The H0–H4 Hypotheses in Colocalization

Why Colocalization Uses Five Hypotheses

Suppose a genome-wide association study reports a signal for a clinical trait near a gene, and a molecular-QTL analysis reports an expression association for that gene in the same region. The overlap is suggestive, but it does not by itself tell us whether the same genetic variant underlies both associations. Nearby variants can be correlated through linkage disequilibrium, and two distinct causal variants can therefore produce association signals in overlapping regions.

Colocalization makes this distinction explicit by describing possible configurations of causal variants for two traits. The five hypotheses, \(H_0\) through \(H_4\), range from neither trait having a causal association in the region to both traits sharing one causal variant. They are statements about the underlying variant configuration, not conclusions that follow merely from seeing one or two small association \(p\)-values.

Definition: Consider two traits measured or analyzed in a genomic region: trait 1 and trait 2. Let \(C_1\) be the causal variant for trait 1 in that region, and \(C_2\) the causal variant for trait 2. Write \(C_j=\varnothing\) if trait \(j\) has no causal variant in the region. Under the usual five-hypothesis, single-signal setup, each trait has either no causal variant in the region or one causal variant there. The hypotheses are:

\(H_0\): \(C_1=\varnothing\) and \(C_2=\varnothing\) (neither trait is associated in the region).
\(H_1\): \(C_1\ne\varnothing\) and \(C_2=\varnothing\) (only trait 1 is associated).
\(H_2\): \(C_1=\varnothing\) and \(C_2\ne\varnothing\) (only trait 2 is associated).
\(H_3\): \(C_1\ne\varnothing\), \(C_2\ne\varnothing\), and \(C_1\ne C_2\) (both are associated, with distinct causal variants).
\(H_4\): \(C_1=C_2\ne\varnothing\) (both are associated and share one causal variant).

What the Hypotheses Mean in a Molecular-QTL Analysis

For a concrete interpretation, let trait 1 be a disease phenotype from a GWAS and trait 2 be the expression level of a particular gene measured in a relevant tissue. Then \(H_1\) describes a regional disease association without a regional expression association for that gene, whereas \(H_2\) describes a regional expression association without a regional disease association. Under \(H_3\), each trait has a causal variant in the region, but they are different variants. Under \(H_4\), the same variant is causal for both the disease association and the expression association.

The labels “associated” in these definitions refer to the hypothesized causal configuration, not a simple rule such as “the trait’s lead-variant \(p\)-value is below a threshold.” In real data, causal variants are not directly observed as such; association statistics across variants provide evidence about which configuration is more compatible with the data. The hypotheses define the alternatives to be compared. Calculating their posterior probabilities, and deciding how to summarize that evidence, requires additional steps beyond defining the hypotheses.

These five possibilities are mutually exclusive and exhaustive within the stated setup. A trait cannot simultaneously have no causal variant in the region and one causal variant there; if both traits have a causal variant, the variants are either the same or distinct. This exhaustive classification depends on the single-causal-variant assumption for each trait in the region. If either trait has multiple independent causal signals, this five-way setup does not describe the full configuration without further modeling.

Worked Example: Only the Disease Trait Has a Regional Association

Consider a hypothetical study of disease risk and gene expression in a relevant tissue. Assume the analysis region and the single-causal-variant setup have been specified. Suppose the causal-configuration assessment is that a variant in the region affects disease risk, while there is no causal expression association for the gene in that region. Which hypothesis describes this configuration?

Write the two states as \(C_1\ne\varnothing\) for the disease trait and \(C_2=\varnothing\) for expression. Comparing them with the definition,

(C_1\ne\varnothing,\ C_2=\varnothing)\ \longrightarrow\ H_1.

Thus the configuration is \(H_1\), not \(H_4\): the disease association has no shared expression-associated causal variant in the region under this configuration. The conclusion is not that gene expression cannot influence disease by any route; it is only a classification of these two traits’ causal-variant states in this region.

Why Both Signals Do Not Automatically Mean Sharing

The central distinction is between “both traits are associated” and “both traits share a causal variant.” The first condition holds under either \(H_3\) or \(H_4\). The second holds only under \(H_4\). Consequently, two convincing regional association signals are not, by themselves, enough to conclude that a molecular trait links the genetic signal to the clinical trait.

Likewise, the variants with the smallest \(p\)-values for the two traits—the lead variants—do not necessarily identify their causal variants. A lead variant can be a proxy for a causal variant because nearby variants are correlated. The two traits may even have different lead variants while sharing a causal variant, or have the same lead variant while their causal variants are distinct. A lead-variant comparison is therefore not a substitute for evaluating the possible causal configurations across the region.

Worked Example: Both Traits Are Associated, but Their Causal Variants Differ

In a hypothetical region, suppose a disease association is caused by variant \(v_a\), while expression of a nearby gene is caused by a different variant \(v_b\). Both variants lie in the region, and \(v_a\ne v_b\). Classify the configuration.

