End-to-End Fine-Mapping Worked Analysis

From Regional Inputs to a Fine-Mapping Result

Fine-mapping combines association evidence with assumptions about how many variants may be causal and how likely different configurations are before seeing the data. A regional analysis therefore does not simply rank the largest association statistics. It compares specified causal models, accounts for prior probabilities, and summarizes the resulting uncertainty. This tutorial follows a small hypothetical region from summary statistics and an LD matrix through model comparison, posterior probabilities, a credible set, calibration checks, and sensitivity analysis.

The worked calculation uses an approximate Bayes-factor method for a single-causal-variant-or-null model. This is a deliberate simplification: it makes the posterior calculation transparent, but it does not represent every possible regional fine-mapping model. In particular, the scalar Bayes factors below do not use the LD matrix to compute a joint likelihood across variants. LD is still essential for checking whether the association pattern is plausible and whether the model’s single-signal assumption is defensible.

Model and assumptions: The region contains \(m\) candidate variants. Either no variant is causal, or exactly one candidate variant is causal. For candidate \(j\), its estimated effect \(\hat{\beta}_j\), with standard error \(SE_j\), is approximated by a normal summary-statistic likelihood, \(\hat{\beta}_j\mid\beta_j\sim N(\beta_j,SE_j^2)\). Under the causal model for that candidate, \(\beta_j\sim N(0,\tau^2)\); under the null model, \(\beta_j=0\). The approximation assumes the reported estimates and standard errors are suitable for this likelihood and that the effect prior is appropriate. The variants, alleles, genome build, and LD reference must be harmonized, and the LD reference should represent the study population. The single-causal assumption must be plausible for the region being analyzed.

Bayes Factors and Posterior Model Probabilities

Write \(z_j=\hat{\beta}_j/SE_j\) and \(W_j=\tau^2/SE_j^2\). Integrating the normal likelihood over the normal effect prior gives a marginal distribution \(\hat{\beta}_j\sim N(0,SE_j^2+\tau^2)\) under the causal model. Under the null, its distribution is \(N(0,SE_j^2)\). The ratio of these two densities is the approximate Bayes factor:

BF_j=\sqrt{\frac{1}{1+W_j}}\exp\left(\frac{z_j^2 W_j}{2(1+W_j)}\right).

A Bayes factor compares the data’s support for one model with its support for another; it is not itself a posterior probability. For example, \(BF_j=4\) means the observed summary statistic is four times as probable under the specified causal model for variant \(j\) as under its null model, given the likelihood assumptions. Posterior probabilities also depend on model priors.

Let the prior probability of no causal variant be \(1-\pi\), and let the total prior probability of one causal variant be \(\pi\). If the causal prior is uniform across \(m\) variants, each single-variant model has prior probability \(\pi/m\). The posterior normalizing constant and model probabilities are then:

D=(1-\pi)+\frac{\pi}{m}\sum_{j=1}^{m}BF_j,\qquad P(H_0\mid\text{data})=\frac{1-\pi}{D},\qquad P(j\text{ causal}\mid\text{data})=\frac{(\pi/m)BF_j}{D}.

The posterior inclusion probability (PIP) for variant \(j\) is its posterior probability of being causal under the model being fit. In this single-causal-or-null setup, the PIPs sum to the posterior probability that at least one candidate is causal, not necessarily to one. If we construct a variant credible set conditional on a causal variant being present, we normalize each PIP by \(1-P(H_0\mid\text{data})\). That distinction matters when the null model retains appreciable posterior probability.

Worked Example: Calculating Posterior Probabilities

Suppose a hypothetical region has three candidate variants with \(z\)-scores \(4\), \(3\), and \(1\). Use \(W_j=1\) for all three, a prior probability \(\pi=0.5\) that exactly one candidate is causal, and a uniform prior over the three candidates if one is causal. The supplied LD matrix is:

R= \begin{pmatrix} 1 & 0.8 & 0.1\\ 0.8 & 1 & 0.1\\ 0.1 & 0.1 & 1 \end{pmatrix}.

This matrix is positive definite (its determinant is \(0.356\)), and its entries are compatible with strong correlation between variants 1 and 2. Assume the summary statistics and LD use harmonized alleles and the same variant order. For this calculation, the one-causal approximate Bayes factors use each variant’s \(z\)-score and \(W_j\); the matrix is retained for regional plausibility checks rather than inserted into a joint likelihood.

