Simulation-WTCCC

Simulate cis-expression

Simulate quantitative phenotype

Simulation 1: $M$ = No. of all SNPs, $J$ = No. all genes

1.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$
1.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

Simulation 2: $M$ = No. of all SNPs, $J$ = ~10% all genes (J=2000)

2.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$
2.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

Simulation 3: $M$ = ~ 10% all SNPs (M=50000), $J$ = No. all genes

3.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$
3.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

Simulation 4: $M$ = ~ 10% all SNPs (M=50000), $J$ = ~10% all genes (J=2000)

4.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$
4.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

Last updated: 2019-11-18

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Rmd	28a299b	simingz	2019-11-17	PVE
html	28a299b	simingz	2019-11-17	PVE
Rmd	e5e3011	simingz	2019-11-11	simulate expression
html	e5e3011	simingz	2019-11-11	simulate expression
Rmd	dcd5252	simingz	2019-11-07	simulate description
html	dcd5252	simingz	2019-11-07	simulate description
Rmd	f7be05a	simingz	2019-11-05	gemma

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 $L$ 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

$\begin{align} PVE_{SNP} = &\frac{var(G\theta)}{var(G\theta) + var(\tilde{X}\gamma) + \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + \Sigma_jvar(\tilde{X_j})\gamma_j^2+ \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}\\ \end{align}$

$PVE_{expr} \approx \frac{Jvar(\tilde{X})\sigma_\gamma^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}$

Here, $var(\tilde{X})$ is the cis-heratbility of gene expression. $M$ and $J$ are number of causal SNP and gene respectively.

Thus in order to get desired $PVE_{SNP}$ and $PVE_expr$ , we set $\sigma_\theta^2$ and $\sigma_\gamma^2$ based on the following formula:

$\sigma_\theta^2 = \frac{PVE_{SNP}}{M(1-PVE_{SNP} - PVE_{expr})}$ $\sigma_\gamma^2 = \frac{PVE_{expr}}{Jvar(\tilde{X})(1-PVE_{SNP} - PVE_{expr})}$

Simulation 1: $M$ = No. of all SNPs, $J$ = No. all genes

We simulate quantitative phenotype under several scenarios. First we make all genes with imputed gene expression as causal genes and all SNPs as causal SNPs. This matches our polygenic version of the model.

1.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

We set $\sigma_e=1$ .

1.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

We set $\sigma_e=1$ .

Simulation 2: $M$ = No. of all SNPs, $J$ = ~10% all genes (J=2000)

2.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

We set $\sigma_e=1$ .

2.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

We set $\sigma_e=1$ .

Simulation 3: $M$ = ~ 10% all SNPs (M=50000), $J$ = No. all genes

3.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

We set $\sigma_e=1$ .

3.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

We set $\sigma_e=1$ .

Simulation 4: $M$ = ~ 10% all SNPs (M=50000), $J$ = ~10% all genes (J=2000)

4.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

We set $\sigma_e=1$ .

4.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

We set $\sigma_e=1$ .

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_1.5.0 Rcpp_1.0.0      digest_0.6.18   later_0.7.5    
 [5] rprojroot_1.3-2 R6_2.3.0        backports_1.1.2 git2r_0.26.1   
 [9] magrittr_1.5    evaluate_0.12   stringi_1.3.1   fs_1.3.1       
[13] promises_1.0.1  whisker_0.3-2   rmarkdown_1.10  tools_3.5.1    
[17] stringr_1.4.0   glue_1.3.0      httpuv_1.4.5    yaml_2.2.0     
[21] compiler_3.5.1  htmltools_0.3.6 knitr_1.20

Simulation-WTCCC

Simulate cis-expression

Simulate quantitative phenotype

Simulation 1: MM = No. of all SNPs, JJ = No. all genes

1.1 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.1PVE_{expr}=0.1

1.2 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.4PVE_{expr}=0.4

Simulation 2: MM = No. of all SNPs, JJ = ~10% all genes (J=2000)

2.1 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.1PVE_{expr}=0.1

2.2 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.4PVE_{expr}=0.4

Simulation 3: MM = ~ 10% all SNPs (M=50000), JJ = No. all genes

3.1 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.1PVE_{expr}=0.1

3.2 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.4PVE_{expr}=0.4

Simulation 4: MM = ~ 10% all SNPs (M=50000), JJ = ~10% all genes (J=2000)

4.1 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.1PVE_{expr}=0.1

4.2 PVESNP=0.1PVE_{SNP}=0.1, PVEexpr=0.4PVE_{expr}=0.4

Simulation 1: $M$ = No. of all SNPs, $J$ = No. all genes

1.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

1.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

Simulation 2: $M$ = No. of all SNPs, $J$ = ~10% all genes (J=2000)

2.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

2.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

Simulation 3: $M$ = ~ 10% all SNPs (M=50000), $J$ = No. all genes

3.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

3.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$

Simulation 4: $M$ = ~ 10% all SNPs (M=50000), $J$ = ~10% all genes (J=2000)

4.1 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.1$

4.2 $PVE_{SNP}=0.1$ , $PVE_{expr}=0.4$