LD score regression

Introduction

LDSC is one of the most commonly used command line tool to estimate inflation, hertability, genetic correlation and cell/tissue type specificity from GWAS summary statistics.

LD: Linkage disequilibrium

Linkage disequilibrium (LD) : non-random association of alleles at different loci in a given population. (Wiki)

LD score

LD score \(l_j\) for a SNP \(j\) is defined as the sum of \(r^2\) for the SNP and other SNPs in a region.

\[ l_j= \Sigma_k{r^2_{j,k}} \]

LD score regression

Key idea: A variant will have higher test statistics if it is in LD with causal variant, and the elevation is proportional to the correlation ( \(r^2\) ) with the causal variant.

\[ E[\chi^2|l_j] = {{Nh^2l_j}\over{M}} + Na + 1 \]

\(N\): sample size.
\(M\) : number of SNPs.
\(h^2\) : observed-scale heritability
\(a\) : the effect of confounding factors, including crytic relatedness and populatiuon stratification.

For more details of LD score regression, please refer to : - Bulik-Sullivan, Brendan K., et al. "LD Score regression distinguishes confounding from polygenicity in genome-wide association studies." Nature genetics 47.3 (2015): 291-295.

Install LDSC

LDSC can be downloaded from github (GPL-3.0 license): https://github.com/bulik/ldsc

For ldsc, we need anaconda to create virtual environment (for python2). If you haven't installed Anaconda, please check how to install anaconda.

# change to your directory for tools
cd ~/tools

# clone the ldsc github repository
git clone https://github.com/bulik/ldsc.git

# create a virtual environment for ldsc (python2)
cd ldsc
conda env create --file environment.yml  

# activate ldsc environment
conda activate ldsc

Data Preparation

In this tutoial, we will use sample summary statistics for HDLC and LDLC from Jenger. - Kanai, Masahiro, et al. "Genetic analysis of quantitative traits in the Japanese population links cell types to complex human diseases." Nature genetics 50.3 (2018): 390-400.

The Miami plot for the two traits:

Download sample summary statistics

# HDL-c and LDL-c in Biobank Japan
wget -O BBJ_LDLC.txt.gz http://jenger.riken.jp/61analysisresult_qtl_download/
wget -O BBJ_HDLC.txt.gz http://jenger.riken.jp/47analysisresult_qtl_download/

Download reference files

# change to your ldsc directory
cd ~/tools/ldsc
mkdir resource
cd ./resource

# snplist
wget https://storage.googleapis.com/broad-alkesgroup-public/LDSCORE/w_hm3.snplist.bz2

# EAS ld score files
wget https://storage.googleapis.com/broad-alkesgroup-public/LDSCORE/eas_ldscores.tar.bz2

# EAS weight
wget https://storage.googleapis.com/broad-alkesgroup-public/LDSCORE/1000G_Phase3_EAS_weights_hm3_no_MHC.tgz

# EAS frequency
wget https://storage.googleapis.com/broad-alkesgroup-public/LDSCORE/1000G_Phase3_EAS_plinkfiles.tgz

# EAS baseline model
wget https://storage.googleapis.com/broad-alkesgroup-public/LDSCORE/1000G_Phase3_EAS_baseline_v1.2_ldscores.tgz

# Cell type ld score files
wget https://storage.googleapis.com/broad-alkesgroup-public/LDSCORE/LDSC_SEG_ldscores/Cahoy_EAS_1000Gv3_ldscores.tar.gz

You can then decompress the files and organize them.

Munge sumstats

Before the analysis, we need to format and clean the raw sumstats.

Note

Rsid is used here. If the sumstats only contained id like CHR:POS:REF:ALT, annotate it first.

snplist=~/tools/ldsc/resource/w_hm3.snplist
munge_sumstats.py \
    --sumstats BBJ_HDLC.txt.gz \
    --merge-alleles $snplist \
    --a1 ALT \
    --a2 REF \
    --chunksize 500000 \
    --out BBJ_HDLC
munge_sumstats.py \
  --sumstats BBJ_LDLC.txt.gz \
    --a1 ALT \
  --a2 REF \
  --chunksize 500000 \
  --merge-alleles $snplist \
  --out BBJ_LDLC

After munging, you will get two munged and formatted files:

BBJ_HDLC.sumstats.gz
BBJ_LDLC.sumstats.gz

And these are the files we will use to run LD score regression.

LD score regression

Univariate LD score regression is utilized to estimate heritbility and confuding factors (cryptic relateness and population stratification) of a certain trait.

Reference : Bulik-Sullivan, Brendan K., et al. "LD Score regression distinguishes confounding from polygenicity in genome-wide association studies." Nature genetics 47.3 (2015): 291-295.