With \(W_j=1\), \(BF_j=2^{-1/2}\exp(z_j^2/4)\). Thus:

BF_1=2^{-1/2}\exp(4)=38.606722,\quad BF_2=2^{-1/2}\exp(2.25)=6.708842,\quad BF_3=2^{-1/2}\exp(0.25)=0.907944.

The null prior is \(0.5\), and each variant model has prior probability \(0.5/3=1/6\). Therefore:

D=0.5+\frac{38.606722+6.708842+0.907944}{6} =8.203918.

The null posterior probability is \(0.5/8.203918=0.06095\). The PIPs are \(38.606722/(6D)=0.7843\), \(6.708842/(6D)=0.1363\), and \(0.907944/(6D)=0.01845\), respectively. They sum to \(0.9391\), which is the posterior probability that at least one variant is causal, up to rounding.

Conditional on a causal variant being present, the variant probabilities are approximately \(0.8352\), \(0.1451\), and \(0.0196\). A 95% credible set formed by adding variants in descending conditional probability includes variants 1 and 2: their cumulative conditional mass is about \(0.9804\). The set has at least 95% posterior mass because variants are added as whole units; this construction need not attain exactly 95%.

Why the LD Matrix and Model Choice Matter

The matrix \(R\) records correlations between genotypes, not posterior probabilities that variants are causal. Strong LD means multiple variants can carry similar association evidence. In the example, variants 1 and 2 are highly correlated, so their different \(z\)-scores should be checked against the actual summary data and reference panel rather than interpreted as independent evidence. A mismatch in ancestry, allele coding, variant order, or sample composition can make the LD matrix inappropriate for the region.

The model choice is also substantive. A single-causal model excludes configurations with two or more causal variants. If several independent signals are present, a single-signal analysis can concentrate posterior mass on a proxy or otherwise misrepresent uncertainty. A multi-causal configuration model instead assigns prior probabilities to sets of variants and compares the evidence for those sets. That can use LD in a joint summary-statistic likelihood, but the results then depend on the configuration prior, effect prior, and correct specification of the LD-based likelihood.

A useful end-to-end workflow is to verify inputs and assumptions first, then choose the causal model and priors, calculate and normalize evidence, construct the set under an explicitly stated rule, and finally examine calibration and sensitivity. Keeping these choices visible prevents a posterior ranking from being mistaken for a model-free result.

Check inputs and scope.
Confirm variant identity, allele alignment, units, standard errors, sample definitions, and LD reference. Decide whether a null-or-one-causal model is scientifically plausible for the region.

Specify priors and calculate evidence.
State the null prior, how causal prior mass is distributed, and the effect-size prior. Compute Bayes factors and normalize them to posterior model probabilities.

Summarize and challenge the result.
Report the null posterior, PIPs, and credible-set rule. Check simulated coverage when possible and assess whether plausible changes to priors or LD assumptions alter the conclusion.

Calibration and Sensitivity Checks

A posterior credible set’s nominal probability is conditional on the model, priors, and data. Calibration asks a different question: across repeated datasets generated under a specified scenario, how often does a set with a stated target contain the true causal variant? Simulation-based calibration of fine-mapping and empirical credible-set coverage are useful concepts for this assessment. They do not guarantee that the model fits a particular observed region, especially if the simulations omit important sources of misspecification.

Worked Example: Reading a Coverage Check

Suppose a hypothetical simulation exercise generates 100 regions, each with one known causal variant under a stated data-generating model. The analysis produces a 95% credible set for each region, and 91 of the 100 sets contain the true variant. The empirical coverage is \(91/100=0.91\), or 91%.

For a rough measure of simulation noise, the estimated standard error of this proportion is:

SE=\sqrt{\frac{0.91(1-0.91)}{100}}=0.0286.

The observed coverage is four percentage points below the nominal 95%, about \(0.04/0.0286=1.40\) of these estimated standard errors. This is a warning worth investigating, not decisive evidence of failure by itself. More simulations would reduce Monte Carlo uncertainty, while simulations that better represent LD, effect sizes, sample size, and causal-variant frequency would test relevance to the intended analysis. Coverage under one simulation design does not establish coverage under different or misspecified conditions.

