Worked Example With Near-Perfect LD

Association Evidence Is Not the Same as Causal Resolution

When two candidate variants are in very high linkage disequilibrium (LD), their genotypes carry nearly the same information about an association signal. Both can therefore have very strong association statistics while the data provide little basis for deciding which one is causal. Fine-mapping expresses this uncertainty through posterior probabilities and credible sets: the set of candidates may be small, yet its members can remain difficult to distinguish.

The calculations below use a single-causal-variant model and approximate Bayes factors (ABFs) from marginal z-statistics. The results are conditional on the candidate variants, the model, the effect-size prior, and the stated prior probabilities. They are not proof that either candidate is causal.

Definition: A posterior inclusion probability (PIP) for variant \(j\) is the posterior probability that variant \(j\) is causal under the specified model. A credible set at level \(1-\alpha\) is a set of variants whose cumulative posterior probability reaches or exceeds \(1-\alpha\), after ranking variants from highest to lowest posterior probability. Its interpretation is conditional on the model, candidate set, and priors used in the analysis.

Model and Bayes-Factor Calculation

Suppose a region contains two candidate variants, and the model assumes that exactly one of them is causal. Let \(H_j\) denote the hypothesis that variant \(j\) is causal, and let \(\pi_j=P(H_j)\) be its prior probability. For a variant with z-statistic \(z_j\), assume the approximate effect estimate has standard error \(s_j\), so its sampling variance is \(V_j=s_j^2\). The effect-size prior under \(H_j\) is a mean-zero normal distribution with variance \(W\). The ABF compares the marginal likelihood under that prior to the likelihood under a zero effect for that variant:

\operatorname{ABF}_j = \sqrt{\frac{V_j}{V_j+W}} \exp\left\{ \frac{z_j^2 W}{2(V_j+W)} \right\}.

This expression uses a normal approximation to the effect estimate and a normal prior on its effect. It treats each candidate's marginal summary statistic as the input to its candidate-specific Bayes factor. In a summary-statistic fine-mapping analysis, the LD structure and the model used to compare candidate causal variants also matter; the ABFs here are used in the specified single-causal comparison, not multiplied together as if the variants were independent observations.

Given prior probabilities \(\pi_j\), posterior probabilities among these hypotheses are proportional to prior probability times Bayes factor. If a null configuration is also allowed, assign it prior probability \(\pi_0\) and Bayes factor \(1\), which sets the evidence scale relative to the null.

P(H_j\mid \text{data}) = \frac{\pi_j\operatorname{ABF}_j} {\pi_0+\sum_k \pi_k\operatorname{ABF}_k}.

For a posterior conditional on there being one causal variant among the candidates, the null is omitted and the denominator is \(\sum_k \pi_k\operatorname{ABF}_k\). A credible set is then formed by accumulating these conditional posterior probabilities in decreasing order until the target level is reached.

Worked Example: Nearly Equal Statistics, Nearly Equal Posterior Probabilities

Consider hypothetical candidates A and B with LD correlation \(r=0.999\). Their z-statistics are \(z_A=10.00\) and \(z_B=9.98\). For both, take \(V=1\) in the standardized effect scale and prior effect variance \(W=0.25\). Assume equal prior probabilities conditional on exactly one of the two being causal.

First calculate the common factor and exponent coefficient:

\sqrt{\frac{1}{1.25}}=0.8944, \qquad \frac{0.25}{2(1.25)}=0.1000.

Thus \(\operatorname{ABF}_j=0.8944\exp(0.1000z_j^2)\). For A,

\log(\operatorname{ABF}_A) = \frac{1}{2}\log(1/1.25)+0.1000(10.00)^2 = -0.1116+10.0000 = 9.8884, $$ $$ \operatorname{ABF}_A\approx \exp(9.8884)\approx 19{,}700.

For B, \(9.98^2=99.6004\), so

\log(\operatorname{ABF}_B) = -0.1116+0.1000(99.6004) = 9.8484, \qquad \operatorname{ABF}_B\approx 18{,}930.

