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Fine-mapping

Introduction

Fine-mapping aims to identify the causal variant(s) within a locus for a disease, given the evidence of the significant association of the locus (or genomic region) in GWAS of a disease.

After identifying a significant association signal in GWAS, fine-mapping helps narrow down which specific variant(s) are likely to be causal, rather than just being in linkage disequilibrium (LD) with the true causal variant.

What is fine-mapping?

Fine-mapping is a statistical approach used to identify the most likely causal variant(s) within a genomic region that shows significant association with a trait. It helps distinguish between: - Causal variants: Variants that directly affect the trait - Tag variants: Variants that are associated only because they are in LD with causal variants

Fine-mapping using individual-level data is usually performed by fitting the multiple linear regression model:

\[y = Xb + e\]

where: - \(y\) is the phenotype vector - \(X\) is the genotype matrix - \(b = (b_1, …, b_J)^T\) is a vector of genetic effects of variants - \(e\) is the error term

Fine-mapping using Bayesian methods aims to estimate the PIP (posterior inclusion probability), which indicates the evidence for variant \(j\) having a non-zero effect (i.e., being causal).

PIP (Posterior Inclusion Probability)

PIP is the probability that a variant is causal, given the observed data. It is calculated by summing the posterior probabilities over all models that include variant \(j\) as causal:

\[ PIP_j := Pr(b_j \neq 0 | X, y) \]
  • PIP ranges from 0 to 1
  • Higher PIP values indicate stronger evidence that the variant is causal
  • Variants with PIP > 0.5 are often considered strong candidates for being causal

Bayesian methods and Posterior probability

Bayesian fine-mapping uses Bayes' theorem to calculate the posterior probability of each model (configuration of causal variants):

\[ Pr(M_m | O) = \frac{Pr(O | M_m) Pr(M_m)}{\sum_{i=1}^n Pr(O | M_i) Pr(M_i)} \]

where: - \(O\): Observed data (genotypes and phenotypes) - \(M\): Models (the configurations of causal variants in the context of fine-mapping) - \(Pr(M_m | O)\): Posterior Probability of model \(m\) (probability that model \(m\) is true given the data) - \(Pr(O | M_m)\): Likelihood (the probability of observing the data given that model \(m\) is true) - \(Pr(M_m)\): Prior distribution of model \(m\) (the prior probability that model \(m\) is true) - \(\sum_{i=1}^n Pr(O | M_i) Pr(M_i)\): Evidence (the probability of observing the data), namely \(Pr(O)\)

Credible sets

A credible set is the minimum set of variants that contains all causal variants with probability \(\alpha\).

Under the single-causal-variant-per-locus assumption, the credible set is calculated by: 1. Ranking variants based on their posterior probabilities 2. Summing these probabilities until the cumulative sum is \(\geq \alpha\)

We usually report 95% credible sets (\(\alpha = 0.95\)) for fine-mapping analysis, meaning there is a 95% probability that the true causal variant is included in the set.

Interpreting credible sets

  • Smaller credible sets indicate better fine-mapping resolution
  • If multiple causal variants exist, methods like SuSiE can identify multiple credible sets

Commonly used tools for fine-mapping

Methods assuming only one causal variant in the locus: - ABF (Approximate Bayes Factor): Fast method for single causal variant fine-mapping

Methods assuming multiple causal variants in the locus: - SuSiE / SuSiE-RSS: Sum of Single Effects model (works with individual-level data or summary statistics) - CAVIAR, CAVIARBF, eCAVIAR: Bayesian fine-mapping methods - FINEMAP: Bayesian fine-mapping using summary statistics

Methods assuming a small number of larger causal effects with many infinitesimal effects: - SuSiE-inf: Extension of SuSiE for mixed effects - FINEMAP-inf: Extension of FINEMAP for mixed effects

Methods for cross-ancestry fine-mapping: - SUSIEX: Extension of SuSiE for cross-ancestry analysis - MESuSiE (Multi-ancestry SuSiE): Uses GWAS summary statistics from multiple ancestries; models both shared and ancestry-specific causal signals; handles diverse LD patterns across ancestries; uses variational inference for scalable computation - MultiSuSiE: Extension of SuSiE to multiple ancestries that allows causal effect sizes to vary across populations; demonstrates improved power and resolution with lower computational cost in large multi-ancestry datasets

For more information, check here.

