The Five Hypotheses In Bayesian Colocalization

Why Five Hypotheses?

Colocalization asks whether the association signal for one trait and the association signal for another trait in a genomic region are compatible with the same causal variant. For example, one signal might come from a genome-wide association study (GWAS) of a disease and the other from an expression quantitative trait locus (eQTL) study of gene expression. The two studies can show association in the same region without the same variant causing both signals.

The standard five-hypothesis model makes that distinction explicit. It treats a region as having at most one causal variant for each trait. The possible configurations are then: neither trait has a causal variant in the region; only the first trait does; only the second does; both do but their causal variants differ; or both do and share one causal variant. These are model states, not conclusions inferred just by looking at the lead SNPs.

Definition: Let trait 1 and trait 2 be the two traits, and let \(Q\) be the set of variants considered in the region. In the standard single-signal Bayesian colocalization model, \(H0\) means neither trait has a causal variant in \(Q\); \(H1\) means only trait 1 has a causal variant; \(H2\) means only trait 2 has a causal variant; \(H3\) means both traits have causal variants, at distinct variants in \(Q\); and \(H4\) means both traits have a causal variant at the same variant in \(Q\).

The labels \(H1\) and \(H2\) depend on which trait is designated trait 1 or trait 2. Swapping the trait labels swaps \(H1\) and \(H2\), but does not change the meanings of \(H0\), \(H3\), or \(H4\). State which data source corresponds to each trait when reporting the hypotheses.

Configurations, Not Lead-SNP Labels

A causal configuration specifies which variant, if any, is causal for each trait. For example, if the region contains three variants \(a\), \(b\), and \(c\), a configuration with trait 1 causal at \(a\) and trait 2 causal at \(b\) belongs to \(H3\). A configuration with both traits causal at \(a\) belongs to \(H4\). A shared lead SNP is not by itself enough to establish the second configuration: linkage disequilibrium (LD) can make several nearby variants show similar association patterns.

The hypotheses are mutually exclusive because a configuration cannot simultaneously have no causal variant and have one, or have the same causal variant for both traits and distinct variants for them. They are exhaustive under the model because each trait is assigned either no causal variant or one causal variant in \(Q\). This is a consequence of the model's at-most-one-causal-variant assumption, not a claim that real genomic regions always have this structure.

Worked Example: Classifying Causal Configurations

Suppose a hypothetical region contains variants \(a\), \(b\), and \(c\). List the hypothesis for each proposed configuration.

Neither trait has a causal variant in the region. This is \(H0\), regardless of whether the observed association statistics happen to have small p-values due to chance or other sources of error.

Trait 1 is causal at \(b\), and trait 2 has no causal variant in the region. Only trait 1 has a causal variant, so this is \(H1\).

Trait 1 is causal at \(a\), and trait 2 is causal at \(c\). Both traits have a causal variant, and \(a\ne c\), so this is \(H3\).

Both traits are causal at \(b\). The causal variant is shared, so this is \(H4\). The key distinction from the preceding case is the causal configuration, not whether the two observed lead SNPs happen to match.

Assumptions Behind the Five-State Model

The hypothesis labels are meaningful only in relation to the model used to define them. The standard formulation assumes a specified region and a set \(Q\) of variants that can be compared across the two traits. It also assumes at most one causal variant per trait in that region. Evidence for each variant is usually summarized by a Bayes factor: the likelihood of the observed association data under a causal-variant model relative to the likelihood under no association for that variant. The two trait-specific sets of evidence are combined across configurations.

For a basic interpretation, the analysis also needs the trait definitions, variant identities, and genomic region to be sufficiently aligned. If a variant is absent or mismatched between data sets, a truly shared causal variant may not be represented as a shared candidate. The conventional calculations further use specified prior probabilities for configurations; the hypotheses themselves do not determine those prior choices.

