Statistical Fine-Mapping

Statistical fine-mapping and colocalization are complementary Bayesian inference frameworks used in human genetics and functional genomics to resolve true causal variants from non-causal tag variants correlated by linkage disequilibrium (LD) at genome-wide association study (GWAS) loci, and to test whether two overlapping genetic association signals share the same underlying molecular mechanism. Because meiotic recombination occurs unevenly across human chromosomes at PRDM9-directed hotspots, alleles inherited on the same ancestral haplotype within a 10 kb to 500 kb genomic window exhibit pairwise squared correlation coefficients (r²) approaching 0.80 to 1.00. Consequently, a single causal mutation altering a transcription factor motif or topologically associating domain enhancer generates a broad "skyscraper" of marginal association p-values spanning dozens or hundreds of neighboring single-nucleotide polymorphisms (SNPs), where the sentinel SNP with the smallest marginal p-value is frequently a non-causal proxy perturbed by sampling noise.

Statistical fine-mapping converts marginal or joint association summary statistics (β̂, s) and the local m × m linkage disequilibrium correlation matrix R into per-variant Posterior Inclusion Probabilities (PIPs) and 95% credible sets (CS₉₅)—the minimal subset of variants guaranteed to contain a causal variant with posterior probability ≥ 0.95. Once candidate variants are prioritized, Bayesian colocalization (coloc, eCAVIAR, HyPrColoc) evaluates whether a clinical GWAS trait signal and a molecular quantitative trait locus—such as an expression QTL (cis-eQTL affecting transcription), splicing QTL (sQTL affecting alternative splicing), or protein abundance QTL (pQTL)—are driven by the same shared causal variant (H₄) rather than two distinct causal variants that happen to reside in linkage disequilibrium (H₃), providing genetic validation prior to Mendelian randomization and wet-lab CRISPR-Cas perturbation.

Derivation of the Wakefield (2009) ABF from Summary Statistics

Under the simplest fine-mapping model assuming at most one causal variant within a genomic locus of m candidate SNPs (Mallick, Maller et al., 2012), each variant j ∈ {1, ..., m} has a marginal effect size estimate β̂_j, standard error s_j (with sampling variance V_j = s_j²), and marginal Wald test statistic z_j = β̂_j / s_j. Rather than requiring individual-level genotype matrices, Jon Wakefield (2009) derived an analytical Approximate Bayes Factor (ABF_j) comparing the alternative model M₁—in which the true effect β_j follows a zero-mean normal prior β_j ~ N(0, W)—against the null model M₀ (β_j = 0), assuming an asymptotic normal likelihood β̂_j | β_j ~ N(β_j, V_j):

ABF_j = P(Data | M₁) / P(Data | M₀) = [ V_j / (V_j + W) ]^1/2 · exp[ (z_j² / 2) · (W / (V_j + W)) ]

Here, the ratio r_j = W / (V_j + W) ∈ (0, 1) represents the Bayesian shrinkage factor comparing the prior effect variance W to the total marginal variance V_j + W. Note that in Wakefield's original definition of the Bayes factor as P(Data | M₀) / P(Data | M₁), the orientation is inverted; modern statistical genetics software (coloc, finemap) standardizes ABF_j as M₁ over M₀ so that larger values of ABF_j indicate stronger evidence of association:

• For case-control binary traits (logistic regression log-odds ratios), the prior SD is typically set to √W = 0.20 (so W = 0.04), encoding a 95% prior belief that causal odds ratios lie between exp(−1.96 × 0.2) ≈ 0.68 and exp(+1.96 × 0.2) ≈ 1.48.
• For quantitative traits standardized to unit phenotypic variance, the prior SD is typically set to √W = 0.15 (so W = 0.0225).

Log-Sum-Exp Normalization, PIPs, and 95% Credible Sets (CS₉₅)

Assuming exactly one causal variant in the locus with uniform prior probability π_j = 1 / m across all m variants, the Posterior Inclusion Probability (PIP_j) that SNP j is the causal variant equals its ABF_j divided by the sum of ABFs across the entire locus. Because z_j² can exceed 100 at strong GWAS hits (causing exp(z_j² r_j / 2) to overflow double-precision floating-point limits), computation is performed in natural log-space using the log-sum-exp identity with ℓ_j = ln(ABF_j) and ℓ_max = max_k(ℓ_k):

PIP_j = ABF_j / ∑_k=1..m ABF_k = exp(ℓ_j − ℓ_max) / ∑_k=1..m exp(ℓ_k − ℓ_max)

Sorting all m variants in descending order of posterior probability (PIP₍₁₎ ≥ PIP₍₂₎ ≥ ... ≥ PIP_(m)), the 95% causal credible set (CS₉₅) is the smallest prefix set {1, ..., K} whose cumulative posterior sum reaches or exceeds α = 0.95:

K = min { k : ∑_i=1..k PIP_(i) ≥ 0.95 }

The resolution of fine-mapping at a locus is quantified both by the cardinality |CS₉₅| = K and by the posterior Shannon entropy H_PIP = −∑_j=1..m PIP_j log₂(PIP_j) (in bits): a resolved single-variant locus where PIP₍₁₎ ≥ 0.95 yields |CS₉₅| = 1 and H_PIP → 0 bits, whereas an unresolved tight LD block of 64 indistinguishable variants yields |CS₉₅| = 61 and H_PIP = 6.0 bits.

