Example 1: RSS Model Fitting via MCMC & SNP Heritability Estimation
Xiang Zhu
Last updated: 2024-09-16
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Knit directory: rss/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
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| html | bab3f58 | Xiang Zhu | 2020-06-24 | Build site. |
| html | 3488014 | Xiang Zhu | 2020-06-23 | Build site. |
| Rmd | 221b7e3 | Xiang Zhu | 2020-06-23 | wflow_publish("rmd/example_1.Rmd") |
Overview
This example illustrates how to fit RSS models using MCMC algorithms. Three types of prior distributions are considered: BVSR in Guan and Stephens (2011), BSLMM in Zhou, Carbonetto and Stephens (2013), and ASH in Stephens (2017). The MCMC output is further used to estimate the SNP heritability. This example is closely related to Section 4.2 of Zhu and Stephens (2017).
The single-SNP summary-level data are computed from a simulated GWAS dataset. The GWAS data are simulated under the Scenario 2.1 in Zhu and Stephens (2017). Specifically, 100 “causal” SNPs are randomly drawn from 12758 SNPs on chromosome 16, with effect sizes coming from standard normal \({\cal N}(0,1)\). Effect sizes of remaining SNPs are zero. The true PVE (SNP heritability) is 0.2.
To reproduce results of Example 1, please read the step-by-step guide
below and run example1.m.
Before running example1.m,
please first install the MCMC
subroutines. Please find installation instructions here.
Step-by-step illustration
Step 1. Download data files.
All data files required to run this example are freely available at
Zenodo 
. Please contact me if you have trouble accessing this
file. After a complete download, you should see the following files.
$ tree ./
./
├── example1.mat
├── example1_rssash.mat
├── example1_rssbslmm.mat
├── example1_rssbvsr.mat
└── readme
0 directories, 5 filesThe data file example1.mat contains the following
elements.
betahat: 12758 by 1 vector, single-SNP effect size estimate for each SNPse: 12758 by 1 vector, standard errors of the single-SNP effect size estimatesNsnp: 12758 by 1 vector, sample size of each SNPR: 12758 by 12758 matrix, LD matrix estimated from a reference panelbwd: integer, bandwidth of the banded matrixRBR: (bwd+1) by 12758 matrix, banded storage of matrixR
Note that only betahat, se,
Nsnp and R are needed for running MCMC. The
other two quantities, bwd and BR, are only
used in SNP heritability calculation.
Step 2. Check the “small effects” model assumption.
Using single-SNP summary data, we can compute squared sample
correlation between phenotype and each SNP, and then check the “small
effect” assumption by looking at these marginal squared correlation
values (please see Table 1 of Zhu and Stephens
(2017) for more details). The following code illustrates the “small
effect” check in example1.m1.
>> chatsqr = (betahat(:).^2) ./ (Nsnp(:).*(se(:).^2) + betahat(:).^2);
>> disp(prctile(log10(chatsqr), 0:25:100));
-11.6029 -4.1154 -3.4721 -2.9962 -1.5982Since our data are generated from genotypes of a single chromosome, the simulated effect sizes per SNP are larger than would be expected in a typical GWAS (see Table 1 Zhu and Stephens (2017)).
Step 3. Fit RSS-BVSR, RSS-BSLMM and RSS-ASH models via MCMC.
To fit RSS-BVSR and RSS-BSLMM models, we only need to specify the length of Markov chains for RSS software.
Ndraw = 2e6;
Nburn = 2e5;
Nthin = 9e1;
[betasam, gammasam, hsam, logpisam, Naccept] = rss_bvsr(betahat, se, R, Nsnp, Ndraw, Nburn, Nthin);
[bsam, zsam, lpsam, hsam, rsam, Naccept] = rss_bslmm(betahat, se, R, Nsnp, Ndraw, Nburn, Nthin);To fit RSS-ASH model, we need to specify the length of Markov chain and a grid for the prior standard deviations of multiple-SNP effect sizes.
Ndraw = 5e7;
Nburn = 1e7;
Nthin = 1e3;
sigma_beta = [0 0.001 0.003 0.01 0.03 0.1 0.3 1 3];
[bsam, zsam, wsam, lsam, Naccept] = rss_ash(betahat, se, R, Nsnp, sigma_beta, Ndraw, Nburn, Nthin);Step 4. Estimate SNP heritability.
We use the posterior sample of multiple-SNP effect sizes
(bsam) to obtain the posterior sample of SNP heritability
(pvesam).
M = length(hsam); % the length of posterior simulations
pvesam = zeros(M,1); % preallocate the pve posterior estimates
for i = 1:M
pvesam(i) = compute_pve(bsam(i,:), betahat, se, Nsnp, bwd, BR, 1);
endRecall that the SNP heritability estimator (Equation 3.8 of Zhu and Stephens
(2017)) involves vector-matrix-vector product. To speed calculation,
we exploit the banded structure of R and use the banded
version of vector-matrix-vector product implemented in
lapack. Hence, the banded storage BR, instead
of the original form R, is used to calculate SNP
heritability.
Step 5. Summarize results.
The dataset is simulated with the true SNP heritability (PVE) being 0.2. The following table summarizes the posterior estimates (with 95% credible interval) and the total computational time (including MCMC iterations and PVE calculations) for three models.
| Model | PVE estimation | Total time |
|---|---|---|
| RSS-BVSR | 0.200 [0.125, 0.290] | 1.38 hours |
| RSS-BSLMM | 0.216 [0.136, 0.306] | 2.52 hours |
| RSS-ASH | 0.197 [0.114, 0.286] | 6.69 hours |
The following histograms show the posterior distributions of estimated SNP heritability under these three models.
More simulations
Simulations in Section 4.2 of Zhu and Stephens
(2017) are essentially “replications” of the example above. The
simulated datasets for Scenarios 2.1 and 2.2 in Section 4.2 of Zhu and Stephens
(2017) are available as
rss_example1_simulations.tar.gz2.
Each simulated dataset contains three files:
genotype.txt, phenotype.txt and
simulated_data.mat. The files genotype.txt and
phenotype.txt are the genotype and phenotype files for GEMMA. The
file simulated_data.mat contains three cells.
true_para = {pve, beta, gamma, sigma};
individual_data = {y, X};
summary_data = {betahat, se, Nsnp};Only the summary_data cell above is used as the input
for RSS methods.
The RSS methods also require an estimated LD matrix R.
This matrix R is provided in the file
genotype.mat.
After applying RSS methods to these simulated data, we obtain the following PVE estimation results.
| Scenario 2.1 (sparse), True PVE = 0.2 | Scenario 2.1 (sparse), True PVE = 0.6 |
|---|---|
 |
 |
| Scenario 2.2 (polygenic), True PVE = 0.2 | Scenario 2.2 (polygenic), True PVE = 0.6 |
|---|---|
 |
 |
Footnotes:
The function
prctileused inexample1.mrequires the Statistics and Machine Learning Toolbox. Please see this commit (courtesy of Dr. Peter Carbonetto) if this required toolbox is not available in your environment.Currently these files are locked, since they contain individual-level genotypes from Wellcome Trust Case Control Consortium (WTCCC, https://www.wtccc.org.uk/). You need to get permission from WTCCC before we can share these files with you.
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