These assumptions do not require the two traits to have the same effect size or direction at a shared variant. \(H4\) says that the modeled causal variant is the same; it does not say that the effects are equal. Nor does \(H4\) establish that one trait mediates the effect on the other.

Model condition: The five hypotheses form an exhaustive partition only for the specified region and the model allowing zero or one causal variant for each trait. If a trait has multiple causal variants in the region, the true configuration may not be represented correctly by this five-state model.

How Variant Evidence Maps to the Hypotheses

The configuration-based definition can be expressed using variant-level evidence. Let \(BF_{1i}\) and \(BF_{2i}\) denote the Bayes factors for variant \(i\) for trait 1 and trait 2, respectively. In a common single-signal formulation, \(p_1\) and \(p_2\) are per-variant prior probabilities of a trait-specific association, and \(p_{12}\) is the per-variant prior probability of a shared association. The unnormalized evidence weights have the form

\begin{aligned} W_0 &= 1,\\ W_1 &= p_1\sum_{i\in Q} BF_{1i},\\ W_2 &= p_2\sum_{i\in Q} BF_{2i},\\ W_3 &= p_1p_2\sum_{\substack{i,j\in Q\\i\ne j}} BF_{1i}BF_{2j},\\ W_4 &= p_{12}\sum_{i\in Q} BF_{1i}BF_{2i}. \end{aligned}

The index restriction \(i\ne j\) in \(W_3\) counts configurations in which the two traits have distinct causal variants. The matching index in \(W_4\) counts configurations in which both trait-specific evidence contributions refer to the same variant. Thus the sums are not merely comparisons of the strongest single-variant Bayes factors; they account for the configurations allowed under each hypothesis. The weights shown are not posterior probabilities: converting weights to posterior probabilities requires normalization across hypotheses, using the specified model and priors.

Worked Example: Same-Variant and Distinct-Variant Contributions

Consider three hypothetical variants, \(a\), \(b\), and \(c\), with Bayes factors for trait 1 of \(4\), \(1\), and \(0.5\), and for trait 2 of \(0.5\), \(3\), and \(1\), respectively. For illustration, take \(p_1=0.01\), \(p_2=0.01\), and \(p_{12}=0.001\). Compute the unnormalized evidence weights for \(H1\) through \(H4\).

For \(H1\), the sum of trait 1 Bayes factors is \(4+1+0.5=5.5\). Therefore \(W_1=0.01(5.5)=0.055\). For \(H2\), the sum for trait 2 is \(0.5+3+1=4.5\), so \(W_2=0.01(4.5)=0.045\).

For \(H4\), multiply the two trait Bayes factors at each same variant and sum: \(4(0.5)+1(3)+0.5(1)=2+3+0.5=5.5\). Thus \(W_4=0.001(5.5)=0.0055\).

For \(H3\), include all ordered pairs of distinct variants. The sum of all cross-trait products is \((4+1+0.5)(0.5+3+1)=5.5(4.5)=24.75\). The same-variant products sum to \(5.5\), so the distinct-variant sum is \(24.75-5.5=19.25\). Hence \(W_3=(0.01)(0.01)(19.25)=0.001925\). Finally \(W_0=1\) by the reference scaling. These are evidence weights, not probabilities; the calculation illustrates which configurations each hypothesis includes.

Worked Interpretation and Checks

Worked Example: Interpreting a Hypothesis Statement

A hypothetical analysis compares a disease GWAS as trait 1 with an eQTL study of a specified gene as trait 2. An analyst states that “the data support \(H3\) because both traits have strong associations in the region.” What does \(H3\) assert, and what can be concluded from that statement alone?

By definition, \(H3\) asserts that both traits have a causal variant in the region but that the two causal variants are distinct. It does not assert a shared causal variant; that is \(H4\). The presence of strong associations for both traits is not sufficient by itself to identify \(H3\), because both traits could instead share a variant, have evidence arising from LD with other variants, or have more than one signal in the region.