The disease trait has a causal variant, so \(C_1\ne\varnothing\). The expression trait also has a causal variant, so \(C_2\ne\varnothing\). Since the specified variants differ, \(C_1\ne C_2\). Substitution into the hypothesis definition gives

(C_1\ne\varnothing,\ C_2\ne\varnothing,\ C_1\ne C_2)\ \longrightarrow\ H_3.

The answer is \(H_3\): both traits have regional causal associations, but those associations are not driven by the same causal variant. Even if the variants are close together or correlated, they remain distinct causal variants under the stated configuration. This hypothesis does not say that the gene is irrelevant to disease biology; it says that the two associations do not share a causal variant in this region.

A Short Derivation: Partitioning the Causal States

The five hypotheses can be derived by enumerating the possible states of \(C_1\) and \(C_2\). First consider whether each trait has a causal variant in the region. If neither does, the configuration is \(H_0\). If exactly one does, the configuration is \(H_1\) or \(H_2\), depending on which trait. If both do, compare the variant identities: equality gives \(H_4\), and inequality gives \(H_3\).

\begin{array}{c|c|c} \text{Trait 1 state} & \text{Trait 2 state} & \text{Hypothesis}\\ C_1=\varnothing & C_2=\varnothing & H_0\\ C_1\ne\varnothing & C_2=\varnothing & H_1\\ C_1=\varnothing & C_2\ne\varnothing & H_2\\ C_1\ne\varnothing & C_2\ne\varnothing,\ C_1\ne C_2 & H_3\\ C_1=C_2\ne\varnothing & & H_4 \end{array}

This partition is complete under the assumption that each trait has at most one causal variant in the region: there are four presence-or-absence combinations, and the “both present” combination separates into same-variant and different-variant cases. There is no overlap between the rows because their conditions cannot hold simultaneously. This is a classification of causal configurations; it does not establish which row is supported by observed data.

Worked Example: Both Traits Share a Causal Variant

Suppose a hypothetical GWAS of an inflammatory phenotype and an eQTL analysis for gene \(G\) in a specified tissue are consistent with the same regional variant, \(v_c\), being causal for both traits. Write the causal states as \(C_1=v_c\) and \(C_2=v_c\). What hypothesis describes this state?

Both states are non-null because \(v_c\) lies in the region, and they are equal:

C_1=v_c,\qquad C_2=v_c,\qquad C_1=C_2\ne\varnothing.

The definition therefore assigns this configuration to \(H_4\). In words, the disease and gene-expression associations share a causal variant in the region. This supports shared genetic origin of the two associations, but does not, on its own, show that altered expression of gene \(G\) mediates the variant’s effect on the inflammatory phenotype. The variant could affect both traits through other biological routes.

Assumptions and Scope of the Five-Way Classification

Interpreting the five hypotheses requires clarity about the region, the traits, and the model. The region must be specified: a variant outside it is not counted as a causal variant for this comparison. The two association analyses must refer to the intended traits and be sufficiently comparable in genomic location and variant identity for a shared-variant claim to be meaningful. The causal-variant statements also assume that each trait has at most one causal variant in the region. If the region contains multiple independent signals for either trait, treating the entire region as a single signal can blur distinct configurations.

The hypotheses do not encode the direction or size of an effect. For example, \(H_4\) says the same variant is causal for both traits; it does not require that the variant increase both expression and disease risk, nor does it state how large either effect is. Nor does \(H_4\) specify a causal pathway from expression to disease. These are separate scientific questions.

Key takeaway: \(H_3\) and \(H_4\) both allow associations for both traits. Their defining difference is whether the causal variants are distinct (\(H_3\)) or shared (\(H_4\)). Colocalization concerns that shared-variant question, not proof of molecular mediation.

Common Mistakes

Equating two association signals with colocalization. Wrong reasoning: “Both traits have small \(p\)-values in the region, so they share a causal variant.” Correction: both signals are compatible with \(H_3\) as well as \(H_4\); the shared-versus-distinct causal-variant question must be evaluated.
Reading different lead variants as proof of \(H_3\). Wrong reasoning: “The lead variants differ, so the causal variants differ.” Correction: lead variants can be correlated proxies; distinct lead variants do not by themselves establish distinct causal variants.
Interpreting \(H_4\) as mediation. Wrong reasoning: “A shared causal variant proves the expression change causes the disease.” Correction: \(H_4\) specifies a shared genetic variant, not the pathway connecting the traits.
Ignoring the single-signal assumption. Wrong reasoning: “The five hypotheses fully describe any number of signals in a region.” Correction: their usual interpretation assumes at most one causal variant per trait in the region. Multiple signals require a model that represents them.

Check Your Understanding

For each item, identify the hypothesis or explain the distinction being tested.

In a hypothetical region, neither a disease trait nor expression of a candidate gene has a causal variant. Which hypothesis applies?
Expression has a regional causal variant, while the disease trait has none. Which hypothesis applies?
Both traits have causal variants, but the variants are different. Which hypothesis applies, and how does it differ from \(H_4\)?
Why is it not sufficient to observe small association \(p\)-values for both traits to conclude they share a causal variant?
What does \(H_4\) establish about a disease and an eQTL association, and what does it not establish about mediation?