With equal priors, the conditional posterior probability for A is

P(H_A\mid \text{data}, H_A\text{ or }H_B) = \frac{19{,}700}{19{,}700+18{,}930} \approx 0.5100.

The corresponding probability for B is about \(0.4900\). A's evidence is strong relative to the null under this effect prior, but it is only slightly stronger than B's. The 95% credible set contains both: the first variant contributes about \(0.5100\), below 0.95, and adding the second brings the cumulative probability to 1.000. The result is strong locus-level evidence with poor resolution between these two candidates.

Why the Normalization Gives the Posterior

The posterior calculation follows directly from Bayes' rule. Under hypothesis \(H_j\), the data likelihood relative to the null likelihood is \(\operatorname{ABF}_j\). Consequently, the unnormalized posterior weight for \(H_j\) is its prior weight multiplied by that likelihood ratio. Dividing each weight by the sum of all hypothesis weights makes the probabilities add to one:

P(H_j\mid \text{data}) = \frac{P(H_j)P(\text{data}\mid H_j)} {\sum_k P(H_k)P(\text{data}\mid H_k)} = \frac{\pi_j\operatorname{ABF}_j} {\sum_k\pi_k\operatorname{ABF}_k},

where the final expression is conditional on the candidate causal hypotheses and omits a null hypothesis. If the null is included, its likelihood ratio is 1 and its prior weight appears in the denominator. This distinction matters: posterior probabilities conditional on a causal variant being present answer which candidate is favored, while posterior probabilities that include the null also reflect uncertainty about whether the region contains a causal variant under the model.

Worked Example: Strong Evidence for the Region but Not for One Candidate

Use the ABFs from the previous calculation, but now assign prior probability \(0.01\) to the hypothesis that neither candidate is causal and probability \(0.99\) to there being exactly one causal candidate. Within the causal hypothesis, give A and B equal prior probabilities, so their unconditional priors are each \(0.495\).

The Bayes factor for the one-causal-variant configuration, averaged over its two equally weighted candidates, is

\operatorname{BF}_{\text{causal vs null}} = \frac{19{,}700+18{,}930}{2} = 19{,}315.

The prior odds of a causal configuration versus the null are \(0.99/0.01=99\). The posterior odds are therefore

$$ 99(19{,}315)=1{,}912{,}185. $$

Thus the posterior probability of the causal configuration is approximately

\frac{1{,}912{,}185}{1+1{,}912{,}185} \approx 0.9999995.

This very large value is driven by the chosen priors and ABFs; it indicates overwhelming evidence against the null in this hypothetical calculation. Conditional on that causal configuration, the probabilities remain about 0.5100 for A and 0.4900 for B. The high posterior probability that there is a signal does not identify which variant generated it.

LD, Prior Weights, and the Credible Set

LD is the correlation between variant genotypes in the analyzed population. When \(r\) is near 1 or -1, the variants tend to carry almost the same information about an association signal. With finite data and a single-signal model, their marginal association statistics may consequently be very similar. If their Bayes factors are similar, the posterior odds are driven largely by the prior odds:

\frac{P(H_A\mid \text{data})}{P(H_B\mid \text{data})} = \frac{\pi_A}{\pi_B} \frac{\operatorname{ABF}_A}{\operatorname{ABF}_B}.

This identity also shows why a small difference in z-statistics need not yield a decisive distinction. In the first example, the ABF ratio is only about \(19{,}700/18{,}930=1.041\). The data favor A by only about four percent in likelihood-ratio terms. A 95% set must therefore include both variants even though either marginal z-statistic would conventionally be described as highly significant.

Worked Example: A Prior Preference Changes the Ranking, Not the Resolution

Keep the ABFs from the first example, but use prior probabilities \(0.60\) for A and \(0.40\) for B, conditional on exactly one being causal. The unnormalized posterior weights are \(0.60(19{,}700)=11{,}820\) for A and \(0.40(18{,}930)=7{,}572\) for B. Therefore,

P(H_A\mid \text{data}) = \frac{11{,}820}{11{,}820+7{,}572} \approx 0.6095, \qquad P(H_B\mid \text{data})\approx 0.3905.

