When the Single-Causal-Variant Assumption Fails

Why the Single-Signal Assumption Matters

Colocalization asks whether genetic association signals for two traits in a region are consistent with a shared causal variant. In this setting, the traits might be a disease phenotype and gene expression, or two molecular phenotypes. A single-signal analysis makes a strong simplifying assumption: each trait has at most one causal variant in the region. When that assumption is false, the analysis must represent several signals using a model designed for one, and its posterior probabilities can be difficult to interpret.

The problem is not simply that multiple associations exist. Nearby variants can be correlated through linkage disequilibrium (LD), so association statistics may reflect both causal effects and the correlation structure among variants. A single-signal method can then mistake overlapping association patterns for evidence of a shared cause, or fail to represent a shared signal alongside a second, distinct signal. The posterior remains conditional on the model and priors that were supplied; it does not automatically diagnose a violated assumption.

Definition: In single-signal colocalization, the model assumes that each trait has either no causal variant or exactly one causal variant among the variants in the region. The hypotheses are \(H_0\): neither trait is associated; \(H_1\): only trait 1 is associated; \(H_2\): only trait 2 is associated; \(H_3\): both traits are associated, with distinct causal variants; and \(H_4\): both traits are associated with the same causal variant. These hypotheses describe the region under the single-signal model, not every possible pattern of multiple causal effects.

A posterior probability such as \(P(H_4\mid D)\) is the probability assigned to \(H_4\), conditional on the observed data \(D\), the model, and the priors. If the true region contains several causal variants, \(H_0\)–\(H_4\) do not include the full truth. The posterior then allocates support among the available simplified explanations. A large \(P(H_4\mid D)\) is not, by itself, evidence that the one-causal-variant-per-trait assumption is satisfied.

How the Single-Signal Weights Represent a Region

For a region containing \(m\) candidate variants, let \(BF_{1i}\) and \(BF_{2i}\) denote the association Bayes factors for traits 1 and 2 at variant \(i\). A Bayes factor compares the evidence under association at a variant with the evidence under no association. Under the usual single-signal construction, let \(p_1\) and \(p_2\) be per-variant prior probabilities of association for the two traits, and let \(p_{12}\) be the prior probability that a variant is causal for both. The unnormalized hypothesis weights are:

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

The distinction between the last two sums is central. \(W_3\) pairs evidence at different variants, whereas \(W_4\) pairs evidence at the same variant. Posterior probabilities are obtained by dividing each weight by \(W_0+W_1+W_2+W_3+W_4\). This construction assumes that the relevant causal configuration for each trait has at most one variant. If a trait has two causal variants, its evidence cannot generally be represented as a choice of just one index \(i\).

Worked Examples: Reading Single-Signal Posterior Weights

Worked Example: Moderate H4 Support From Hypothetical Weights

Suppose a hypothetical single-signal analysis of two traits returns the following unnormalized weights: \(W_0=1\), \(W_1=0.1301\), \(W_2=0.1301\), \(W_3=0.016006\), and \(W_4=0.92\). These are illustrative weights, not results from a real study. Find the posterior probability for \(H_4\).

First, add all five weights to obtain the normalizing total:

$$ 1+0.1301+0.1301+0.016006+0.92=2.196206. $$

Then divide the \(H_4\) weight by that total:

P(H_4\mid D)=\frac{0.92}{2.196206}\approx 0.4189.

The single-signal model assigns about 41.89% posterior probability to its shared-variant hypothesis. This is not a 41.89% probability that the null is false, nor does it establish that the region has exactly one causal variant for each trait. It is the normalized support for \(H_4\) among these five modelled configurations, given the specified weights.

Worked Example: High H4 Support Despite Distinct Causal Variants

Consider two candidate variants, A and B, and hypothetical marginal Bayes factors \(BF_{1A}=1000\), \(BF_{1B}=10\), \(BF_{2A}=1000\), and \(BF_{2B}=10\). Let \(p_1=p_2=0.01\) and \(p_{12}=0.0001\). Suppose the underlying hypothetical truth is that trait 1 is caused by A and trait 2 by B, but LD makes the single-variant association evidence appear strongest at A for both traits. Calculate the single-signal posterior weights and explain the limitation.

For trait 1 and trait 2 alone, respectively:

W_1=0.01(1000+10)=10.1,\qquad W_2=0.01(1000+10)=10.1.

For \(H_3\), the distinct-variant pairings are A for trait 1 with B for trait 2, and B for trait 1 with A for trait 2:

$$ W_3=0.0001[(1000)(10)+(10)(1000)]=2. $$

For \(H_4\), the same-variant pairings are A with A and B with B:

$$ W_4=0.0001[(1000)(1000)+(10)(10)]=100.01. $$

Including \(W_0=1\), the total is \(1+10.1+10.1+2+100.01=123.21\). Therefore:

P(H_4\mid D)=\frac{100.01}{123.21}\approx 0.8117.

The single-signal analysis gives about 81.17% posterior probability to \(H_4\), even though the hypothetical causal variants differ. The example illustrates how similar, LD-influenced association evidence can favor a shared-variant explanation in a restricted model. It does not show that every high \(H_4\) posterior is wrong; it shows that the posterior alone cannot verify the single-signal assumption or distinguish this hypothetical truth from a genuinely shared cause.

