Worked Single-Causal-Variant Fine-Mapping Analysis

From Association Evidence to a Fine-Mapping Result

A locus may contain several variants whose association statistics are compatible with a genetic effect. Fine-mapping under a single-causal-variant model turns the evidence for each candidate variant into posterior probabilities, then summarizes those probabilities in a credible set. The calculation is short, but its interpretation depends on being explicit about the model, the prior probabilities, and which variants are included.

This tutorial works through that calculation using Wakefield approximate Bayes factors. The examples use hypothetical summary statistics. A Bayes factor measures how much the data favor a variant-specific alternative over the null of no effect for that candidate. The posterior inclusion probabilities (PIPs) then compare candidates under the assumption that exactly one candidate variant in the locus is causal.

Definition: Suppose a locus has candidate variants $1,\ldots,m$, and let $C_i$ denote the hypothesis that variant $i$ is the single causal variant. Under the single-causal-variant model, exactly one of the $C_i$ is true. If $\pi_i=P(C_i)$ is the prior probability for variant $i$, with $\sum_{i=1}^{m}\pi_i=1$, and $BF_i$ is its Bayes factor, then $$ \mathrm{PIP}_i=P(C_i\mid\text{data})= \frac{\pi_i BF_i}{\sum_{j=1}^{m}\pi_j BF_j}. $$ The PIPs are posterior probabilities conditional on the model and the candidate variants under consideration; they sum to 1.

Approximate Bayes Factors from Summary Statistics

For candidate variant $i$, let $\hat{\beta}_i$ be its estimated effect and $SE_i$ its standard error, so the estimated variance is $V_i=SE_i^2$. Let $W$ be the prior variance of the effect under the alternative. The standard Wakefield approximation assumes the sampling distribution $\hat{\beta}_i\mid\beta_i$ is approximately normal with mean $\beta_i$ and variance $V_i$, and uses a zero-mean normal prior $\beta_i\sim N(0,W)$. The null fixes $\beta_i=0$. In these formulas $W$ and $V_i$ must use the square of the effect's units.

Formula: Define $z_i=\hat{\beta}_i/SE_i$ and $r_i=W/(V_i+W)$. The approximate Bayes factor for the alternative versus the null is $$ BF_i=\sqrt{1-r_i}\exp\left(\frac{z_i^2r_i}{2}\right). $$ A Bayes factor greater than 1 favors the alternative over the null for that candidate, while a value below 1 favors the null. It is evidence on a ratio scale, not a posterior probability.

The formula follows by integrating over the effect prior. Under the alternative, the normal sampling model and normal prior imply $\hat{\beta}_i\sim N(0,V_i+W)$; under the null, $\hat{\beta}_i\sim N(0,V_i)$. The Bayes factor is the ratio of these two densities at the observed estimate. Taking that ratio gives

\begin{aligned} BF_i &=\sqrt{\frac{V_i}{V_i+W}} \exp\left\{\frac{\hat{\beta}_i^2}{2} \left(\frac{1}{V_i}-\frac{1}{V_i+W}\right)\right\}\\ &=\sqrt{1-r_i}\exp\left(\frac{z_i^2r_i}{2}\right). \end{aligned}

The second line uses $r_i=W/(V_i+W)$, $z_i=\hat{\beta}_i/SE_i$, and $V_i=SE_i^2$. This expression shows why the evidence depends both on the strength of the association statistic and on its precision relative to the effect-size prior.

Worked Example: Calculate Bayes Factors for Four Variants

Suppose a hypothetical locus has four candidates, A through D. The effect estimates are in a common unit, and the prior effect variance is $W=0.04$ in squared units. The summary statistics are:

Variant	$\hat{\beta}$	$SE$	$z=\hat{\beta}/SE$
A	0.200	0.100	2.00
B	0.150	0.050	3.00
C	-0.120	0.060	-2.00
D	0.080	0.080	1.00

For A, $V_A=0.100^2=0.0100$, so $r_A=0.04/(0.0100+0.04)=0.8000$. Its Bayes factor is

BF_A=\sqrt{1-0.8000}\exp\left(\frac{2.00^2(0.8000)}{2}\right) =\sqrt{0.2}\exp(1.600)\approx2.2151.

For B, $V_B=0.050^2=0.0025$ and $r_B=0.04/(0.0025+0.04)=0.9411765$. Thus

BF_B=\sqrt{1-0.9411765}\exp\left(\frac{3.00^2(0.9411765)}{2}\right) \approx16.7548.

For C, $V_C=0.060^2=0.0036$ and $r_C=0.04/0.0436=0.9174312$. Since the statistic is squared, its negative sign does not change this Bayes factor:

BF_C=\sqrt{1-0.9174312}\exp\left(\frac{(-2.00)^2(0.9174312)}{2}\right) \approx1.8000.

For D, $V_D=0.080^2=0.0064$ and $r_D=0.04/0.0464=0.8620690$, giving

BF_D=\sqrt{1-0.8620690}\exp\left(\frac{1.00^2(0.8620690)}{2}\right) \approx0.57151.

These values compare each candidate's alternative with its null. They are not yet PIPs: the next step is to combine them with prior probabilities across the candidate variants.

Normalize Evidence to Obtain PIPs

For a single-causal-variant model, each candidate's unnormalized posterior weight is its prior probability multiplied by its Bayes factor. Dividing that weight by the sum of all candidate weights gives its PIP. If all candidates have equal prior probability, their common prior factor cancels, so the PIPs are the Bayes factors divided by their sum. This normalization relies on the mutually exclusive, exhaustive hypotheses that exactly one of the listed candidates is causal.

