Resolution Limits From Local LD Structure

What Does It Mean to Resolve a Variant?

Fine-mapping asks which variant, or set of variants, could explain an association signal. A strong association can coexist with poor variant-level resolution: nearby variants may have nearly the same genotype pattern across the sampled individuals. If so, the data can support the region or signal while providing little information about which of those variants is causal.

The key distinction is between uncertainty that more informative data could reduce and a resolution limit imposed by the local LD structure. Linkage disequilibrium (LD) describes correlation between genotypes at different variants. When two variants are highly correlated, their association statistics tend to be similar. That similarity is not proof that the variants have equal causal probabilities, nor that one is causal; it means the observed genetic data may not distinguish their explanations well.

Definition: Variant-level resolution is the extent to which the data distinguish one candidate causal variant from other candidates in the same signal. A local LD resolution limit occurs when the genotype patterns of candidates are identical or so similar that their likelihoods remain difficult to separate, even when the signal itself is well supported.

To make this precise, let \(C=j\) mean that variant \(j\) is causal under a specified fine-mapping model, and let \(D\) denote the observed data. Within that model, the posterior probability of candidate \(j\) is proportional to its prior probability and the likelihood of the data under that candidate. A posterior inclusion probability (PIP) is the posterior probability that a variant is included as causal under the model; when the model permits one causal variant, it is the posterior probability assigned to that candidate.

P(C=j\mid D,M) = \frac{\mathrm{BF}_j\pi_j}{\sum_{\ell}\mathrm{BF}_{\ell}\pi_{\ell}}

Here \(M\) is the assumed fine-mapping model, \(\pi_j\) is the prior probability for candidate \(j\) under that model, and \(\mathrm{BF}_j\) is its Bayes factor relative to a common null model. This expression makes an important point: a posterior allocation depends on both the likelihood evidence and the prior. When candidates have nearly equal likelihoods, prior differences can have a large influence on their posterior probabilities.

Perfect LD: A Structural Limit

Consider a quantitative-trait model with one causal variant:

y=\beta x_j+\varepsilon,\qquad \varepsilon\sim N(0,\sigma^2 I)

The vector \(y\) contains the observed outcomes, \(x_j\) is the genotype vector at candidate variant \(j\), and \(\beta\) is its effect. For this illustration, assume the genotype vectors are measured without error, the outcome errors are independent normal variables with common variance, and the candidate causal variants are compared using the same effect-size treatment and the same prior probability. These assumptions isolate the role of LD; real analyses must also account for their chosen association and fine-mapping models.

If \(x_1=x_2\), then the predicted outcomes under \(C=1\) and \(C=2\) are identical for every value of \(\beta\). The likelihoods are therefore identical for every possible observed \(y\). With equal priors, the posterior probabilities of the two variants are equal; with unequal priors, their posterior odds equal their prior odds. No increase in the number of observations that preserves the same identical genotype patterns can make the outcome data distinguish these two explanations.

Worked Example: Two Variants in Perfect LD

Suppose a hypothetical fine-mapping analysis considers two variants as alternative explanations for one signal. Their genotype vectors are identical in the study sample. The model assigns each variant prior probability \(0.5\), and each has Bayes factor 100 relative to the same null model.

The posterior probability for variant 1, conditional on these two candidates in the one-causal-variant model, is \(100(0.5)/[100(0.5)+100(0.5)]=0.5\). The same calculation gives \(0.5\) for variant 2. Both candidates can be strongly supported relative to the null, while neither is preferred over the other.

If instead the prior probabilities were \(0.8\) and \(0.2\), the posterior probabilities would remain \(0.8\) and \(0.2\), because the likelihoods are still equal. That allocation would reflect the prior distinction, not a distinction supplied by the observed genotypes and outcomes.

Near-Perfect LD and Posterior Evidence

Real candidates are often highly correlated rather than exactly identical. Their likelihoods can then differ, but the difference may be too small to support a stable variant-level conclusion. For a simple illustration using summary statistics, an approximate Bayes factor for a variant can be calculated from its association estimate \(\hat{\beta}\), standard error \(SE\), and a normal prior on its effect with mean zero and variance \(W\). Write \(V=SE^2\) and \(z=\hat{\beta}/SE\). Under the usual normal approximation and a common \(W\), the approximate Bayes factor is

\mathrm{BF} = \sqrt{\frac{V}{V+W}} \exp\left(\frac{z^2W}{2(V+W)}\right).

