simulation-WTCCC

Simulate cis-expression

Simulate quantitative phenotype

Last updated: 2019-11-07

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Use WTCCC data obtained from Peter on CRI.

Simulate cis-expression

We simulate cis- expression based on the following model:

$\tilde{X} = G\alpha + \xi$ For each gene, we sample $K$ eQTLs for this gene, for eQTL $k$ , we have $\alpha_k \sim N(0,\sigma_\alpha^2)$ $\xi \sim N(0,1)$ Then we have the heritability of the gene:

$\begin{aligned} h^2_{eQTL} &= \frac{var(G\alpha)}{var(G\alpha)+var(\xi)} \\ &= \frac{\Sigma_kvar(G_k) \alpha_k^2}{\Sigma_kvar(G_k) \alpha_k^2 + var(\xi)}\\ &= \frac{\Sigma_k\alpha_k^2}{\Sigma_k\alpha_k^2 + var(\xi)}\\ &\approx \frac{\Sigma_k\sigma_\alpha^2}{\Sigma_k\sigma_\alpha^2 + var(\xi)} \\ &=\frac{K\sigma_\alpha^2}{1+K\sigma_\alpha^2} \end{aligned}$

Here, we use scaled genotype data, so $var(G)=1$ . We also have $\alpha^2_k \approx E(\alpha_k^2)=var(\alpha_k^2)=\sigma_\alpha^2$ . Usually, $K$ ranges from 1 to 5, let $\sigma_\alpha = 0.3$ , then $h^2_{eQTL}$ ranges from 0.08 to 0.31, this is consistent with gene cis-heritability in reality.

Simulate quantitative phenotype

Simulate different $PVE_{SNP}$ , $PVE_{gene}$ .

$PVE_{SNP} = \frac{M\sigma_\theta^2}{J\sigma_\gamma^2 + M\sigma_\theta^2 + \sigma_e^2}$ $PVE_{gene} = \frac{J\sigma_\gamma^2}{J\sigma_\gamma^2 + M\sigma_\theta^2 + \sigma_e^2}$ Scenario 1: $PVE_{SNP}=0.2$ , $PVE_{gene}=0.2$

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)

Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.4.0 Rcpp_1.0.0      digest_0.6.18   rprojroot_1.3-2
 [5] backports_1.1.2 git2r_0.25.2    magrittr_1.5    evaluate_0.12  
 [9] stringi_1.3.1   fs_1.3.1        whisker_0.3-2   rmarkdown_1.10 
[13] tools_3.5.1     stringr_1.4.0   glue_1.3.0      yaml_2.2.0     
[17] compiler_3.5.1  htmltools_0.3.6 knitr_1.20