Using the munged sumstats, we can run:

ldsc.py \
  --h2 BBJ_HDLC.sumstats.gz \
  --ref-ld-chr ~/tools/ldsc/resource/eas_ldscores/ \
  --w-ld-chr ~/tools/ldsc/resource/eas_ldscores/ \
  --out BBJ_HDLC

ldsc.py \
  --h2 BBJ_LDLC.sumstats.gz \
  --ref-ld-chr ~/tools/ldsc/resource/eas_ldscores/ \
  --w-ld-chr ~/tools/ldsc/resource/eas_ldscores/ \
  --out BBJ_LDLC

Lest's check the results for HDLC:

cat BBJ_HDLC.log
*********************************************************************
* LD Score Regression (LDSC)
* Version 1.0.1
* (C) 2014-2019 Brendan Bulik-Sullivan and Hilary Finucane
* Broad Institute of MIT and Harvard / MIT Department of Mathematics
* GNU General Public License v3
*********************************************************************
Call: 
./ldsc.py \
--h2 BBJ_HDLC.sumstats.gz \
--ref-ld-chr /home/he/tools/ldsc/resource/eas_ldscores/ \
--out BBJ_HDLC \
--w-ld-chr /home/he/tools/ldsc/resource/eas_ldscores/ 

Beginning analysis at Sat Dec 24 20:40:34 2022
Reading summary statistics from BBJ_HDLC.sumstats.gz ...
Read summary statistics for 1020377 SNPs.
Reading reference panel LD Score from /home/he/tools/ldsc/resource/eas_ldscores/[1-22] ... (ldscore_fromlist)
Read reference panel LD Scores for 1208050 SNPs.
Removing partitioned LD Scores with zero variance.
Reading regression weight LD Score from /home/he/tools/ldsc/resource/eas_ldscores/[1-22] ... (ldscore_fromlist)
Read regression weight LD Scores for 1208050 SNPs.
After merging with reference panel LD, 1012040 SNPs remain.
After merging with regression SNP LD, 1012040 SNPs remain.
Using two-step estimator with cutoff at 30.
Total Observed scale h2: 0.1583 (0.0281)
Lambda GC: 1.1523
Mean Chi^2: 1.2843
Intercept: 1.0563 (0.0114)
Ratio: 0.1981 (0.0402)
Analysis finished at Sat Dec 24 20:40:41 2022
Total time elapsed: 6.57s

We can see that from the log:

Observed scale h2 = 0.1583
lambda GC = 1.1523
intercept = 1.0563
Ratio = 0.1981

According to LDSC documents, Ratio measures the proportion of the inflation in the mean chi^2 that the LD Score regression intercept ascribes to causes other than polygenic heritability. The value of ratio should be close to zero, though in practice values of 10-20% are not uncommon.

\[ Ratio = {{intercept-1}\over{mean(\chi^2)-1}} \]

Cross-trait LD score regression

Cross-trait LD score regression is employed to estimate the genetic correlation between a pair of traits.

Key idea: replace \chi^2 in univariate LD score regression and the relationship (SNPs with high LD ) still holds.

\[ E[z_{1j}z_{2j}] = {{\sqrt{N_1N_2}\rho_g}\over{M}}l_j + {{\rho N_s}\over{\sqrt{N_1N_2}}} \]

\(z_ij\) : z score of trait i for SNP j
\(N_i\) : sample size of trait i
\(\rho\) : phenotypic correlation
\(\rho_g\) : genetic covariance
\(l_j\) : LD score
\(M\) : number of SNPs

Then we can get the genetic correlation by :

\[ r_g = {{\rho_g}\over{\sqrt{h_1^2h_2^2}}} \]

Reference: Bulik-Sullivan, Brendan, et al. "An atlas of genetic correlations across human diseases and traits." Nature genetics 47.11 (2015): 1236-1241.

ldsc.py \
  --rg BBJ_HDLC.sumstats.gz,BBJ_LDLC.sumstats.gz \
  --ref-ld-chr ~/tools/ldsc/resource/eas_ldscores/ \
  --w-ld-chr ~/tools/ldsc/resource/eas_ldscores/ \
  --out BBJ_HDLC_LDLC

Let's check the results:

*********************************************************************
* LD Score Regression (LDSC)
* Version 1.0.1
* (C) 2014-2019 Brendan Bulik-Sullivan and Hilary Finucane
* Broad Institute of MIT and Harvard / MIT Department of Mathematics
* GNU General Public License v3
*********************************************************************
Call: 
./ldsc.py \
--ref-ld-chr /home/he/tools/ldsc/resource/eas_ldscores/ \
--out BBJ_HDLC_LDLC \
--rg BBJ_HDLC.sumstats.gz,BBJ_LDLC.sumstats.gz \
--w-ld-chr /home/he/tools/ldsc/resource/eas_ldscores/ 

Beginning analysis at Thu Dec 29 21:02:37 2022
Reading summary statistics from BBJ_HDLC.sumstats.gz ...
Read summary statistics for 1020377 SNPs.
Reading reference panel LD Score from /home/he/tools/ldsc/resource/eas_ldscores/[1-22] ... (ldscore_fromlist)
Read reference panel LD Scores for 1208050 SNPs.
Removing partitioned LD Scores with zero variance.
Reading regression weight LD Score from /home/he/tools/ldsc/resource/eas_ldscores/[1-22] ... (ldscore_fromlist)
Read regression weight LD Scores for 1208050 SNPs.
After merging with reference panel LD, 1012040 SNPs remain.
After merging with regression SNP LD, 1012040 SNPs remain.
Computing rg for phenotype 2/2
Reading summary statistics from BBJ_LDLC.sumstats.gz ...
Read summary statistics for 1217311 SNPs.
After merging with summary statistics, 1012040 SNPs remain.
1012040 SNPs with valid alleles.