In this tutorial, we will introduce SuSiE as an example. SuSiE stands for "Sum of Single Effects" model.

SuSiE model

The key idea behind SuSiE is to decompose the genetic effect vector as a sum of single effects:

\[b = \sum_{l=1}^L b_l \]

where: - Each vector \(b_l = (b_{l1}, …, b_{lJ})^T\) is a single effect vector (a vector with only one non-zero element) - \(L\) is the upper bound on the number of causal variants - This model can be fitted using Iterative Bayesian Stepwise Selection (IBSS)

Why 'Sum of Single Effects'?

SuSiE models the genetic effect as a sum of multiple single effects, where each single effect corresponds to one causal variant. This allows SuSiE to: - Identify multiple causal variants in a region - Account for LD between variants - Provide PIPs and credible sets for each causal variant

SuSiE-RSS (Summary Statistics)

For fine-mapping with summary statistics using SuSiE (SuSiE-RSS), IBSS was modified (IBSS-ss) to take sufficient statistics as input. SuSiE-RSS approximates the sufficient statistics from summary statistics (effect sizes, standard errors, and LD matrix) to perform fine-mapping without requiring individual-level data.

SuSiE and SuSiE-RSS references

For details of SuSiE and SuSiE-RSS, please check:

  • SuSiE (individual-level data): Wang, G., Sarkar, A., Carbonetto, P., & Stephens, M. (2020). A simple new approach to variable selection in regression, with application to genetic fine mapping. Journal of the Royal Statistical Society: Series B, 82(5), 1273-1300.

  • SuSiE-RSS (summary statistics): Zou, Y., Carbonetto, P., Wang, G., & Stephens, M. (2022). Fine-mapping from summary data with the "Sum of Single Effects" model. PLoS Genetics, 18(7), e1010299. Link

File Preparation

Before performing fine-mapping, you need to prepare:

  1. Summary statistics for the locus: GWAS summary statistics (effect sizes, standard errors, p-values) for variants in the region of interest
  2. SNP list for calculating LD matrix: List of variant IDs to extract from reference panel for LD calculation
  3. Reference panel: Genotype data (e.g., 1000 Genomes) for calculating LD matrix

Locus selection

Typically, you would: - Identify lead variants from GWAS (e.g., variants with \(p < 5 \times 10^{-8}\)) - Define a region around each lead variant (e.g., ±500 kb or ±1 Mb) - Extract summary statistics for all variants in that region

Using Python (gwaslab) to identify lead variants and extract locus data

import gwaslab as gl
import pandas as pd
import numpy as np

# Load summary statistics
sumstats = gl.Sumstats("../06_Association_tests/1kgeas.B1.glm.firth", fmt="plink2")

# Perform basic quality checks
sumstats.basic_check()

# Identify lead variants (independent significant variants)
sumstats.get_lead()

# Output example:
# Fri Jan 13 23:31:43 2023 Start to extract lead variants...
# Fri Jan 13 23:31:43 2023  -Processing 1122285 variants...
# Fri Jan 13 23:31:43 2023  -Significance threshold : 5e-08
# Fri Jan 13 23:31:43 2023  -Sliding window size: 500  kb
# Fri Jan 13 23:31:44 2023  -Found 59 significant variants in total...
# Fri Jan 13 23:31:44 2023  -Identified 3 lead variants!
# Fri Jan 13 23:31:44 2023 Finished extracting lead variants successfully!

# Display lead variants
print(sumstats.data[sumstats.data["STATUS"] == "LEAD"])
# SNPID CHR POS EA  NEA SE  Z P OR  N STATUS
# 110723  2:55574452:G:C  2 55574452  C G 0.160948  -5.98392  2.178320e-09  0.381707  503 9960099
# 424615  6:29919659:T:C  6 29919659  T C 0.155457  -5.89341  3.782970e-09  0.400048  503 9960099
# 635128  9:36660672:A:G  9 36660672  G A 0.160275  5.63422 1.758540e-08  2.467060  503 9960099

We will perform fine-mapping for the first significant locus whose lead variant is 2:55574452:G:C (chromosome 2, position 55574452).