Worked Example: Equal Priors and a 95% Credible Set

Use the four Bayes factors just calculated and assign each variant prior probability $0.25$. Because these priors are equal, normalize the Bayes factors directly. Retaining unrounded Bayes factors for normalization, their sum is approximately $21.3414$. The resulting PIPs are:

\begin{aligned} \mathrm{PIP}_A&=\frac{2.2151}{21.3414}\approx0.1038,\\ \mathrm{PIP}_B&=\frac{16.7548}{21.3414}\approx0.7851,\\ \mathrm{PIP}_C&=\frac{1.8000}{21.3414}\approx0.0843,\\ \mathrm{PIP}_D&=\frac{0.57151}{21.3414}\approx0.0268. \end{aligned}

The PIPs sum to approximately 1, allowing for rounding. To construct a 95% thresholded credible set, rank the variants from largest to smallest PIP and add them until the cumulative posterior probability reaches or exceeds 0.95. The order is B, A, C, D: B alone has 0.7851, B and A together have $0.7851+0.1038=0.8889$, and adding C gives $0.8889+0.0843=0.9732$. The 95% set is therefore $\{B,A,C\}$, with posterior mass approximately 0.9732.

The set exceeds the target because variants are added as whole units; in this case, including only B and A would leave the cumulative mass below 0.95. The result is a posterior-mass summary under the stated model and prior, not a claim that each included variant has the same probability.

Variant-Specific Priors Change the Ranking

Equal prior probabilities are not required. If prior probabilities differ, they enter the numerator as weights and can change both the PIPs and the ordering used for a credible set. For example, annotation-informed priors may assign different probabilities to candidates. Whatever their source, the prior probabilities in this single-causal model must be defined over the candidate variants and sum to 1.

Worked Example: Unequal Variant Priors

Keep the same four Bayes factors, but assign prior probabilities $(\pi_A,\pi_B,\pi_C,\pi_D)=(0.40,0.30,0.20,0.10)$. The unnormalized weights, calculated using the unrounded Bayes factors, are approximately

\begin{aligned} w_A&=0.40(2.2151)\approx0.8860,\\ w_B&=0.30(16.7548)\approx5.0265,\\ w_C&=0.20(1.8000)\approx0.3600,\\ w_D&=0.10(0.57151)\approx0.05715. \end{aligned}

The weights sum to approximately $6.3296$. Dividing each weight by this total gives $\mathrm{PIP}_A\approx0.1400$, $\mathrm{PIP}_B\approx0.7941$, $\mathrm{PIP}_C\approx0.0569$, and $\mathrm{PIP}_D\approx0.0090$. The ranking remains B, A, C, D. The cumulative mass for B and A is approximately $0.7941+0.1400=0.9341$, which is below 0.95; adding C gives approximately $0.9910$. Thus the 95% set is again $\{B,A,C\}$, but its posterior mass is now about 0.9910. The changed PIPs reflect the changed prior probabilities, not changed association data.

The Candidate Set Is Part of the Calculation

PIPs are normalized over the variants included in the model. Changing the candidate set can change a PIP even when the summary statistics for a retained variant are unchanged, because the denominator and the allocation of prior probability change. The next example isolates this point by considering only two candidates from the four-variant example.

Worked Example: Recalculating PIPs for Two Candidates

Suppose the candidate set contains only A and B, with equal prior probabilities. Their Bayes factors remain approximately 2.2151 and 16.7548. The normalizing total is $2.2151+16.7548=18.9699$, so

\mathrm{PIP}_B=\frac{16.7548}{18.9699}\approx0.8832, \qquad \mathrm{PIP}_A=\frac{2.2151}{18.9699}\approx0.1168.

The two PIPs sum to 1. Under this two-candidate model, a thresholded 95% credible set must contain both A and B: B alone contributes only about 0.8832 posterior probability, while the pair contributes 1. In the earlier four-candidate calculation, B's PIP was about 0.7851. This difference is expected: the two calculations normalize over different candidate sets, and the two-candidate result assumes that one of A or B is causal.

Assumptions, Checks, and a Common Pitfall

Before interpreting a worked result, check the conditions that make its calculation meaningful. The summary-statistic approximation treats the effect estimate as approximately normal with the reported standard error. The effect prior is zero-mean normal with variance $W$, and the same $W$ is used for candidates here. The single-causal model assumes exactly one causal variant among the listed candidates. The variant priors are nonnegative and sum to 1, and the effect estimates and prior variance use compatible units. If these assumptions or the candidate set are unsuitable, the normalized PIPs need not answer the intended scientific question.

A frequent mistake is to report the largest Bayes factor as though it were the probability that the variant is causal. For example, $BF_B\approx16.7548$ is not a PIP of 16.7548 or 0.7851. The latter is B's PIP only after weighting by priors and normalizing against all candidates in the equal-prior four-variant example. Similarly, a 95% thresholded set is assembled by accumulating ranked PIPs; it is not generally the set of variants each having a PIP of at least 0.95.

Key takeaway: For a single-causal-variant analysis, calculate a Bayes factor for each candidate, multiply by its prior probability, normalize the weights to obtain PIPs, and accumulate the largest PIPs to form the thresholded credible set. State the candidate set and model assumptions alongside the result.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

In the Wakefield approximate Bayes factor, what are $V_i$, $W$, and $r_i$, and why must $V_i$ and $W$ use compatible squared units?
If all candidate variants have equal prior probabilities, why can the common prior factor be omitted when calculating PIPs?
For the equal-prior four-variant example, what is the cumulative PIP after adding B and A, and why must C also be included in the 95% thresholded set?
How does the formula for a PIP change when candidate variants have unequal prior probabilities?
Why can B's PIP differ between the four-candidate and two-candidate calculations even though its Bayes factor is unchanged?