A is now more strongly favored, partly because it had the larger prior probability. Nevertheless, the 95% credible set still requires both variants: the leading posterior probability is only \(0.6095\), and the two probabilities sum to 1.000. The ranking has changed, but the data and model still do not resolve the causal variant at the 95% level.

At the limiting case of perfect LD, \(r=1\), standardized genotype columns are identical. Under a model that differs only by which of those identical columns is labelled causal, the likelihoods are identical. With equal priors, no analysis of those data can favor one label over the other: their posterior probabilities are equal. Near-perfect rather than perfect LD allows some distinction in principle, but the information separating the variants can remain very small.

Diagnostics, Comparisons, and Limitations

Before interpreting a near-tie in posterior probabilities as a biological resolution limit, check that the evidence being compared is coherent. Confirm that effect alleles and allele frequencies are harmonized, that the LD reference is appropriate for the analyzed population, and that the summary statistics and LD refer to compatible samples and variants. Inconsistent summary statistics or LD, or LD reference mismatch, can distort the relative evidence. These checks do not create information that the genotypes lack, but they help distinguish genuine ambiguity from a data-processing problem.

Compare several quantities rather than relying on one number:

Association statistic or p-value: measures evidence against a no-association null under its testing assumptions; it does not identify a causal variant.
Bayes factor: compares the data likelihood under a specified hypothesis with a reference hypothesis. Its value depends on the effect-size prior and model.
PIP: gives a candidate's posterior probability under the stated model and priors. A moderate PIP can coexist with strong locus-level evidence when candidates are difficult to distinguish.
Credible set: reports the group of candidates needed to attain a target cumulative posterior probability. It does not say that each member has that target probability, nor that the set is guaranteed to contain the true causal variant when the model is misspecified.

The single-causal model used here is deliberately narrow. If more than one causal signal is present, the candidate probabilities and credible sets must be obtained from a model that represents that possibility; forcing a single-causal analysis can misrepresent the evidence. The ABF calculation also depends on the normal approximation and effect-size prior, while PIPs depend on the variant priors and candidate set. A credible set's nominal posterior mass is not an unconditional guarantee of empirical coverage if those assumptions are wrong.

Common pitfall: “Both variants have tiny association p-values, so both are causal,” or “the variant with the smaller p-value is the causal one.” Neither conclusion follows. Strong association statistics indicate evidence for association, not causal identification; when LD makes the evidence nearly interchangeable, posterior probabilities can remain split and the credible set can contain both variants.

The practical lesson is to report the distinction explicitly: a region can have compelling evidence for an association and still have poor variant-level resolution. In that setting, the size and composition of the credible set are scientifically informative, and a near-equal posterior split should be treated as uncertainty rather than as a tie-breaking contest.

Check Your Understanding

Use the model and calculations in this tutorial to answer the questions. Solutions follow the questions.

If two variants have equal prior probabilities and equal Bayes factors, what are their posterior probabilities conditional on exactly one being causal?
In the first worked example, why does the 95% credible set contain both variants even though A has the higher posterior probability?
Suppose the Bayes-factor ratio \(\operatorname{ABF}_A/\operatorname{ABF}_B\) is 2 and the prior probabilities are equal. What are the conditional posterior probabilities?
What does a very large Bayes factor for the causal configuration versus the null establish, and what does it not establish in a near-perfect LD example?

Solutions. (1) Each has posterior probability \(0.5\), because equal prior weights multiplied by equal Bayes factors remain equal before normalization. (2) A contributes only about \(0.5100\), which is below 0.95; B must be added, bringing cumulative posterior probability to 1.000. (3) Normalize the weights 2 and 1: A has probability \(2/(2+1)=0.6667\), and B has probability \(1/3=0.3333\). (4) It supports the causal configuration over the null under the specified model and priors. It does not establish which of two highly correlated candidates is causal; their relative posterior probabilities determine that resolution.