This formula is useful here as a comparison of candidate evidence, not as a universal fine-mapping procedure. It assumes that the summary estimate is approximately normal with variance \(V\), and that the effect prior and its variance are appropriate for both candidates. Comparing the resulting Bayes factors must also respect the LD among variants; marginal association statistics are correlated when genotypes are correlated.

Worked Example: Similar Evidence Under Near-Perfect LD

Suppose two hypothetical variants have \(SE=0.05\), with \(z_1=6.0\) and \(z_2=5.9\). Let the common prior effect variance be \(W=0.01\), and assume equal prior probabilities for the two candidates. The squared standard error is \(V=0.05^2=0.0025\), so the common leading factor is \(\sqrt{0.0025/(0.0025+0.01)}=\sqrt{0.2}=0.4472\). The coefficient multiplying \(z^2\) in the exponent is \(0.01/[2(0.0125)]=0.4\).

Thus \(\mathrm{BF}_1=0.4472\exp(0.4\times36)=0.4472\exp(14.4)\), and \(\mathrm{BF}_2=0.4472\exp(0.4\times34.81)=0.4472\exp(13.924)\). The Bayes-factor ratio is \(\mathrm{BF}_1/\mathrm{BF}_2=\exp[0.4(36-34.81)]=\exp(0.476)=1.61\).

With equal priors and only these two candidates in the comparison, the posterior probability for variant 1 is \(1.61/(1+1.61)=0.617\); variant 2 receives \(0.383\). Both \(z\)-scores indicate a strong association, but their difference gives only modest relative evidence for variant 1. If the variants are in near-perfect LD, such similar statistics are expected. These values do not establish that either variant is causal; they illustrate how a strong signal can still have limited resolution between candidates.

Why Sample Size, Effect, and LD Matter

A direct way to quantify the separability of two candidate explanations is to compare their expected data distributions. Suppose the genotype vectors are standardized to have squared length \(n\), their correlation is \(r\), and the same fixed effect \(\beta\) is assumed under either candidate. Under \(C=1\), the mean outcome is \(\beta x_1\); under \(C=2\), it is \(\beta x_2\). For normal errors with variance \(\sigma^2 I\), the Kullback–Leibler divergence between these two outcome distributions is

D_{\mathrm{KL}}(P_1\Vert P_2) = \frac{\|\beta x_1-\beta x_2\|^2}{2\sigma^2} = \frac{\beta^2\{2n-2nr\}}{2\sigma^2} = \frac{n\beta^2(1-r)}{\sigma^2}.

The first equality is the normal-distribution result for two distributions with the same covariance matrix. The second uses \(\|x_1\|^2=\|x_2\|^2=n\) and \(x_1^Tx_2=nr\). The divergence is the expected log-likelihood ratio under candidate 1 relative to candidate 2. It is not a posterior probability, a test statistic with a universal cutoff, or a guarantee about the result in a particular sample. It does show the directions in which expected discrimination improves: with larger \(n\), larger \(|\beta|/\sigma\), or lower positive correlation \(r\).

Worked Example: An Information-Based Resolution Check

Consider hypothetical standardized genotypes for \(n=1{,}000\) individuals, with \(r=0.999\) between two candidates and an effect-to-noise ratio \(\beta/\sigma=0.1\). The expected divergence is \(1{,}000(0.1)^2(1-0.999)=1{,}000(0.01)(0.001)=0.01\) nats. This is very small: under these assumptions, the expected log-likelihood separation is slight.

Keeping the sample size and effect-to-noise ratio fixed but changing the correlation to \(r=0.9\) gives \(1{,}000(0.01)(0.1)=1\) nat, a larger expected separation. Alternatively, keeping \(r=0.999\) and the effect fixed but increasing the sample size to \(100{,}000\) also gives \(100{,}000(0.01)(0.001)=1\) nat. These calculations are comparative diagnostics, not thresholds for declaring a variant resolved. They show why a small effect and very high LD may require far more data to distinguish candidates.

The derivation assumes a known, common effect under the two candidate models and a fixed error variance. In practical fine-mapping, effect sizes are generally uncertain and assigned priors, and multiple variants may contribute to a signal. The calculation should therefore be read as an explanation of the basic information limit, not as a replacement for the full model’s posterior. The same principle remains: if candidate genotypes are almost redundant, the data contain little contrast with which to choose between their explanations.

