Colocalization With Multiple Causal Signals

Why Colocalization Needs a Multiple-Signal Model

A genomic region can contain more than one association signal for a trait. For example, a GWAS and an expression quantitative trait locus (eQTL) analysis might each identify two signals in the same region. One GWAS signal could share a causal variant with one eQTL signal, while the other signals are distinct. A method that treats each entire region as having at most one causal variant cannot represent that arrangement.

The earlier material on Bayesian colocalization introduced the five hypotheses for a pair of traits, including the distinction between distinct causal variants and one shared causal variant. Those hypotheses remain useful, but with multiple signals they must be applied at the signal-pair level. A whole-locus analysis under a single-causal-variant assumption may blend evidence from different signals, obscuring a genuine shared pair or making distinct signals appear to compete with one another.

SuSiE-coloc is a multi-signal strategy: it uses fine-mapping results that represent each trait’s association as a set of signals, then compares signal pairs across traits. It does not assume that signal 1 for one trait must correspond to signal 1 for the other. Instead, it considers the possible pairs and asks which, if any, show evidence of a shared causal variant.

How Fine-Mapping Represents Multiple Signals

A simplified SuSiE model writes the genetic contribution to a trait as a sum of \(L\) components. If \(X\) is a matrix of variant genotypes and \(y\) is a vector of trait measurements, the model can be written as:

Each component \(b_{\ell}\) is modeled as having at most one nonzero variant effect, or being null. The sum allows the trait to have multiple signals in the region. \(L\) is an upper bound on the number of components, not a claim that every component is a true signal. Fine-mapping estimates, for each component, how its association evidence is distributed across variants. These probabilities are often summarized as a credible set: a set of variants intended to contain the causal variant for that component at a specified coverage level.

For a valid comparison, the two traits’ data must refer to the same genomic variants, genome build, and allele coding. The fine-mapping results must also be based on appropriate linkage disequilibrium (LD) information, and the model must allow enough components to represent the signals present. If the LD reference is poorly matched to the study population, or the component limit is too small, the resulting signal assignments and variant probabilities may be misleading.

After fitting a multi-signal model separately to each trait, compare every relevant signal from the first trait with every relevant signal from the second. For each pair, evaluate evidence for a shared causal variant against evidence for distinct causal variants. The pairwise results describe local relationships between signals. They do not, by themselves, establish that all signals at the locus are shared, or that a shared genetic signal proves a causal effect of expression on the phenotype.

What Same-Variant Evidence Measures

A useful way to understand a signal-pair comparison is to start with variant-level evidence. Suppose the first signal has Bayes factors \(BF_{1i}\) across variants \(i=1,\ldots,m\), and the second has \(BF_{2i}\). A Bayes factor measures how well the data at a variant support an association relative to a null model. Under a shared-variant hypothesis, the same variant index contributes evidence for both traits. Under a distinct-variant hypothesis, different indices contribute.

With per-variant priors \(p_{12}\) for a shared association, \(p_1\) for association with trait 1, and \(p_2\) for association with trait 2, the evidence contributions for the shared and distinct hypotheses are proportional to:

The sum in \(W_4\) uses the same variant index for both traits. The sum in \(W_3\) uses different indices. These are contributions to the shared-signal and distinct-signal hypotheses; a complete colocalization analysis also accounts for the other hypotheses and their prior probabilities. In particular, \(W_4/(W_3+W_4)\) is a conditional comparison of these two hypotheses only, not the full posterior probability of colocalization.

There is also a simple normalized interpretation. If \(q_{1i}\) and \(q_{2i}\) are posterior probabilities for the causal variant of each signal, then, under independent trait data and the corresponding one-causal-variant-per-signal setup, the posterior probability that both signals use the same variant is:

This follows by adding the probabilities of the mutually exclusive events “both signals use variant \(i\)” over all variants. It is a useful overlap summary, but it is not automatically the full SuSiE-coloc posterior: the latter’s hypothesis probabilities also depend on the specified priors and the full model. Treat the overlap as a way to understand the alignment of two signal distributions, not as a replacement for the method’s posterior calculation.

Comparing All Signal Pairs

The central advantage of a multi-signal approach is that it can identify a shared pair without forcing every association in the region into one explanation. Consider hypothetical variant-level weights for two signals from each trait. The weights are proportional to evidence within each signal; dividing each same-variant sum by the product of the corresponding weight totals gives the normalized overlap summary.

Assumptions, Diagnostics, and Failure Modes

Signal-pair interpretation depends on several assumptions. The variants must be harmonized so that the same index refers to the same genomic position and effect allele in both traits. Each trait’s fine-mapping model must adequately represent the number and structure of its signals. The LD information used in fine-mapping must be appropriate, and the priors must be plausible for the analysis. The Bayesian coloc comparison also relies on its specified evidence and prior model; changing priors can change posterior probabilities, especially when evidence is modest.

Check the component-level results before interpreting colocalization. Inspect whether each signal has a reasonably informative credible set, whether its posterior is concentrated or diffuse, and whether signal estimates change substantially under reasonable model settings. A broad credible set may produce an uncertain match even when the signals overlap. A missing or weak eQTL signal is not evidence that the GWAS signal is distinct; it may simply leave little information for comparison. LD mismatch, allele errors, or too few allowed components can also produce misleading signal assignments.

A common mistake is to run a single-causal-variant colocalization analysis on a region with multiple signals and interpret the resulting posterior as a complete locus-wide answer. That reasoning forces the evidence for several effects into one pair of hypotheses. Instead, fine-map multiple signals and compare the signal pairs; then describe which pairs have evidence for sharing, which appear distinct, and where uncertainty remains. Do not add pairwise posterior probabilities as if they were independent, or report the largest pairwise value as the probability that the entire locus colocalizes.

The main alternative is a single-signal colocalization model, which is simpler and can be suitable when each region genuinely contains only one well-resolved signal. Conditional analyses and other fine-mapping-based comparisons are also used to address multiple signals, but they rely on different modeling choices. SuSiE-coloc’s appeal is that it carries the multiple-component fine-mapping representation into signal-pair comparisons. Its limitations remain important: it depends on fine-mapping quality, model and prior choices, and correct alignment; pairwise results can be difficult to summarize when many components are weak or overlapping. Colocalization indicates shared genetic evidence, not proof that one trait mediates the effect on the other.