Over 30% of genome-wide significant GWAS loci and >50% of cis-eQTL loci harbor allelic heterogeneity—two or more distinct causal variants (L ≥ 2) within the same 1 Mb window. When multiple causal variants exist in partial linkage disequilibrium, single-causal Wakefield fine-mapping can misplace the credible set onto a non-causal tagging SNP that correlates moderately with both true causal variants.

To model multi-causal loci without combinatorial explosion across ∑_k=1..L (m choose k) subsets, Gao Wang, Abhishek Sarkar, Peter Carbonetto, and Matthew Stephens (2020) formulated SuSiE (Sum of Single Effects). SuSiE writes the sparse m-vector of true genetic effects b as the sum of L "single-effect" vectors b_l, each of which has exactly one non-zero entry determined by a multinomial indicator vector γ_l ∈ {0, 1}^m (with ∑_j γ_lj = 1):

y = X b + ε, where b = ∑_l=1..L b_l, b_l = b_l γ_l, γ_l ~ Mult(1, π), b_l ~ N(0, σ_0l²)

Using Iterative Bayesian Stepwise Selection (IBSS)—which fits each single-effect regression model (b_l) to the partial residuals after subtracting the expected effects of the other L − 1 components—SuSiE (and its summary-statistic extension SuSiE-RSS operating on z-scores and the LD matrix R = X^TX / n) produces L separate posterior probability vectors α₁, ..., α_L ∈ [0, 1]^m. Each component l with non-zero estimated prior variance yields its own distinct 95% credible set CS₉₅^(l), and the overall Posterior Inclusion Probability for variant j across all L effects is combined via the complement rule:

PIP_j = 1 − ∏_l=1..L (1 − α_lj)

When summary statistics are paired with an external LD reference panel (such as 1000 Genomes or UK Biobank), care must be taken to verify alignment between summary z-scores and the LD matrix R using the DENTIST / SuSiE-RSS diagnostic (z_j | z_−j) to prevent allele-flip errors from generating false-positive multi-causal credible sets.

Given summary statistics across an m-SNP genomic region for Trait 1 (e.g., a disease GWAS) and Trait 2 (e.g., a tissue-specific cis-eQTL or sQTL), Claudia Giambartolomei, Chris Wallace, and colleagues (2014) formulated coloc.abf to partition the space of joint configurations into five mutually exclusive global hypotheses (H₀–H₄):

• H₀: Neither Trait 1 nor Trait 2 has a genetic association in the genomic region.
• H₁: Only Trait 1 has a causal variant in the region (and Trait 2 has none).
• H₂: Only Trait 2 has a causal variant in the region (and Trait 1 has none).
• H₃ (Linkage / Distinct Causal Variants): Both Trait 1 and Trait 2 are associated in the region, but they are driven by two distinct causal variants (j ≠ k) that reside in linkage disequilibrium.
• H₄ (Colocalization / Shared Causal Variant): Both Trait 1 and Trait 2 are associated and share the exact same single causal variant (j = k).

Specify three prior probabilities: p₁ (the prior probability that any given SNP is causal for Trait 1 alone, default 10⁻⁴), p₂ (the prior probability that a SNP is causal for Trait 2 alone, default 10⁻⁴), and p₁₂ (the prior probability that a single SNP is simultaneously causal for both Trait 1 and Trait 2, typically set to 10⁻⁵ or more conservatively 5 × 10⁻⁶). Let ABF_j⁽¹⁾ and ABF_j⁽²⁾ denote the Wakefield Approximate Bayes Factors at SNP j for Trait 1 and Trait 2. Summing over all m variants yields the five regional Bayes factor sums S₀ through S₄:

• S₀ = 1
• S₁ = p₁ ∑_j=1..m ABF_j⁽¹⁾
• S₂ = p₂ ∑_j=1..m ABF_j⁽²⁾
• S₃ = p₁ p₂ ∑_j=1..m ∑_k≠j ABF_j⁽¹⁾ ABF_k⁽²⁾ = ( S₁ · S₂ ) − p₁ p₂ ∑_j=1..m (ABF_j⁽¹⁾ ABF_j⁽²⁾)
• S₄ = p₁₂ ∑_j=1..m (ABF_j⁽¹⁾ · ABF_j⁽²⁾)

Normalizing each S_h by the total partition sum ∑_k=0..4 S_k yields the five posterior probabilities PP.H0 through PP.H4. In functional genomics studies, PP.H4 ≥ 0.80 (combined with PP.H4 / PP.H3 ≥ 5) is the standard threshold demonstrating that the GWAS disease locus and the molecular eQTL/pQTL colocalize to a shared causal variant, linking the non-coding genetic variant to its target effector gene.

1. Wakefield, J. "Bayes factors for genome-wide association studies: comparison with P-values." Genetic Epidemiology 33, 79–86 (2009). https://doi.org/10.1002/gepi.20359
2. Maller, J. B., et al. "Bayesian refinement of association signals for 14 loci in 3 common diseases." Nature Genetics 44, 1294–1301 (2012). https://doi.org/10.1038/ng.2435
3. Giambartolomei, C., et al. "Bayesian test for colocalisation between pairs of genetic association studies using summary statistics." PLOS Genetics 10, e1004383 (2014). https://doi.org/10.1371/journal.pgen.1004383
4. Wang, G., Sarkar, A., Carbonetto, P., & Stephens, M. "A simple new approach to variable selection in regression, with application to genetic fine mapping." Journal of the Royal Statistical Society Series B 82, 1273–1300 (2020). https://doi.org/10.1111/rssb.12388