Colocalizing Multiple Molecular Traits

Why Colocalize Several Molecular Traits?

A genetic region may be associated with more than one molecular phenotype: for example, gene expression, transcript splicing, and protein abundance. Asking whether each phenotype has an association is not the same as asking whether the associations are driven by the same causal variant. Multi-trait colocalization addresses the latter question across several phenotypes at once.

The joint view matters because patterns can differ. One variant might be shared by all measured traits, while another pattern might involve a shared variant for expression and splicing but a different variant for protein abundance. A third possibility is that each trait has a distinct causal variant. Reporting only a collection of pairwise results can obscure these distinctions.

Configurations and Joint Posterior Evidence

Let \(D\) denote the genetic association data for the selected traits in a region. A joint analysis compares configurations \(C\) using their prior probabilities and how well each configuration explains the data. In a Bayesian analysis, the posterior probability of a configuration is proportional to its prior probability multiplied by its likelihood:

Here \(P(C)\) is the prior probability assigned to configuration \(C\), and \(P(D\mid C)\) is the likelihood of the observed data under that configuration. A Bayes factor compares the likelihood under one configuration with the likelihood under a reference configuration. The posterior probability for a broader pattern—such as “at least two traits share a variant”—is found by summing the posterior probabilities of all configurations that satisfy that pattern.

The model must state what constitutes a signal. A simple illustration may assume that each trait has one causal variant in the region, conditional on having an association to explain. This assumption makes the possible sharing patterns easier to display, but can be inadequate when a trait has multiple independent signals. In that setting, the analysis needs to represent multiple signals rather than treating all association evidence as if it arose from one variant. Earlier material on multi-signal colocalization addresses that issue.

A joint likelihood must also reflect the data structure. If the traits were measured in overlapping samples, their association estimates may be correlated. Treating them as independent when they are not can misrepresent the evidence. The precise likelihood and prior structure depend on the method; posterior probabilities are interpretable only in relation to those assumptions and the traits included in the analysis.

Why Pairwise Results Do Not Determine the Joint Pattern

For three traits, a pairwise analysis addresses one pair at a time. Even if each pair has appreciable posterior support for sharing, the evidence need not imply that one variant is shared by all three. Pairwise probabilities are marginal summaries: each sums over configurations involving the trait or traits not in that pair.

For example, E and S can share one variant in some configurations, while E and P share a different variant in others. Those possibilities contribute to the respective pairwise probabilities but not to the probability that all three share one variant. Thus a joint configuration analysis is needed when the scientific claim concerns an all-trait or subset-sharing pattern.

Linkage Disequilibrium and Variant-Level Ambiguity

Linkage disequilibrium (LD) means that alleles at nearby variants are correlated. As a result, association data may not distinguish a causal variant from correlated variants that tag it. A fine-mapping analysis can distribute posterior probability across several variants in LD. Two traits may therefore have overlapping credible sets—sets of variants containing substantial posterior mass—without the evidence establishing that the same variant drives both traits.

A simple overlap calculation can help illustrate the issue, but it is not a substitute for a joint colocalization model. Suppose each trait has a posterior distribution over candidate causal variants, conditional on the model used to obtain that distribution. If one independently draws a candidate variant from each distribution, the probability of drawing the same variant is the sum of the products of their variant-specific probabilities. This calculation describes matching under those independent draws; by itself, it is not the posterior probability of colocalization.

Trait Selection Changes the Question

The set of molecular traits in a joint analysis is part of the scientific question, not a neutral formatting choice. An analysis of expression, splicing, and protein abundance asks whether those particular phenotypes share variants in the region. Adding a trait introduces additional possible configurations, changes which sharing patterns can be summarized, and may change the prior probability assigned to each pattern.

Selection also matters when traits or pairs are chosen after screening many results. A posterior calculated for a selected pair is conditional on the model and data used for that pair; it does not automatically account for having searched many pairs and reported only the most compelling one. When the goal is to make a set of discovery claims across many tested traits or pairs, the selection and multiplicity process should be addressed. Posterior probabilities and frequentist multiple-testing adjustments are different quantities: one should not be substituted for the other without a clearly defined inferential procedure.

A Short Derivation: Summing Configuration Probabilities

A joint model assigns posterior probability to mutually exclusive configurations. To obtain the posterior probability of a pattern, add the probabilities of all configurations that satisfy it. This follows by partitioning the event of interest into disjoint configuration events.

Here \(\mathcal{S}\) is the set of configurations containing at least one shared pair. The sum is valid because any observed configuration belongs to at most one member of the mutually exclusive set. For three traits, this event includes the all-shared configuration and each configuration in which exactly one pair shares. It does not include the all-distinct configuration. This posterior is different from the posterior probability that all traits share one variant, which sums only the all-shared configurations.

Interpreting the Result Carefully

Multi-trait colocalization is useful for describing how molecular association signals align across phenotypes. A strong all-shared posterior supports a shared-variant pattern under the specified model; a strong subset-sharing posterior points to a more limited pattern. Neither conclusion, on its own, establishes that one molecular phenotype causes another or that the shared variant acts through a particular biological pathway.

Interpretation should name the traits and the pattern explicitly. “The traits colocalize” can be ambiguous: it may mean all traits share one variant, that at least one pair shares, or merely that their credible sets overlap. State which posterior event was evaluated, how many traits were selected, and whether the analysis allowed multiple signals. Where traits have correlated measurements or overlapping samples, explain how their dependence was handled. These details determine what the reported evidence supports.

Common Mistakes

• Inferring an all-trait pattern from pairwise results. Pairwise sharing probabilities are marginal summaries and can be high even when the all-shared configuration has much lower probability. Use the joint configuration probability for an all-trait claim.
• Treating credible-set overlap as colocalization. LD can spread probability across correlated variants. Overlap indicates shared candidate variants, not necessarily a shared causal variant.
• Ignoring which traits were selected. A posterior for a chosen set of traits does not automatically account for screening many traits and reporting only a selected result. Define the target set and address selection when making claims across a search.
• Reading a posterior as certainty or as a causal effect. A posterior is conditional on the data, model, and priors. Evidence for a shared variant is not by itself evidence that one molecular trait causes another.