Bayesian Fine-Mapping with BayesC

GWAS Regions Rarely Identify a Unique Causal Variant

• A GWAS association usually identifies an associated region, not a unique causal variant.

• Because nearby SNPs are correlated through linkage disequilibrium (LD), several SNPs can show strong association even when only one is causal.

Fine-mapping moves from

\[ \text{associated region} \quad\longrightarrow\quad \text{small set of plausible causal variants}. \]

The goal is to quantify which variants remain plausible causal explanations after accounting for LD.

Fine-Mapping Connects Association Signals to Biology

• Biological interpretation: connect associated variants to genes, regulatory elements, and molecular mechanisms.
• Functional follow-up: prioritize variants, genes, and pathways for experimental studies.
• Quantifying uncertainty: distinguish one strongly supported variant from several LD-linked alternatives.
• Functional information: combine genetic evidence with eQTL, chromatin, conservation, and coding annotations.

Standard GWAS Tests One SNP at a Time

In a standard genome-wide association study (GWAS), each SNP is tested separately for association with the phenotype.

For SNP \(j\), fit a separate regression:

\[ y_i = \mu + x_{ij}\beta_j + e_i, \qquad e_i \sim \mathcal{N}(0,\sigma^2). \]

Test whether the SNP is associated with the phenotype:

\[ H_0:\beta_j=0 \qquad\text{versus}\qquad H_1:\beta_j\neq0. \]

Each SNP is tested one at a time.

Standard GWAS Reports Marginal Evidence

For every SNP, GWAS typically reports:

• estimated effect \(\widehat{\beta}_j\)
• standard error \(\operatorname{SE}(\widehat{\beta}_j)\)
• \(z\)- or \(t\)-statistic
• \(p\)-value

The result is a marker-by-marker summary of marginal association evidence.

Important: GWAS identifies associated SNPs, not necessarily the causal SNP.

LD Makes Multiple SNPs Look Associated

Nearby SNPs are often correlated because of linkage disequilibrium (LD). If SNP \(j^\ast\) is causal,

\[ y=x_{j^\ast}\beta_{j^\ast}+e. \]

Another SNP \(j\) can still show strong marginal association when

\[ \operatorname{Cor}(x_j,x_{j^\ast})\neq0 \qquad\text{but}\qquad \beta_j=0. \]

A GWAS peak can therefore represent an associated region, not a unique causal variant.

LD Explains the Cluster of Marginal Associations

The same simulated 30-SNP region is shown in both panels. The causal SNP is marked in red.

Fine-Mapping: Key Inputs

Fine-mapping aims to identify which SNPs within a GWAS-associated region remain plausible causal variants after accounting for LD.

• Marker-level association evidence: e.g. marginal effect estimates \(\hat{\beta}_j\) and their standard errors \(\mathrm{SE}(\hat{\beta}_j)\).

• LD between candidate SNPs: represented by the LD correlation matrix, e.g. \(R=\operatorname{cor}(X)\).

• A sparse joint model for genetic effects: e.g. BayesC / spike-and-slab.

Here we use individual-level genotypes and phenotypes. Summary-statistic methods instead combine compatible GWAS evidence with external LD.

Fine-Mapping: Key Outputs

• Posterior inclusion probabilities (PIPs): probability that each SNP has a non-zero effect.

• Posterior effect estimates: model-averaged SNP effects.

• Prioritized variant sets: small sets of plausible causal variants.

• Fine-mapping uncertainty: one leading SNP or several LD-linked alternatives.

Fine-Mapping Fits SNP Effects Jointly

All SNPs are fitted jointly, allowing correlated SNPs to compete to explain the same association signal:

\[ y_i = \mu + \sum_{j=1}^{m}x_{ij}b_j + e_i, \qquad e_i\sim\mathcal{N}(0,\sigma_e^2). \]

BayesC Uses Inclusion Indicators

BayesC assumes that each SNP is either inactive or active. The point mass at zero creates a sparse model in which many SNP effects are exactly zero.

Introduce an inclusion indicator:

\[ d_j \in \{0,1\}, \qquad d_j \sim \operatorname{Bernoulli}(\pi). \]

Conditional on the indicator:

\[ b_j\mid(d_j=0)=0, \]

\[ b_j\mid(d_j=1) \sim \mathcal{N}(0,\sigma_b^2). \]

BayesC Is a Spike-and-Slab Prior

Equivalently,

\[ b_j \sim (1-\pi)\delta_0 + \pi\mathcal{N}(0,\sigma_b^2). \]

\(\pi\) controls how often SNPs are active; \(\sigma_b^2\) controls the variance of non-zero effects.

Correlated SNPs Compete Across Posterior Models

The sampler repeatedly explores possible explanations of the association signal.

Posterior Samples Become Marker-Level Evidence

Joint Modelling Redistributes the GWAS Signal

The marginal GWAS signal is compared with support from the joint BayesC model.

PIP and Effect Size Answer Different Questions

Constructing a Local 0.9 Credible Set

A practical local credible set (LCS) summarizes the leading signal as a small set of plausible causal variants.

1. Identify the SNP with the highest PIP.

2. Select its LD neighborhood, for example \(r^2\ge0.5\).

3. Rank these SNPs by decreasing PIP.

4. Add their PIPs until

   \[ \sum_{j\in\mathcal C}\operatorname{PIP}_j \ge 0.90. \]

Interpreting the Local Credible Set

• PIPs come from the joint model and account for LD.
• A small LCS indicates a well-localized signal.
• A larger LCS indicates uncertainty among correlated SNPs.
• Raw PIPs are used without normalization.

Because BayesC allows multiple active SNPs, this is a practical local summary, not a formal \(90\%\) posterior coverage set.

Fine-Mapping Result: Local 0.9 Credible Set

The LCS contains 1 SNPs and includes the simulated causal SNP: yes.

Functional Information Can Refine Fine-Mapping

When LD leaves several plausible SNPs, functional information can refine their prior support.

Annotation-Informed Inclusion Probabilities

Standard BayesC gives every SNP the same prior probability of being active:

\[ d_j \sim \operatorname{Bernoulli}(\pi). \]

Annotations can make that probability SNP-specific:

\[ P(d_j=1)=\pi_j. \]

A relevant annotation may therefore raise a SNP’s prior probability of having a non-zero effect.

Annotation-Informed Effect Variances

Standard BayesC uses a common variance for active effects:

\[ b_j\mid d_j=1 \sim \mathcal{N}(0,\sigma_b^2). \]

Annotations can make the active-effect variance SNP-specific:

\[ b_j\mid d_j=1 \sim \mathcal{N}(0,\sigma_{b,j}^2). \]

An annotated SNP may therefore be assigned a larger prior variance for its non-zero effect.

Annotations Reweight Competing Causal Explanations

Suppose two SNPs have similar statistical evidence because they are in strong LD.

Annotation-Informed Fine-Mapping Workflow

Both approaches report PIPs, ranked candidates, posterior effects, and local credible sets.

Only the prior assigned to each SNP changes—through its inclusion probability, non-zero effect variance, or both.

Summary