Sum of Single Effects Models

From One Signal to a Sum of Signals

A single-causal-variant model represents an association as one variant carrying an effect. That representation can be too restrictive when a region contains more than one signal: a variant associated with the outcome may partly reflect another signal correlated with it through linkage disequilibrium (LD). A multiple-signal model needs to represent several effects while still allowing uncertainty about which variants carry them.

A sum of single effects (SuSiE) does this by writing the total genetic effect as a sum of component effects. Each component has its own variant choice and effect size. The components are not the same thing as a list of variants that have already been identified: their selections and effect sizes are uncertain, and the observed data inform them jointly.

For clarity, suppose the outcome has been centered and the genotype columns are centered and scaled to have squared length \(n\). The model assumes independent Gaussian errors, \(\varepsilon\sim N(0,\sigma^2 I_n)\), conditional on the genotype matrix and the modeled effects. If covariates such as ancestry principal components are required, they must be included or appropriately accounted for; otherwise their effects can be mistaken for genetic associations. The Gaussian model is a working assumption, not a guarantee that every phenotype has Gaussian residuals.

Component Choices, Effect Sizes, and the Total Effect

One way to express component \(l\) is with a one-hot indicator vector \(\gamma_l\): its \(j\)th entry is 1 if that component selects variant \(j\), and all other entries are 0. If \(\theta_l\) is its effect size, then \(b_l=\gamma_l\theta_l\). A common prior assigns variant-selection probabilities \(\pi_j\), where \(\pi_j\geq 0\) and \(\sum_{j=1}^{p}\pi_j=1\), and gives the effect size a mean-zero normal prior, \(\theta_l\sim N(0,\tau^2)\). The prior choices are made before examining the association evidence; the likelihood then updates uncertainty about the component selections and effects.

In this basic formulation, each component selects one variant. Some implementations also allow an explicit null choice, so a component may contribute no effect. A component with an effect close to zero can also contribute little to the total even if it selects a variant. These details matter when interpreting \(L\): it is not automatically a statement that exactly \(L\) distinct variants are causal.

The distinction between component effects and total effects is important. If two components select the same variant, their effect sizes add in \(b\). The model therefore allows multiple components to select the same variant unless an additional constraint rules that out. In practical fine-mapping, component labels are also not inherently meaningful: relabeling components does not change the sum \(b\).

Why the Components Are Coupled by the Data

Although the model is written as a sum of component effects, the components are generally not statistically independent after observing the data. The reason is that the same outcome is explained by their combined prediction. Expanding the Gaussian likelihood makes this explicit.

The final expression contains cross-products between components. When two selected genotype columns are correlated, the corresponding cross-product need not be zero. Consequently, evidence for one component can depend on the choices and effects of the others. In the special case of orthogonal genotype columns, those cross-products between distinct variants vanish; real genomic data commonly have LD, so that simplifying case should not be assumed.

This likelihood also explains why “fit each variant separately, then add the strongest results” is not equivalent to fitting a multiple-signal model. Separate marginal associations do not account for how variants jointly explain the same outcome. A configuration model can represent combinations directly, but the number of possible configurations grows rapidly with the number of variants and signals. The sum-of-single-effects representation organizes a multiple-signal model as a collection of simple components; it does not make the likelihood independent across those components.

From Component Uncertainty to Variant Inclusion

A component’s selection probability describes the probability that it assigns its effect to a particular variant. A variant-level inclusion probability instead asks whether that variant is selected by at least one component. These are different summaries: a variant may have modest probability in several components, or substantial probability in just one.

For example, if a posterior approximation treats the component selections as independent and gives component \(l\) probability \(\alpha_{lj}\) of selecting variant \(j\), then the probability that no component selects \(j\) is \(\prod_l(1-\alpha_{lj})\). The resulting approximate inclusion probability is one minus this quantity. This product formula requires independence under the posterior being used; it is not a general identity for a dependent joint posterior.

Component-specific effect-size uncertainty is also needed to describe the total effect. If \(\alpha_{lj}\) is a component’s selection probability and \(m_{lj}\) is its posterior mean effect conditional on selecting variant \(j\), then under that component’s posterior representation its expected contribution to variant \(j\) is \(\alpha_{lj}m_{lj}\). Summing across components gives the expected total coefficient, provided the component effects are defined on the same scale. This mean is not a substitute for the full posterior distribution: components may have uncertainty and dependence that affect the total-effect interval.

LD, Diagnostics, and Limits of Interpretation

LD creates a central difficulty for multiple-signal fine-mapping. If two variants have highly correlated genotype columns, the data may have difficulty distinguishing which one a component selects. The component representation can describe this uncertainty, but it cannot create information that is absent from the observed genotypes. A sharp-looking component assignment should therefore be considered alongside the LD structure and the assumptions behind the model.

Useful warning signs include components repeatedly assigning substantial probability to the same group of highly correlated variants, unstable selections under modest changes in prior settings, and poor agreement between the genotype information used for fitting and the information used to describe LD. These signs call for investigation, not automatic rejection: correlated alternatives can be a genuine feature of the data. Conversely, a model with too few components can force several signals into one component, while too many components can yield weak or redundant allocations. A model’s component count and prior assumptions should be reported rather than treated as invisible defaults.

A sum of single effects is a modeling representation, not a guarantee of causal discovery. A causal interpretation depends on the study design and assumptions connecting genotype to outcome, including control of population structure and other sources of bias. The method also relies on an adequate variant set, an appropriate likelihood, and suitable priors. If important variants are missing, the fitted effects may absorb their associations; if phenotype or genotype data are misspecified, posterior probabilities can be misleading.

Common Mistake and Practical Comparison

• Mistake: “There are \(L\) components, so the model has identified \(L\) distinct causal variants.” Correction: \(L\) is the number of modeled components, not necessarily the number of distinct, well-supported causal variants. Components can be uncertain, weak, redundant, or—in some implementations—null.
• Mistake: Treating a component’s selection probability as the variant’s overall inclusion probability. Correction: Inclusion concerns selection by any component; combining component probabilities requires accounting for their joint posterior distribution.
• Mistake: Reading a high inclusion probability as proof of causality. Correction: It is conditional on the data, model, and priors, and does not remove bias or identify a causal variant when LD prevents distinction.

Compared with enumerating multi-variant configurations, the component representation expresses effects in a structured way that is useful when many variants are under consideration. Direct configuration models make joint variant combinations explicit but can become computationally burdensome as the number of possible combinations grows. Compared with one-at-a-time marginal association, a multiple-component model aims to represent several signals together rather than confusing correlated evidence with independent effects. The trade-off is that inference still requires handling dependence among components; the representation alone is not a fitting algorithm or a diagnostic guarantee.