# Filter variants in the locus (±500 kb around lead variant)
locus = sumstats.filter_value('CHR==2 & POS>55074452 & POS<56074452')

# Fill missing data (e.g., calculate BETA from OR if needed)
locus.fill_data(to_fill=["BETA"])

# Harmonize alleles (ensure consistent strand and allele coding)
locus.harmonize(basic_check=False, ref_seq="/path/to/reference/genome.fasta")

# Export summary statistics for the locus
locus.data.to_csv("sig_locus.tsv", sep="\t", index=None)

# Export SNP list for LD matrix calculation
locus.data["SNPID"].to_csv("sig_locus.snplist", sep="\t", index=None, header=None)

Check in terminal: 

head sig_locus.tsv
SNPID   CHR     POS     EA      NEA     BETA    SE      Z       P       OR      N       STATUS
2:54535206:C:T  2       54535206        T       C       0.30028978      0.142461        2.10786 0.0350429       1.35025 503     9960099
2:54536167:C:G  2       54536167        G       C       0.14885099      0.246871        0.602952        0.546541        1.1605  503     9960099
2:54539096:A:G  2       54539096        G       A       -0.0038474211   0.288489        -0.0133355      0.98936 0.99616 503     9960099
2:54540264:G:A  2       54540264        A       G       -0.1536723      0.165879        -0.926409       0.354234        0.857553        503     9960099
2:54540614:G:T  2       54540614        T       G       -0.1536723      0.165879        -0.926409       0.354234        0.857553        503     9960099
2:54540621:A:G  2       54540621        G       A       -0.1536723      0.165879        -0.926409       0.354234        0.857553        503     9960099
2:54540970:T:C  2       54540970        C       T       -0.049506452    0.149053        -0.332144       0.739781        0.951699        503     9960099
2:54544229:T:C  2       54544229        C       T       -0.14338203     0.151172        -0.948468       0.342891        0.866423        503     9960099
2:54545593:T:C  2       54545593        C       T       -0.1536723      0.165879        -0.926409       0.354234        0.857553        503     9960099

head sig_locus.snplist
2:54535206:C:T
2:54536167:C:G
2:54539096:A:G
2:54540264:G:A
2:54540614:G:T
2:54540621:A:G
2:54540970:T:C
2:54544229:T:C
2:54545593:T:C
2:54546032:C:G

LD Matrix Calculation

The LD (linkage disequilibrium) matrix is essential for fine-mapping, as it captures the correlation structure between variants in the region. Fine-mapping methods use the LD matrix to distinguish between causal variants and variants that are associated only due to LD.

Why is LD matrix important?

  • Fine-mapping methods need to account for LD to identify which variants are truly causal
  • The LD matrix is typically calculated from a reference panel (e.g., 1000 Genomes Project)
  • The reference panel should match the ancestry of your study population

Calculate LD matrix using PLINK

#!/bin/bash

# Path to reference panel (PLINK binary format)
plinkFile="../01_Dataset/1KG.EAS.auto.snp.norm.nodup.split.rare002.common015.missing/1KG.EAS.auto.snp.norm.nodup.split.rare002.common015.missing"

# Calculate LD correlation (r) matrix
plink \
  --bfile ${plinkFile} \
  --keep-allele-order \
  --r square \
  --extract sig_locus.snplist \
  --out sig_locus_mt

# Calculate LD r2 matrix (optional, for visualization)
plink \
  --bfile ${plinkFile} \
  --keep-allele-order \
  --r2 square \
  --extract sig_locus.snplist \
  --out sig_locus_mt_r2

PLINK LD matrix options

  • --r square: Calculate correlation coefficient (r) matrix
  • --r2 square: Calculate squared correlation coefficient (r²) matrix
  • --extract: Only include variants in the SNP list
  • --keep-allele-order: Preserve allele order from the reference panel

Inspect the LD matrix (first 5 rows and columns):

# LD correlation (r) matrix
head -5 sig_locus_mt.ld | cut -f 1-5
# 1       -0.145634       0.252616        -0.0876317      -0.0876317
# -0.145634       1       -0.0916734      -0.159635       -0.159635
# 0.252616        -0.0916734      1       0.452333        0.452333
# -0.0876317      -0.159635       0.452333        1       1
# -0.0876317      -0.159635       0.452333        1       1

