, then cycles through the components. The resulting component-level probabilities and PIPs are approximate posterior summaries. This procedure is a computational strategy for the specified model; it does not turn the variational approximation into an exact joint posterior.

What the Model Allows—and a Common Pitfall

The sparse-component representation is useful because it can describe several signals without enumerating every multi-variant causal configuration directly. It also gives component-level summaries that can help distinguish signals in the presence of linkage disequilibrium. However, the model’s results depend on choices such as the candidate variants, effect-size prior, prior assignment probabilities, residual model, and fixed number \(L\) of components. The same variant can be selected by multiple components, so component count and the count of distinct selected variants need not match.

A common mistake is to read \(L\) as the known number of causal variants, or to treat the component probabilities as if they were automatically independent under the exact posterior. The correction is to distinguish the model’s fixed component capacity from the evidence for non-null assignments, and to identify the product formula as applying to the product-form variational approximation. Uncertainty about how many components should be included in the model is a separate modeling issue.

Finally, a high PIP is posterior support for inclusion under the specified model, not proof of biological causation. Fine-mapping evidence can prioritize variants for follow-up, but causal interpretation still depends on assumptions and evidence beyond this representation.