Worked Example: A Shared Signal Plus a Secondary Signal

Now suppose trait 1 has a primary causal variant A and a second causal variant B, while trait 2 has a causal variant at A. Use the hypothetical Bayes factors \(BF_{1A}=1000\), \(BF_{1B}=200\), \(BF_{2A}=1000\), and \(BF_{2B}=100\), with \(p_1=p_2=0.01\) and \(p_{12}=0.0001\). Calculate the single-signal posterior for \(H_4\).

The single-trait weights are:

W_1=0.01(1000+200)=12,\qquad W_2=0.01(1000+100)=11.

The distinct-variant weight is:

$$ W_3=0.0001[(1000)(100)+(200)(1000)]=30. $$

The same-variant weight is:

$$ W_4=0.0001[(1000)(1000)+(200)(100)]=102. $$

The total, including \(W_0=1\), is \(1+12+11+30+102=156\). Thus:

P(H_4\mid D)=\frac{102}{156}\approx 0.6538.

The single-signal model assigns about 65.38% posterior probability to \(H_4\). In the hypothetical truth, there is a shared causal variant A, but trait 1 also has a secondary causal variant B. Thus \(H_4\) is incomplete as a description of the whole region: it represents a single shared signal and has no term for the additional trait 1 signal. A high posterior for \(H_4\) can be compatible with one shared signal while still failing to describe all causal effects.

Why Multiple Signals Distort Posterior Support

The single-signal weights sum evidence over candidate variants, but they do not fit a model in which each trait can have several causal variants jointly. This matters in at least two ways. First, if distinct causal variants are correlated, both traits may show strong marginal evidence at the same candidate variant. The \(H_4\) sum rewards matching indices, so the shared-variant hypothesis can receive substantial support even when the underlying causal variants differ. Second, if one or both traits have secondary signals, the analysis must compress those signals into a one-variant configuration. It may favor the strongest shared-looking signal and leave the additional effect unrepresented, or distribute support among \(H_3\), \(H_4\), and single-trait hypotheses in ways that depend on the evidence and priors.

A short way to see the structural limitation is to compare the modelled configurations with a two-signal truth. Under \(H_4\), the model selects one index \(i\) for both traits. Under a truth in which trait 1 has causal indices \(i\) and \(k\), with \(i\ne k\), no single choice of \(i\) represents both effects. The analysis can still assign posterior mass to \(H_4\), but that mass answers a restricted question: which one-signal configuration best accounts for the summaries under the specified model? It does not estimate the probability of the full multi-signal configuration.

Changing prior probabilities can alter the posterior allocation among hypotheses, as discussed in the earlier material on calibrating colocalization priors. Prior calibration is important, but it cannot repair a structural mismatch: a well-calibrated prior does not add missing multi-causal configurations to a single-signal model.

Interpreting Results and Avoiding Overclaims

When reading a single-signal result, keep the claim proportional to what was fitted. A high \(P(H_4\mid D)\) supports a shared-variant explanation within the single-signal model. It does not prove that the variant is causal in a mechanistic sense, that the locus has only one signal, or that a molecular trait mediates a disease association. Conversely, a lower \(P(H_4\mid D)\) does not establish that the traits have no shared causal signal: multiple signals, LD, or limited power can complicate the allocation among the five hypotheses.

A useful diagnostic question is whether each trait plausibly has only one association signal across the region. Evidence of multiple peaks, conditional associations, or distinct fine-mapping patterns can warn that the assumption is questionable. Those observations do not themselves settle the colocalization question; they indicate that a single-signal posterior should be interpreted cautiously. The next analytical step is to use a method that can represent multiple signals, rather than treating a single-signal posterior as a complete regional summary.

Common mistake: “The posterior probability of \(H_4\) is high, so the region contains exactly one causal variant shared by both traits.” The correction is that \(P(H_4\mid D)\) is conditional on a model that assumes at most one causal variant per trait. A high value supports the shared-variant configuration among the configurations that model includes; it neither checks the assumption nor rules out secondary signals.

A Practical Interpretation Sequence

Identify the fitted assumption.
Confirm whether the analysis allows one causal variant per trait or can represent multiple signals.

Read the posterior conditionally.
Interpret \(P(H_4\mid D)\) as support for the shared single-variant hypothesis under its priors and model, not as an unconditional probability of the biological truth.

Look for signs of additional signals.
Consider whether regional association patterns are consistent with one signal for each trait, while remembering that LD can make marginal patterns difficult to interpret.

Limit the conclusion.
Report the evidence as single-signal colocalization support and avoid claiming that the full locus has only one shared cause unless the analysis addresses that possibility.

Key takeaway: A single-signal colocalization posterior is only as comprehensive as its hypothesis space. Multiple causal variants can make distinct signals look shared, hide secondary effects behind a dominant shared signal, or otherwise redistribute posterior support among the single-signal hypotheses.

Check Your Understanding

Answer each question by referring to the assumptions and interpretations of the single-signal model.

What does \(H_4\) represent, and what additional assumption about each trait is built into the usual single-signal analysis?
Why can LD make two distinct causal variants appear to support a shared-variant hypothesis?
If \(P(H_4\mid D)=0.90\), what can be concluded within the model, and what cannot be concluded about secondary signals?
In the shared-primary-plus-secondary example, why is a high \(H_4\) posterior not a complete description of the region?
Why can changing priors not, by itself, solve a mismatch caused by multiple causal variants?