The analyst would need to compare the evidence for the permitted configurations under the model, including the \(H3\) and \(H4\) configurations, and assess whether the single-signal assumptions are reasonable. Even if the model favors \(H3\), the interpretation is limited to distinct causal variants under that model, rather than a general claim that the two traits are biologically unrelated.

Worked Example: Nonsignificance Is Not the Definition of H0

In a hypothetical region, neither study reports a variant with a p-value below its chosen threshold. A reader concludes that the region is \(H0\). Explain the error.

\(H0\) is the model state in which neither trait has a causal variant in the specified region. A p-value threshold is a decision rule for observed test statistics, not a definition of the underlying causal configuration. Weak effects, limited sample size, or noisy measurements could leave a study without a threshold-crossing association even when a causal variant is present. Conversely, a nominal association can occur when the model state is \(H0\).

The appropriate conclusion from the stated facts is only that neither study crossed the specified threshold. Determining support for \(H0\) requires comparing the model evidence for \(H0\) with the other allowed configurations; failure to detect an association is not proof of no causal variant.

Why the Distinctions Matter

The practical value of the five hypotheses is that they prevent “both traits associate in the same region” from being confused with “the same variant affects both traits.” \(H1\) and \(H2\) describe one-trait-only configurations, \(H3\) allows two distinct causal variants, and only \(H4\) represents a shared causal variant within the modeled region. \(H0\) represents neither trait having a causal variant there, not merely a lack of statistically significant results.

Several diagnostics help identify when the model may be an imperfect description. Examine whether the region contains evidence of multiple association signals, whether the candidate variants are aligned across studies, and whether LD patterns or variant coverage differ in ways that could obscure the causal configuration. A strong sensitivity of the inference to region boundaries or variant inclusion is also a warning that the stated hypotheses may not capture the relevant signal cleanly.

The main alternative is not a different label for \(H4\), but a model that allows multiple causal variants or uses conditional or fine-mapping-based evidence to represent them. Such approaches can be more appropriate when the single-signal assumption is implausible, but they require a different configuration space. A five-hypothesis result should therefore be described as conditional on the single-signal model, not as a definitive statement about every causal variant in the region.

Common mistake: “Both lead SNPs are the same, so \(H4\) is established.” A lead SNP is the strongest observed association in a study, not necessarily the causal variant. LD can make distinct causal variants produce similar lead signals.
Correction: \(H4\) concerns a shared causal variant in the model’s configuration space. Assess the evidence across same-variant configurations relative to the alternatives.
Common mistake: “Neither association is significant, so \(H0\) is true.” Nonsignificance is not proof that a causal variant is absent.
Correction: Treat \(H0\) as one explicit model hypothesis and distinguish it from the observed outcome of a significance test.

Key takeaway: \(H0\)–\(H4\) classify causal-variant configurations under a specified region and a model allowing at most one causal variant per trait. \(H3\) means two distinct causal variants; \(H4\) means a shared one. Neither label is determined simply by comparing lead SNPs.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

Trait 1 has a causal variant at \(a\), while trait 2 has none in the region. Which hypothesis applies?
Both traits have causal variants, one at \(a\) and one at \(b\), where \(a\ne b\). Which hypothesis applies?
Why is the absence of a p-value below a threshold not enough to establish \(H0\)?
Under the standard five-hypothesis model, what assumption makes the five configurations exhaustive?
What does \(H4\) establish, and what does it not establish about mediation?

Solutions. (1) \(H1\), because only trait 1 has a causal variant. (2) \(H3\), because both traits have causal variants at distinct variants. (3) A significance threshold describes observed test results; weak or imprecise evidence can fail to cross it even when a causal variant is present. (4) Each trait has at most one causal variant in the specified region, and each is either absent or present at a variant in the candidate set. (5) \(H4\) represents the same modeled causal variant for both traits; by itself it does not prove that one trait mediates the effect on the other.