Fine-Mapping With an Omitted Causal Variant

Why an Incomplete Candidate Set Matters

Fine-mapping compares candidate causal explanations for an observed association signal. Its posterior probabilities depend not only on the data, but also on the candidate variants and causal configurations that the model permits. If the true causal variant is absent, the model cannot assign it posterior probability. Instead, it distributes support among the available explanations, which may include correlated variants that track the missing one through linkage disequilibrium (LD).

That redistribution can produce a credible set that reaches its stated posterior threshold using only modeled variants. The set may be internally coherent under the fitted model and still exclude the true causal variant. The key is to read the result as conditional on the candidate list and model, not as proof that the causal variant must be one of the listed variants.

How LD Redirects Evidence

Consider a simple setting with one causal variant \(c\), a quantitative outcome, and standardized genotypes. Suppose the outcome follows \(y=\beta x_c+\varepsilon\), where \(x_c\) is the standardized genotype at \(c\), \(\beta\) is its effect, and the errors have mean zero and are independent of the genotypes. For another standardized genotype \(x_j\), define \(r_{jc}=\operatorname{Corr}(x_j,x_c)\). The marginal association of \(x_j\) with the outcome is proportional to \( \operatorname{Cov}(x_j,y)=\beta\operatorname{Cov}(x_j,x_c)=\beta r_{jc}\). Thus a candidate in LD with the omitted causal variant can show association even if it has no direct effect in the data-generating process.

This calculation explains why a correlated candidate can receive substantial support; it does not by itself determine the posterior. Posterior support also depends on the likelihood used by the fine-mapping method, the prior on variants or configurations, and the LD information supplied to the analysis. The candidate that best approximates the missing signal need not be the one with the largest pairwise correlation alone.

Posterior Probabilities and Credible Sets Under Restriction

For a single-causal-variant model with candidate variants \(1,\ldots,m\), let \(H_j\) denote the hypothesis that candidate \(j\) is causal. Given data \(D\), the posterior probability is \(P(H_j\mid D)=P(D\mid H_j)P(H_j)/\sum_{k=1}^{m}P(D\mid H_k)P(H_k)\). Because the hypotheses in this particular model are mutually exclusive and exhaustive within the candidate list, these posterior probabilities sum to one. If the true variant is not among the candidates, the model nevertheless normalizes across the listed hypotheses unless an alternative state is included. The resulting probabilities describe the relative support among those hypotheses, not the probability that the list contains the truth.

In this single-causal setting, a credible set at level \(q\) can be formed by ordering candidates by posterior probability and taking the smallest set whose probabilities sum to at least \(q\). For a multi-causal model, posterior inclusion probabilities (PIPs) for different variants are not generally mutually exclusive and need not sum to one. Their sum is therefore not, in general, the posterior probability that a set contains a causal variant. A credible set for a particular signal should use the relevant signal-specific posterior distribution or, where appropriate, posterior probabilities over causal configurations.

A Short Derivation: Why a Tag Variant Can Look Causal

The LD mechanism can be made explicit for marginal association. Under the outcome model above, with standardized \(x_j\) and \(x_c\), the population regression coefficient from regressing \(y\) on \(x_j\) alone is \(\operatorname{Cov}(x_j,y)/\operatorname{Var}(x_j)\). Since \(\operatorname{Var}(x_j)=1\), and \(y=\beta x_c+\varepsilon\), we have \(\operatorname{Cov}(x_j,y)=\beta\operatorname{Cov}(x_j,x_c)+\operatorname{Cov}(x_j,\varepsilon)\). If genotype and error are independent, the second covariance is zero; standardization gives \(\operatorname{Cov}(x_j,x_c)=r_{jc}\). Therefore the marginal coefficient is \(\beta r_{jc}\). A tag variant with nonzero LD can consequently have a nonzero marginal association even though its direct effect is zero.

This is a population-level justification, not a guarantee that a particular sample will rank candidates in the same order. Sampling variation, allele coding, imputation quality, and the model’s joint use of LD and association data all matter. In particular, this derivation does not justify treating several correlated marginal association statistics as independent evidence.

Diagnosing a Misleading Credible Set

A restricted model can look reassuring: its posterior probabilities are normalized, and its credible set reaches the chosen threshold. The central diagnostic question is therefore not whether the arithmetic is internally consistent, but whether the candidate list and model plausibly represent the causal explanations that generated the data.

• Check variant availability. Review whether variants in the associated region were excluded by genotyping or imputation filters, quality control, allele-frequency thresholds, or data harmonization. Confirm genomic build, position, reference and alternate alleles, and allele coding.
• Check the LD reference. The LD matrix should represent the relevant population and align with the analyzed variants. Missing, misaligned, or poorly estimated LD can distort which candidates appear to explain the signal.
• Inspect the association pattern. A strong regional signal with no plausible candidate, or a poor match between observed statistics and model expectations, can motivate scrutiny of the variant list and LD inputs. This pattern is a clue, not proof that a causal variant was omitted.
• Vary the candidate set where scientifically justified. If adding a plausible variant substantially changes posterior support, the original result was sensitive to the restricted list. Such a change is informative about the model’s dependence on its candidate universe, not automatic confirmation that the added variant is causal.

No single diagnostic establishes that an omitted variant exists. A poorly fitting model could also reflect an incorrect LD reference, multiple causal effects, data errors, or other model misspecification. The appropriate conclusion is that the result may be conditional on an inadequate representation, followed by targeted checks of the inputs and assumptions.

Modeling Alternatives and Limitations

The most direct response to a suspected omission is to verify the variant data and, when justified, rerun the analysis with the relevant candidate added and the LD information updated. If the identity of a missing causal variant is unknown, one can also compare the restricted model with a model that includes an explicit unmodeled or residual-signal state. Such a state must have a defined likelihood and prior; adding a label without specifying how it explains the data does not repair the model.

These approaches answer different questions. Adding a candidate asks how the posterior changes when that variant is a permitted explanation. An unmodeled state asks whether the data support an explanation outside the listed candidates, according to the state’s specified model. Neither approach guarantees correct inference: a broad residual state may be hard to distinguish from other forms of misspecification, while adding candidates with poor data or LD can introduce further errors. A restricted analysis remains useful for describing relative support among its candidates, provided that conditional scope is stated clearly.