Multi-Signal Colocalization With SuSiE

Why Colocalization Must Account for Multiple Signals

A genomic region can contain several association signals for a molecular trait and several for a complex trait. If each trait is analyzed as though it had only one causal variant, a strong signal in one trait can be incorrectly paired with a different signal in the other. The result may obscure a genuinely shared signal or suggest sharing where the evidence instead supports distinct causal variants.

SuSiE provides a way to represent multiple signals when fine-mapping each trait. Colocalization can then compare signals component by component, asking whether a particular signal for one trait is consistent with sharing a causal variant with a particular signal for the other. The unit of interpretation is therefore a pair of signals, not just a genomic region.

Definition: In the sum-of-single-effects model used by SuSiE, the association pattern for a trait is represented as a sum of \(L\) single-effect components. Each component represents one modeled signal, with posterior probabilities assigned to the variants that could underlie that signal. A component's credible set is a set of variants selected to contain the signal's causal variant with a stated posterior coverage, subject to the model and its assumptions.

What a SuSiE Component Represents

For an outcome vector \(y\), a simplified SuSiE regression model is

y = X\left(b_1+b_2+\cdots+b_L\right)+\varepsilon,

where \(X\) is the genotype matrix, \(b_l\) is the effect vector for component \(l\), and \(\varepsilon\) is residual variation. Each component is modeled as a single effect: it assigns its effect to one variant among those considered. The model is useful when a locus may contain multiple signals because the total association pattern can be explained by several such components rather than by one variant alone.

For each component, SuSiE reports variant-level posterior inclusion probabilities (PIPs). A PIP is the posterior probability, under the fitted model, that a variant is included as the causal variant for that component. A credible set summarizes the variants accounting for a chosen amount of that component's posterior probability. It does not say that every variant in the set is causal, nor that the set itself is a colocalization result.

The component labels are bookkeeping labels, not biological identities: “component 1” in one trait is not automatically the same signal as “component 1” in another. The comparison has to be made using the evidence for each possible component pair. The fitted model and its interpretation rely on appropriate input data, a sufficiently rich choice of \(L\), and a suitable linkage-disequilibrium (LD) representation when summary statistics are used. Poorly resolved signals, inaccurate LD, or an inadequate model can affect the component results.

Worked Example: Reading Two SuSiE Fits

Suppose a hypothetical region has two SuSiE components for an expression trait and two for a disease trait. The component credible sets are as follows:

Trait	Component	95% credible set
Expression	E1	rs101, rs102
Expression	E2	rs201, rs202
Disease	D1	rs102, rs103
Disease	D2	rs201, rs203

There are four candidate component pairs: E1–D1, E1–D2, E2–D1, and E2–D2. The fact that E1 and D1 share rs102 makes that pair worth evaluating, but it does not establish colocalization. The component-specific association evidence and the hypotheses about shared versus distinct causal variants must also be compared. Similarly, the shared rs201 makes E2–D2 a candidate pair, not a conclusion.

This locus therefore cannot be summarized reliably by asking only whether the two traits are associated in the region, or whether any credible sets overlap. The question is which signal pairs have evidence for a shared causal variant, and which are better explained by distinct variants.

Component-Level Colocalization

For a single pair of signals, colocalization compares hypotheses that the region contains no association for either trait (\(H_0\)); an association only for trait 1 (\(H_1\)); an association only for trait 2 (\(H_2\)); associations for both traits at distinct variants (\(H_3\)); or associations for both traits at the same variant (\(H_4\)). The component-level comparison asks these questions for one component from each trait at a time.

One way to express the evidence for a component pair is through variant-level Bayes factors. Let \(B_{1j}\) and \(B_{2j}\) be the Bayes factors for variant \(j\) in the two signals being compared. With priors \(p_1\) and \(p_2\) for association in each trait and \(p_{12}\) for a shared association, the unnormalized hypothesis weights are:

\begin{aligned} W_0 &= 1,\\ W_1 &= p_1\sum_j B_{1j},\\ W_2 &= p_2\sum_j B_{2j},\\ W_3 &= p_1p_2\sum_{j\ne k}B_{1j}B_{2k},\\ W_4 &= p_{12}\sum_j B_{1j}B_{2j}. \end{aligned}

Here \(H_3\) sums over pairs of distinct causal variants, whereas \(H_4\) sums over the same variant in both traits. The posterior probability of a hypothesis is its weight divided by the sum of all five weights. In a SuSiE-based analysis, the comparison is applied to the evidence for each component pair rather than treating all signals in the locus as one signal. The precise implementation can depend on the software and inputs; the interpretation remains conditional on the component-level model and the hypothesis comparison.

Worked Example: Correctly Computing Distinct-Variant Evidence

Consider a hypothetical component pair with three candidate variants and Bayes factors \(B_1=(2000,2,1)\) and \(B_2=(1500,1,1)\). Let \(p_1=p_2=0.001\) and \(p_{12}=0.00001\). We calculate the five weights to see how shared and distinct-variant evidence compare.

The sums of Bayes factors are \(\sum_j B_{1j}=2003\) and \(\sum_j B_{2j}=1502\). Thus,

W_1=0.001(2003)=2.003,\qquad W_2=0.001(1502)=1.502.

For \(H_4\), the same-variant products are \(2000(1500)\), \(2(1)\), and \(1(1)\). Therefore,

W_4=0.00001\left[2000(1500)+2(1)+1(1)\right] =0.00001(3{,}000{,}003)=30.00003.

