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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 the simulated single-SNP summary-level data example1.mat. Please contact me if you have trouble accessing this file.

The data file example1.mat contains the following elements.

  • betahat: 12758 by 1 vector, single-SNP effect size estimate for each SNP
  • se: 12758 by 1 vector, standard errors of the single-SNP effect size estimates
  • Nsnp: 12758 by 1 vector, sample size of each SNP
  • R: 12758 by 12758 matrix, LD matrix estimated from a reference panel
  • bwd: integer, bandwidth of the banded matrix R
  • BR: (bwd+1) by 12758 matrix, banded storage of matrix R

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   -1.5982

Since 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);
end

Recall 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.

example1_pve

More simulations

Simulations in Section 4.2 of Zhu and Stephens (2017) are essentially “replications” of the example above. To facilitate reproducible research, we make the simulated datasets for Scenarios 2.1 and 2.2 in Section 4.2 of Zhu and Stephens (2017) available (rss_example1_simulations.tar.gz)2.

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:

  1. The function prctile used in example1.m requires 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.

  2. 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.