Diagnosing a Local Resolution Limit

A resolution assessment should examine both the posterior allocation and the local genotype structure. A large PIP for one variant can be informative, but it does not by itself establish that the data distinguish that variant from nearby alternatives. Check whether high-posterior candidates have nearly identical genotype patterns or high pairwise LD, and whether their evidence changes substantially under reasonable prior or LD choices. Sensitivity to LD reference mismatch and effect-size priors is relevant here; the point is to interpret those sensitivities as evidence about the stability of the allocation, not merely as a reporting detail.

Posterior allocation: Several correlated candidates with comparable posterior support indicate that the signal may be localized more reliably than the causal variant.
LD and genotype information: Near-identical genotype vectors imply little observed contrast. LD estimated from an imperfectly matched reference sample can misstate that contrast.
Signal strength versus resolution: Strong association evidence against a null does not imply strong evidence for one candidate over its correlated alternatives.
Prior sensitivity: If likelihoods are similar, moderate changes in prior weights can noticeably change posterior shares. A change in allocation should not be presented as new data evidence.
Credible sets: A credible set summarizes posterior coverage under its model, but its size or purity alone does not prove that its members are distinguishable. A small set can still contain variants with nearly identical genotype patterns.

A useful comparison is between strategies that target different sources of uncertainty. Increasing sample size can improve separation when LD is high but not perfect. Data from a population with a different LD pattern may provide a new genotype contrast, provided the populations and association estimates are analyzed appropriately. Functional experiments or annotations can add evidence about biological plausibility, but they are not a substitute for distinguishing what the association data themselves identify. Under exact genotype identity in the analyzed data, simply collecting more outcomes from the same unchanged genotype pattern cannot resolve the variants from those outcomes alone.

The main failure is to equate a numerical ranking with definitive identification. The candidate with the largest PIP is the most supported under the specified model and priors; it is not automatically proven causal. A second failure is to call posterior spreading a weak association. Posterior support spread across highly correlated candidates may coexist with compelling evidence that the locus contains a signal. The interpretation should state separately what is learned about the signal and what is learned about the particular variant.

Common Mistake and Practical Interpretation

Common mistake: “Variant A has the largest posterior probability, so the data have resolved the causal variant.” This is not justified when nearby candidates have nearly identical genotype patterns and similar likelihoods. The leading variant may rank first because of a small, sample-specific difference or its prior weight. A more careful conclusion reports the posterior allocation, describes the local LD among candidates, and states whether the evidence distinguishes the leading variant from its alternatives.

The practical goal is not to force every signal to a single-variant answer. It is to identify the scale at which the data support a conclusion: a locus, a signal, a group of correlated variants, or one variant with meaningfully stronger evidence than its alternatives. Making that distinction helps prevent a credible-set result from being interpreted more specifically than the data allow.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Two variants have identical genotypes in the analyzed sample, equal priors, and equal likelihoods. What are their posterior probabilities conditional on these two candidates being the one causal variant?
Under the approximate Bayes-factor calculation, two candidates have equal standard errors and equal priors, with \(z_1=4\), \(z_2=3\), and \(W/[2(V+W)]=0.2\). What is \(\mathrm{BF}_1/\mathrm{BF}_2\), and what is the posterior probability of candidate 1 conditional on the pair?
For standardized genotype vectors, \(n=2{,}000\), \(r=0.995\), and \(\beta/\sigma=0.2\). Calculate the expected KL divergence in the fixed-effect illustration.
Why can a strong association fail to identify one variant among several candidates in high LD?

Solutions. (1) Each posterior probability is \(0.5\): equal priors multiplied by equal likelihoods give equal posterior weights. (2) The Bayes-factor ratio is \(\exp[0.2(4^2-3^2)]=\exp(1.4)=4.055\). With equal priors and only this pair, candidate 1 has posterior probability \(4.055/(1+4.055)=0.802\). (3) The divergence is \(n(\beta/\sigma)^2(1-r)=2{,}000(0.2)^2(0.005)=0.4\) nats. This is an expected separation measure, not a posterior probability or a universal resolution cutoff. (4) A strong association shows that the observed data support a signal relative to a null, but highly correlated candidate genotypes produce similar likelihoods for alternative causal variants. The data may therefore support the signal without distinguishing which correlated variant generated it.