Heritability of phenotype 1
---------------------------
Total Observed scale h2: 0.1054 (0.0383)
Lambda GC: 1.1523
Mean Chi^2: 1.2843
Intercept: 1.1234 (0.0607)
Ratio: 0.4342 (0.2134)

Heritability of phenotype 2/2
-----------------------------
Total Observed scale h2: 0.0543 (0.0211)
Lambda GC: 1.0833
Mean Chi^2: 1.1465
Intercept: 1.0583 (0.0335)
Ratio: 0.398 (0.2286)

Genetic Covariance
------------------
Total Observed scale gencov: 0.0121 (0.0106)
Mean z1*z2: -0.001
Intercept: -0.0198 (0.0121)

Genetic Correlation
-------------------
Genetic Correlation: 0.1601 (0.1821)
Z-score: 0.8794
P: 0.3792


Summary of Genetic Correlation Results
p1                    p2      rg      se       z       p  h2_obs  h2_obs_se  h2_int  h2_int_se  gcov_int  gcov_int_se
BBJ_HDLC.sumstats.gz  BBJ_LDLC.sumstats.gz  0.1601  0.1821  0.8794  0.3792  0.0543     0.0211  1.0583     0.0335   -0.0198       0.0121

Analysis finished at Thu Dec 29 21:02:47 2022
Total time elapsed: 10.39s

Partitioned LD regression

Partitioned LD regression is utilized to evaluate the contribution of each functional group to the total SNP heriatbility.

\[ E[\chi^2] = N \sum\limits_C \tau_C l(j,C) + Na + 1 \]

\(C\) : functional categories
\(l(j,C)\) : LD score for SNP j with respect to C.
\(\tau_C\) : per-SNP contribution of category C to heritability
Reference: Finucane, Hilary K., et al. "Partitioning heritability by functional annotation using genome-wide association summary statistics." Nature genetics 47.11 (2015): 1228-1235.

ldsc.py \
  --h2 BBJ_HDLC.sumstats.gz \
  --overlap-annot \
  --ref-ld-chr ~/tools/ldsc/resource/1000G_Phase3_EAS_baseline_v1_2_ldscores/baseline. \
  --frqfile-chr ~/tools/ldsc/resource/1000G_Phase3_EAS_plinkfiles/1000G.EAS.QC. \
  --w-ld-chr ~/tools/ldsc/resource/1000G_Phase3_EAS_weights_hm3_no_MHC/weights.EAS.hm3_noMHC. \
  --out BBJ_HDLC_baseline

Celltype specificity LD regression

LDSC-SEG : LD score regression applied to specifically expressed genes

An extension of Partitioned LD regression. Categories are defined by tissue or cell-type specific genes.

Reference: Finucane, Hilary K., et al. "Heritability enrichment of specifically expressed genes identifies disease-relevant tissues and cell types." Nature genetics 50.4 (2018): 621-629.

ldsc.py \
  --h2-cts BBJ_HDLC.sumstats.gz \
  --ref-ld-chr-cts ~/tools/ldsc/resource/Cahoy_EAS_1000Gv3_ldscores/Cahoy.EAS.ldcts \
  --ref-ld-chr ~/tools/ldsc/resource/1000G_Phase3_EAS_baseline_v1_2_ldscores/baseline. \
  --w-ld-chr ~/tools/ldsc/resource/1000G_Phase3_EAS_weights_hm3_no_MHC/weights.EAS.hm3_noMHC. \
  --out BBJ_HDLC_baseline_cts

Reference

Bulik-Sullivan, Brendan K., et al. "LD Score regression distinguishes confounding from polygenicity in genome-wide association studies." Nature genetics 47.3 (2015): 291-295.
Bulik-Sullivan, Brendan, et al. "An atlas of genetic correlations across human diseases and traits." Nature genetics 47.11 (2015): 1236-1241.
Finucane, Hilary K., et al. "Partitioning heritability by functional annotation using genome-wide association summary statistics." Nature genetics 47.11 (2015): 1228-1235.
Finucane, Hilary K., et al. "Heritability enrichment of specifically expressed genes identifies disease-relevant tissues and cell types." Nature genetics 50.4 (2018): 621-629.
Kanai, Masahiro, et al. "Genetic analysis of quantitative traits in the Japanese population links cell types to complex human diseases." Nature genetics 50.3 (2018): 390-400.

LD score regression

Table of Contents

Introduction

LD: Linkage disequilibrium

LD score

LD score regression

Install LDSC

Data Preparation

Download sample summary statistics

Download reference files

Munge sumstats

LD score regression

Cross-trait LD score regression

Partitioned LD regression

Celltype specificity LD regression

Reference