Worked Example: Changing the Effect-Size Prior

Return to the three-variant data and keep the model priors and \(z\)-scores unchanged, but reduce the prior variance so that \(W_j=0.5\) for all variants. The Bayes factor formula becomes \(BF_j=\sqrt{1/1.5}\exp(z_j^2/6)\). This gives approximate Bayes factors \(11.751\), \(3.659\), and \(0.9646\). The normalizing constant is:

D=0.5+\frac{11.751+3.659+0.9646}{6}\approx3.229.

The posterior null probability is approximately \(0.5/3.229=0.1548\); the three PIPs are approximately \(0.6065\), \(0.1889\), and \(0.0498\). Conditional on a causal variant being present, their probabilities are about \(0.7177\), \(0.2234\), and \(0.0589\). The first two sum to about \(0.9411\), so this 95% set includes all three variants. Compared with the earlier prior, the posterior is less concentrated on variant 1 and assigns more probability to the null. The data did not change; the specified effect-size prior did.

Diagnostics, Alternatives, and Limitations

Before reporting a result, check that the calculation is numerically coherent: posterior model probabilities should sum to one, the PIPs should sum to the posterior probability of a non-null configuration in this model, and every reported set should meet its stated mass threshold under its stated conditioning. Check the z-scores against the regional association pattern and the LD matrix. Unexpected allele directions, implausibly large or small correlations, or a variant ordering discrepancy can invalidate an otherwise correct calculation.

Also examine whether one-causal assumptions are reasonable. Evidence for multiple signals, a poor fit between the observed association pattern and the LD structure, or a result that changes substantially under plausible priors should make the interpretation more cautious. An LD sensitivity analysis should use a coherent likelihood recalculation under each alternative LD matrix. In the scalar approximate Bayes-factor calculation above, changing the LD matrix while holding all \(z\)-scores and standard errors fixed does not change the calculated PIPs; claiming otherwise would be incorrect. The changed matrix can still reveal that the inputs or the single-signal interpretation are questionable.

The main alternatives answer somewhat different questions. A multi-causal configuration analysis allows more than one causal variant and can model their joint evidence, but is more computationally demanding and requires additional prior choices. Conditional association analysis can help identify distinct signals, but its conclusions depend on the conditioning procedure and available LD. Individual-level genotype analysis can avoid some summary-statistic approximations when the data and model are appropriate, but it has its own assumptions and data requirements. None of these approaches makes association evidence equivalent to proof of biological causation.

A high PIP is conditional on the candidate variants, data quality, likelihood, causal model, and priors. If the true causal variant is absent from the analyzed region or candidate list, the posterior may favor an observed proxy; it cannot assign probability to a variant that was not considered. Report the model and key assumptions alongside PIPs and credible sets, and describe a credible set as a set of variants with the stated posterior mass—not as a guarantee that every member is causal or that the set has that coverage in every real-world setting.

Key takeaway: A fine-mapping result is a conditional posterior summary, not a standalone discovery of causality. Its meaning depends on the regional inputs, LD representation, causal model, and priors; credible-set construction, calibration, and sensitivity checks should all be reported with those conditions in view.

Check Your Understanding

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

In a null-or-one-causal model, why might the PIPs sum to less than one?
For a set formed by adding whole variants until its conditional posterior mass reaches 95%, must its mass equal exactly 95%?
In the worked analysis, would changing only the LD matrix alter the displayed scalar approximate Bayes factors or PIPs if the \(z\)-scores and standard errors were held fixed?
What does an empirical coverage of 91% across 100 simulated 95% sets establish, and what does it not establish?
Why should a high PIP not be described as proof that a variant causes the trait?

Solutions. (1) The remaining posterior probability can belong to the null model; here the PIPs sum to the posterior probability of at least one causal variant. (2) No. Adding whole variants generally produces mass at least 95%, and can exceed the target, as the first two variants do in the worked example. (3) No. Those scalar Bayes factors depend on each \(z\)-score and \(W_j\); changing LD alone does not enter their formula. A joint LD-based model would need to be recalculated. (4) It reports 91% coverage in that particular finite simulation exercise. It is subject to Monte Carlo uncertainty and does not establish coverage under other data-generating conditions or for the observed region. (5) The PIP is conditional on the candidate set, likelihood, causal model, and priors. Association evidence and posterior inclusion do not by themselves establish a biological causal mechanism.