Baseline results of Zhu and Stephens (2017)

What is “baseline model”?

The baseline multiple-SNP model considered here consists of a multiple regression likelihood based on single-SNP association summary statistics (Zhu and Stephens, 2017) and a previously proposed spike-and-slab prior (Guan and Stephens, 2011). Specifically, the model has the following form:

\[ \begin{aligned} \widehat{\beta} &\sim \mbox{Normal}~\left(\widehat{S}\widehat{R}\widehat{S}^{-1}\beta,~\widehat{S}\widehat{R}\widehat{S}\right),\\ \beta_j &\sim \pi\cdot \mbox{Normal}~\left(0,~\sigma_\beta^2\right)+(1-\pi)\cdot \delta_0,\\ \sigma_\beta^2 &= h \cdot \left(\sum_{j=1}^p \pi n^{-1}\hat{s}_j^{-2}\right)^{-1},\\ \pi_j &= \left(1+10^{-\theta_0}\right)^{-1}, \end{aligned} \]

where \(\widehat{\beta}:=(\hat{\beta}_1,\ldots,\hat{\beta}_p)^\intercal\), \(\widehat{S}:=\mbox{diag}(\hat{s})\), \(\hat{s}:=(\hat{s}_1,\ldots, \hat{s}_p)^\intercal\), \(\{\hat{\beta}_j,\hat{s}_j\}\) are single-SNP association summary data, \(\widehat{R}\) is the LD matrix estimated from an external reference panel (Wen and Stephens, 2010), and \(\beta:=(\beta_1,\ldots,\beta_p)^\intercal\) are the true genetic effects of SNPs on a target trait under the multiple-SNP model.

What are “Round 1 & 2”?

For each trait, we fit the baseline model to its GWAS summary statistics twice:

• “Round 1” analysis, using a wide and coarse grid of hyperparameters \(\{h,\theta_0\}\);
• “Round 2” analysis, using a narrow and fine grid of \(\{h,\theta_0\}\) (informed by Round 1 analysis).

For details of these grids, please see Supplementary Table 3 of the corresponding manuscript.

What is “locus”?

To summarize the locus-level results of baseline model fitting, we consider two ways to define “locus”.

1. Divide the entire genome into overlapped loci of 50 SNPs (with an overlap of 25 SNPs between neighboring loci). This approach is used in Carbonetto and Stephens (2013). We term this approach “50-SNP loci”.

2. Download gene definitions from Homo sapiens reference genome GRCh37, and then define the transcribed region \(\pm\) 100 kb of each gene as a locus. We term this approach “genes”.

What are abz, p[n]p and ens?

For a given locus, we summarize the association signal via three statistics: abz, p[n]p and ens.

1. abz: the maximum absolute single-SNP \(z\)-score in the locus \[\max_{j\in\mbox{locus}}|z_j|\]

2. p[n]p: the posterior probability that the locus contains at least \(n\) trait-associated SNPs \[\mbox{Pr}(\#\{j:\beta_j\neq 0\}\geq n~|~\mbox{Data})\]

3. ens: the posterior expected number of trait-associated SNPs in the locus \[\sum_{j\in\mbox{locus}}\mbox{Pr}(\beta_j\neq 0~|~\mbox{Data})\]

Note that abz is a direct summary of single-SNP GWAS summary statistics, whereas p[n]p and ens depend on the multiple-SNP baseline model fitting.

Anthropometric traits

• Adult human height (Wood et al. 2014)’
• Body mass index (Locke et al. 2015)
• Waist-to-hip ratio (Shungin et al. 2015)

Hematopoietic traits

• Haemoglobin (Harst et al. 2012)
• Mean cell haemoglobin (Harst et al. 2012)
• Mean cell haemoglobin concentration (Harst et al. 2012)
• Mean cell volume (Harst et al. 2012)
• Packed cell volume (Harst et al. 2012)
• Red blood cell count (Harst et al. 2012)

Immune-related traits

• Rheumatoid arthritis (Okada et al. 2014)
• Inflammatory bowel disease (Liu et al. 2015)
• Crohn’s disease (Liu et al. 2015)
• Ulcerative colitis (Liu et al. 2015)

Metabolic traits

• High-density lipoprotein (Teslovich et al. 2010)
• Low-density lipoprotein (Teslovich et al. 2010)
• Total cholesterol (Teslovich et al. 2010)
• Triglycerides (Teslovich et al. 2010)
• Type 2 diabetes (Morris et al. 2012)
• Fasting glucose (Manning et al. 2012)
• Fasting insulin (Manning et al. 2012)
• Heart rate (Den Hoed et al. 2013)
• Gout (Köttgen et al. 2013)
• Serum urate (Köttgen et al. 2013)
• Age at natural menopause (Day et al. 2015)
• Coronary artery disease (Nikpay et al. 2015)
• Myocardial infarction (Nikpay et al. 2015)

Neurological traits

• Schizophrenia (Ripke et al. 2014)
• Alzheimer’s disease (Lambert et al. 2013)
• Amyotrophic lateral sclerosis (Rheenen et al. 2016)
• Depressive symptoms (Okbay et al. 2016)
• Neuroticism (Okbay et al. 2016)