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Markov chain Monte Carlo (MCMC)

Details of MCMC algorithms for rss_bvsr.m and rss_bslmm.m are available in Supplementary Appendix B of Zhu and Stephens (2017) Details of MCMC algorithms for rss_ash.m are available in this unpublished note1. Note that only rss_bvsr.m and rss_bslmm.m were used to generate results in Zhu and Stephens (2017).

rss_bvsr.m

Fit the RSS-BVSR model that consists of RSS likelihood and BVSR prior (Guan and Stephens, 2011):

\[ \begin{aligned} \widehat{\boldsymbol\beta} &\sim {\cal N}({\bf SRS}^{-1}{\boldsymbol\beta},{\bf SRS}),\\ \beta_j &\sim \pi\cdot{\cal N}(0,\sigma_B^2) + (1-\pi)\cdot\delta_0, \end{aligned} \]

using a Metropolis-Hastings algorithm.

rss_bslmm.m

Fit the RSS-BSLMM model that consists of RSS likelihood and BSLMM prior (Zhou et al, 2013):

\[ \begin{aligned} \widehat{\boldsymbol\beta} &\sim {\cal N}({\bf SRS}^{-1}{\boldsymbol\beta},{\bf SRS}),\\ \beta_j &\sim \pi\cdot{\cal N}(0,\sigma_B^2+\sigma_P^2) + (1-\pi)\cdot{\cal N}(0,\sigma_P^2), \end{aligned} \]

using a component-wise MCMC algorithm.

rss_ash.m

Fit the RSS-ASH model that consists of RSS likelihood and ASH prior (Stephens, 2017):

\[ \begin{aligned} \widehat{\boldsymbol\beta} &\sim {\cal N}({\bf SRS}^{-1}{\boldsymbol\beta},{\bf SRS}),\\ \beta_j &\sim \pi_0 \cdot \delta_0 + {\textstyle\sum}_{k=1}^K \pi_k \cdot {\cal N}(0,\sigma_k^2), \end{aligned} \]

using a component-wise MCMC algorithm.

Variational Bayes (VB)

Details of VB algorithms, SQUAREM accelerator and parallel implementation are available in Supplementary Notes of Zhu and Stephens (2018). Note that only rss_varbvsr_squarem.m and rss_varbvsr_bigmem_squarem.m were used to generate results in Zhu and Stephens (2018). The other functions were developed merely for testing and benchmarking.

rss_varbvsr.m

Fit the following extended RSS-BVSR model

\[ \begin{aligned} \widehat{\boldsymbol\beta} &\sim {\cal N}({\bf SRS}^{-1}{\boldsymbol\beta},{\bf SRS}),\\ \beta_j &\sim \pi_j\cdot{\cal N}(0,\sigma_j^2) + (1-\pi_j)\cdot\delta_0, \end{aligned} \]

using a mean-field VB algorithm. The VB algorithm largely follows Carbonetto and Stephens (2012). This is an extended RSS-BVSR model because each SNP \(j\) can have its own hyper-parameters \(\{\pi_j,\sigma_j^2\}\), whereas the standard RSS-BVSR model assumes that all SNPs share the same hyper-parameters \(\{\pi,\sigma_B^2\}\).

rss_varbvsr_squarem.m

This is a variant of rss_varbvsr.m with the SQUAREM accelerator (Varadhan and Roland, 2008) added.

rss_varbvsr_parallel.m

This is a parallel implementation of rss_varbvsr.m based on MATLAB Parallel Computing Toolbox.

rss_varbvsr_pasquarem.m

This is a parallel implementation of rss_varbvsr_squarem.m based on MATLAB Parallel Computing Toolbox.

rss_varbvsr_bigmem.m

This is a memory-efficient implementation of rss_varbvsr_parallel.m.

rss_varbvsr_bigmem_squarem.m

This is a memory-efficient implementation of rss_varbvsr_pasquarem.m.

Miscellaneous

import_1000g_vcf.sh

Output 1000 Genomes phased haplotypes of a given list of SNPs in IMPUTE reference-panel format.

compute_pve.m

Use GWAS summary data to estimate PVE (or SNP heritability), a quantity defined by Equation 2.10 in Guan and Stephens (2011). This function corresponds to Equation 3.7 in Zhu and Stephens (2017).

band_storage.m

Convert a symmetric, banded matrix to a compact matrix in such a way that only the main diagonal and the nonzero super-diagonals are stored. This function is used to reduce the file size of a large LD matrix.

find_bandwidth.m

Find the bandwidth of a symmetric, banded matrix.

get_corr.m

Compute linkage disequilibrium (LD) matrix using the shrinkage estimator proposed in Wen and Stephens (2010). This function is also implemented in an R package ldshrink.

data_maker.m

Simulate phenotype data from the genome-wide multiple-SNP model described in Zhou et al (2013), and then compute the single-SNP summary statistics for each SNP. This function was used in some simulation studies of Zhu and Stephens (2017).

enrich_datamaker.m

Simulate phenotype data from the genetic association enrichment model described in Carbonetto and Stephens (2013), and then compute the single-SNP summary statistics for each SNP. This function was used in some simulation studies of Zhu and Stephens (2018).

null_single.m & null_template.m

Fit genome-wide multiple-SNP “baseline model” to single-SNP summary data, using rss/src_vb functions. These scripts were used in data analyses of Zhu and Stephens (2018).

gsea_wrapper.m & gsea_template.m

Fit genome-wide multiple-SNP “enrichment model” to single-SNP summary data, using rss/src_vb functions. These scripts were used in data analyses of Zhu and Stephens (2018).

null_wrapper_fixsb.m & gsea_wrapper_fixsb.m

Fit genome-wide multiple-SNP “baseline model” and “enrichment model” to single-SNP summary data, using a fixed prior variance of causal genetic effects (\(\sigma_B^2\)) in rss/src_vb functions. These scripts were used in simulation studies of Zhu and Stephens (2018).

ash_lrt_31traits.R

Compute a simple likelihood ratio as a sanity check for the more complicated enrichment analysis method developed in Zhu and Stephens (2018). This likelihood ratio calculation is based on an R package ashr.

Footnotes:

  1. There is a missing multiplier term \({\omega_k}/{\omega_1}\) on the right-hand side of Equation 5.1 in this note (page 4). However, this missing term was correctly reflected in the source codes of rss_ash.m, in particular, Lines 80-81 of update_bz.m. We thank Geyu Zhou for pointing out this typo on 01/14/2019.