Posterior Odds For Shared Versus Distinct Signals

From Variant-Level Evidence to Regional Odds

Colocalization asks whether genetic association with two traits in a region is consistent with one shared causal variant or with separate causal variants. The evidence does not come from comparing only the two strongest association statistics: several candidate variants may each be compatible with the data, particularly when their evidence is similar. In the single-signal model, the posterior odds compare the total evidence for all shared-variant configurations with the total evidence for all distinct-variant configurations.

The earlier material on the five hypotheses in Bayesian colocalization defines \(H_0\) as neither trait associated in the region, \(H_1\) as trait 1 only, \(H_2\) as trait 2 only, \(H_3\) as both traits associated through distinct causal variants, and \(H_4\) as both associated through the same causal variant. Here the focus is the odds \(H_4\) versus \(H_3\), not the probability of \(H_4\) against every alternative.

The model assumes at most one causal variant for each trait in the region. Thus \(H_3\) assigns one causal variant \(i\) to trait 1 and a different variant \(j\) to trait 2, while \(H_4\) assigns the same variant \(i\) to both. For the standard calculation, the candidate variants are the same for both traits, the variant-level prior weights are equal within each hypothesis, and the two traits' evidence can be factored into trait-specific Bayes factors conditional on a configuration. The Bayes factors must be computed under an appropriate association model; the odds formula does not repair misspecified likelihoods or poorly matched inputs.

Deriving the Posterior Odds

For a particular configuration, Bayesian updating multiplies its prior weight by its Bayes factor. Since each hypothesis contains several possible variant configurations, its regional evidence is the sum of the configuration-specific evidence. Under \(H_4\), each possible shared variant \(i\) contributes \(B_{1i}B_{2i}\). Under \(H_3\), each ordered pair of distinct variants \(i,j\) contributes \(B_{1i}B_{2j}\). The order matters because \(i\) is assigned to trait 1 and \(j\) to trait 2.

Let \(p_1\) and \(p_2\) denote the per-variant prior weights for association with trait 1 and trait 2, and let \(p_{12}\) denote the per-variant prior weight for a shared causal variant. In the common rare-variant prior parameterization, the unnormalized evidence weights for these two hypotheses are:

These weights yield the posterior odds:

Here \(D\) denotes the data from both traits. The prior odds compare the weight assigned to a shared configuration with the weight assigned to distinct configurations; the evidence ratio compares how well the data support those configurations. The expressions are the standard unnormalized weights used in this parameterization. If variant-specific prior weights are used instead of equal weights, each configuration's Bayes-factor product must be multiplied by its corresponding prior weight before summing.

A short derivation makes clear why the sums, rather than just the best variant pair, appear. If \(C\) indexes the configurations within a hypothesis, Bayes' rule gives \(P(H\mid D)\) proportional to \(\sum_{C\in H}P(C)P(D\mid C)\). Relative to the null likelihood for both traits, a shared configuration at \(i\) has likelihood ratio \(B_{1i}B_{2i}\), and a distinct configuration \((i,j)\) has likelihood ratio \(B_{1i}B_{2j}\). Substituting the respective per-configuration prior weights \(p_{12}\) and \(p_1p_2\), then taking the ratio of the two sums, gives the displayed odds. This also shows why every allowed configuration must be counted once and why \(i=j\) is excluded from \(H_3\).

Reading the Evidence Ratio

The shared evidence sum is large when the same variants receive substantial Bayes factors for both traits. The distinct evidence sum is large when strong evidence can be paired across different variants. This is a regional comparison: many modestly supported configurations can collectively outweigh one striking configuration, and the total also depends on how many candidate configurations are available.

A posterior odds ratio is not the same as a posterior probability. If only \(H_3\) and \(H_4\) are being compared, their conditional probability is obtained by converting the odds: \(P(H_4\mid D,H_3\text{ or }H_4)=O/(1+O)\), where \(O=P(H_4\mid D)/P(H_3\mid D)\). But the overall \(P(H_4\mid D)\) also depends on the evidence for \(H_0\), \(H_1\), and \(H_2\).

Assumptions, Diagnostics, and Alternatives

The calculation relies on the single-signal assumption: each trait has at most one causal variant in the region. If either trait has multiple causal signals, a configuration with one variant per trait does not represent the relevant possibilities, and the resulting odds can be misleading. It also relies on using comparable candidate variants and valid trait-specific Bayes factors. Missing candidate variants, inconsistent variant representation, or poorly calibrated association evidence can alter the sums. Conditional factorization of the two traits' likelihood evidence is another assumption; dependence between the two data sources can make a product of separate Bayes factors inappropriate.

Useful checks include confirming that the \(H_3\) sum uses all ordered pairs with \(i\ne j\), checking the magnitudes of the shared and distinct sums separately, and verifying that the reported posterior odds equal prior odds times the evidence ratio. Repeat calculations with reasonable alternative prior weights when the conclusion is sensitive to \(p_{12}/(p_1p_2)\). A large posterior odds ratio does not diagnose whether the single-signal assumption or the evidence inputs are adequate.

If \(M=1\), there are no distinct-variant configurations, so the \(H_3\) sum is empty and equals zero. The pairwise odds are then undefined or formally infinite when \(W_4>0\); this does not establish colocalization from data, because the candidate set cannot represent distinct signals. More generally, a zero or numerically negligible denominator should prompt a check of the candidate set and the model, rather than an unqualified claim of certainty.

The main alternative summary is the posterior probability of \(H_4\) among all five hypotheses, which answers a broader question than the pairwise odds. Comparing \(H_4\) only with \(H_3\) isolates shared-versus-distinct evidence when both traits are associated, but it leaves out the possibility that one or both traits have no association in the region. Likewise, selecting the top variant for each trait and comparing those two variants discards the accumulated evidence across configurations. The regional sum is the appropriate comparison under the stated model.

Common Pitfall

For transparent reporting, state which hypotheses are being compared, give the prior odds and evidence ratio when feasible, and identify whether the reported quantity is pairwise posterior odds, a conditional probability, or the overall posterior probability of \(H_4\).