# LD r2 matrix
head -5 sig_locus_mt_r2.ld | cut -f 1-5
# 1       0.0212091       0.0638148       0.00767931      0.00767931
# 0.0212091       1       0.00840401      0.0254833       0.0254833
# 0.0638148       0.00840401      1       0.204605        0.204605
# 0.00767931      0.0254833       0.204605        1       1
# 0.00767931      0.0254833       0.204605        1       1

LD matrix format

  • The LD matrix is symmetric (LD between variant i and j equals LD between j and i)
  • Diagonal elements are 1 (perfect correlation with itself)
  • Off-diagonal elements range from -1 to 1 for r, and 0 to 1 for r²

Heatmap visualization of the LD matrix:

LD matrix heatmap

Fine-mapping with summary statistics using SuSiE-R

Installation

Install SuSiE-R package

# Install from CRAN
install.packages("susieR")

# Or install from GitHub for latest version
# install.packages("devtools")
# devtools::install_github("stephenslab/susieR")

Running SuSiE-RSS

Fine-mapping with summary statistics using SuSiE-RSS

library(susieR)

# Load summary statistics and LD matrix
sumstats <- read.table("sig_locus.tsv", header=TRUE, sep="\t")
R <- as.matrix(read.table("sig_locus_mt.ld", header=FALSE))

# Extract required data
bhat <- sumstats$BETA  # Effect sizes
shat <- sumstats$SE    # Standard errors
n <- sumstats$N[1]     # Sample size (should be the same for all variants)

# Run SuSiE-RSS
fitted_rss <- susie_rss(
  bhat = bhat,      # Estimated effects (vector of length p)
  shat = shat,      # Standard errors (vector of length p)
  R = R,            # LD correlation matrix (p x p)
  n = n,            # Sample size
  L = 10            # Maximum number of causal variants
)

# View results
summary(fitted_rss)

SuSiE-RSS parameters

  • bhat: Estimated effect sizes (BETA) from GWAS (vector of length \(p\))
  • shat: Standard errors of effect sizes (vector of length \(p\))
  • R: LD correlation (r) matrix (\(p \times p\)), where \(p\) is the number of variants
  • n: Sample size (scalar, should be the same for all variants)
  • L: Maximum number of causal variants to consider (default: L = 10)

Alternative input: z-scores

Instead of bhat and shat, you can provide z-scores directly: 

z <- sumstats$BETA / sumstats$SE  # Calculate z-scores
fitted_rss <- susie_rss(z = z, R = R, n = n, L = 10)

Interpreting SuSiE results

Key outputs from SuSiE

  • PIPs: Posterior inclusion probabilities for each variant
  • Credible sets: Sets of variants that contain causal variants with specified probability
  • Effect estimates: Estimated effect sizes for each causal variant
# Extract PIPs
pip <- fitted_rss$pip  # Posterior inclusion probabilities

# Extract credible sets
sets <- fitted_rss$sets

# View credible sets
print(sets)

# Get variants in credible sets
credible_set_variants <- sumstats$SNPID[sets$cs[[1]]]  # First credible set

SuSiE-R tutorial

For more details and advanced usage, please check the SuSiE-R tutorial - Fine-mapping with susieR using summary statistics

Using SuSiE in Python (Jupyter notebook)

You can also use SuSiE in Python through the susieR R package via rpy2, or use the Python implementation.

For a complete Python example, check: finemapping_susie.ipynb

Example output visualization:

SuSiE fine-mapping results

References

SuSiE method papers

  • SuSiE (individual-level data): Wang, G., Sarkar, A., Carbonetto, P., & Stephens, M. (2020). A simple new approach to variable selection in regression, with application to genetic fine mapping. Journal of the Royal Statistical Society: Series B, 82(5), 1273-1300. https://doi.org/10.1111/rssb.12388

  • SuSiE-RSS (summary statistics): Zou, Y., Carbonetto, P., Wang, G., & Stephens, M. (2022). Fine-mapping from summary data with the "Sum of Single Effects" model. PLoS Genetics, 18(7), e1010299. https://doi.org/10.1371/journal.pgen.1010299

Fine-mapping reviews

  • Schaid, D. J., Chen, W., & Larson, N. B. (2018). From genome-wide associations to candidate causal variants by statistical fine-mapping. Nature Reviews Genetics, 19(8), 491-504. https://doi.org/10.1038/s41576-018-0016-z

Software and tools