For \(H_3\), include every cross-trait variant pairing except those with the same variant. The sum of all pairwise products is the product of the two sums; subtracting the same-variant products gives

\begin{aligned} \sum_{j\ne k}B_{1j}B_{2k} &=(2003)(1502)-3{,}000{,}003\\ &=3{,}008{,}506-3{,}000{,}003\\ &=8{,}503. \end{aligned}

Consequently,

W_3=0.001(0.001)(8{,}503)=0.008503.

With \(W_0=1\), the total weight is \(1+2.003+1.502+0.008503+30.00003=34.513533\). Hence,

P(H_4\mid\text{data}) =\frac{30.00003}{34.513533} \approx 0.8692.

For this hypothetical component pair, the posterior probability assigned to \(H_4\) is about 0.869 under the stated priors and model. This calculation does not establish that the entire locus has one shared signal: it evaluates one pair of signals. In particular, the distinct-variant sum is 8,503, not 1,510,503; the difference is obtained by correctly subtracting the same-variant products from the product of the sums.

From Component Pairs to a Locus Interpretation

When trait 1 has \(L_1\) components and trait 2 has \(L_2\), there are \(L_1L_2\) possible pairs to assess. Results are commonly inspected as a matrix, with each cell summarizing the evidence for one pair. This makes it possible to distinguish, for example, one shared signal from another signal in the same region that is trait-specific.

Worked Example: Interpreting a Component-Pair Matrix

Suppose a hypothetical analysis fits two expression components (E1 and E2) and three disease components (D1, D2, and D3). The following values are the component-pair posterior probabilities for \(H_4\) returned by the analysis:

Expression component	D1	D2	D3
E1	0.94	0.08	0.03
E2	0.06	0.91	0.12

E1–D1 and E2–D2 each have strong component-level evidence for \(H_4\) in this hypothetical output. The other displayed pairs have much lower \(H_4\) probabilities. The appropriate interpretation is that the analysis supports two candidate shared signal pairs, not that every expression and disease association in the region colocalizes.

The entries are pair-specific posterior probabilities. They should not simply be added to produce a locus-wide probability: pairs can depend on overlapping variants and on the fitted component structure, and the sum is not automatically a calibrated posterior probability for a locus-level event. A broad conclusion should describe the supported pairs and their model-based uncertainty, while noting any weakly resolved components.

Derivation: Why the \(H_3\) Sum Excludes the Diagonal

Under \(H_3\), the two traits have causal variants, but those variants must be different. Start with all possible cross-trait variant pairings. Their total Bayes-factor product is

\sum_j\sum_k B_{1j}B_{2k} =\left(\sum_j B_{1j}\right)\left(\sum_k B_{2k}\right).

This includes both pairs with different variants and pairs where \(j=k\). The latter are precisely the same-variant pairings that contribute to \(H_4\), so subtracting them gives

\sum_{j\ne k}B_{1j}B_{2k} = \left(\sum_j B_{1j}\right)\left(\sum_k B_{2k}\right) -\sum_j B_{1j}B_{2j}.

This identity is useful both computationally and as a check: \(H_3\)'s unweighted sum cannot exceed the product of the two total Bayes-factor sums, and the difference must equal the sum of the excluded same-variant products. It also shows why careful arithmetic matters when one variant has very large evidence.

Assumptions, Interpretation, and Common Mistakes

Component-level colocalization is only as reliable as its inputs and fitted models. The traits must refer to a comparable genomic region and the same set of consistently aligned variants; each SuSiE fit must represent the relevant association signals adequately. For summary-statistic fine-mapping, the LD information must appropriately represent the study data. If a signal is split across components, poorly resolved, or omitted because too few components were allowed, component-pair results can be misleading or incomplete.

A high \(P(H_4\mid\text{data})\) is evidence, under the priors and model, that the particular pair of signals is consistent with a shared causal variant rather than distinct variants. It is not proof of a causal mechanism, and it does not show that the molecular trait mediates the effect on the complex trait. Colocalization concerns shared genetic association evidence; a causal pathway requires additional assumptions and evidence.

Mistake: treating any overlap between credible sets as colocalization. Overlap identifies a candidate component pair, but the posterior comparison must weigh same-variant evidence against distinct-variant evidence and the other hypotheses.
Mistake: matching components by their labels. Component numbers are not shared identifiers across traits. Compare all relevant component pairs using their evidence.
Mistake: collapsing a multi-signal locus to one association per trait. This can pair evidence from different signals and conceal signal-specific results. Fine-map and compare the components instead.
Mistake: summing pairwise \(H_4\) probabilities as a locus-wide probability. Pairwise results are not automatically mutually exclusive or calibrated for that aggregation. Report the supported pairs and their individual uncertainty.

Key takeaway: SuSiE represents multiple association signals as components, and multi-signal colocalization evaluates pairs of components. Interpret each \(H_4\) result as evidence for a shared causal variant for that signal pair, conditional on the fitted models and priors—not as an automatic conclusion about the whole locus.

Check Your Understanding

Answer each question using the component-level interpretation developed above.

A trait has three SuSiE components and another trait has two. How many component pairs are available for colocalization assessment?
Why does a shared variant in two component credible sets not, by itself, establish colocalization?
In the worked Bayes-factor example, explain why the \(H_3\) sum subtracts same-variant products from the product of the two sums.
If one component pair has a high posterior probability for \(H_4\), what does that support, and what does it not establish about the molecular trait's role?
Why is it generally inappropriate to add all pairwise \(H_4\) posterior probabilities and call the result the probability that the locus colocalizes?