Simulation-WTCCC

Simulate cis-expression

Simulate quantitative phenotype

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

1.1 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$
1.2 $PVE_{SNP}\approx 0.1$ , $PVE_{expr}\approx 0.4$

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

2.1 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$
2.2 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.4$

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

3.1 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$
3.2 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.4$

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

4.1 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$
4.2 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.4$

Last updated: 2019-11-22

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Rmd	28a299b	simingz	2019-11-17	PVE
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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 numbers 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:

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

simudir <- "/home/simingz/causalTWAS/simulations/simulation_WTCCC_20191111/"
show_res <- function(simutag){
  load(Sys.glob(paste0(simudir,simutag,"*phenotype.Rd")))
  outdf1 <- data.frame(truth = c(sigma_gamma^2*J.c, sigma_theta^2*M.c , 1))
  row.names(outdf1) <- c("sigma_gamma^2","sigma_theta^2","sigma_e^2")
  res <- readLines(paste0(simudir, simutag, "_VC/output/", simutag, ".log.txt"))
  outdf1$est <- as.numeric(strsplit(res[24], "  ")[[1]][2:4])
  outdf1$est.se <- as.numeric(strsplit(res[25], "  ")[[1]][2:4])
  print(outdf1)
  outdf2 <- data.frame("est"= c(as.numeric(strsplit(res[20], "  ")[[1]][2:3]), as.numeric(strsplit(res[22], "=")[[1]][2])))
  row.names(outdf2) <- c("PVE.expr","PVE.snp","PVE.total")
  outdf2$est.se <- c(as.numeric(strsplit(res[21], "  ")[[1]][2:3]), as.numeric(strsplit(res[23], "=")[[1]][2]))
  print(outdf2)
}

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 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$

show_res("S1.1")

                  truth      est    est.se
sigma_gamma^2 0.1005991 0.122543 0.0395220
sigma_theta^2 0.1250000 0.273277 0.2092710
sigma_e^2     1.0000000 1.007600 0.0804118
                est    est.se
PVE.expr  0.1225340 0.0395189
PVE.snp   0.0705488 0.0540252
PVE.total 0.1930830 0.0643964

1.2 $PVE_{SNP}\approx 0.1$ , $PVE_{expr}\approx 0.4$

show_res("S1.2")

                  truth      est    est.se
sigma_gamma^2 0.6438345 0.659297 0.0715076
sigma_theta^2 0.2000000 0.666863 0.3450190
sigma_e^2     1.0000000 0.975871 0.1385370
               est    est.se
PVE.expr  0.408730 0.0443310
PVE.snp   0.106737 0.0552229
PVE.total 0.515467 0.0687856

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

2.1 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$

show_res("S2.1")

                  truth       est    est.se
sigma_gamma^2 0.1005991 0.0878109 0.0386394
sigma_theta^2 0.1250000 0.0131843 0.1975690
sigma_e^2     1.0000000 1.1136600 0.0758622
                 est    est.se
PVE.expr  0.08931650 0.0393019
PVE.snp   0.00346228 0.0518827
PVE.total 0.09277880 0.0617994

2.2 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.4$

show_res("S2.2")

                  truth       est    est.se
sigma_gamma^2 0.6438345  0.609504 0.0687422
sigma_theta^2 0.2000000 -0.271802 0.3082090
sigma_e^2     1.0000000  1.240020 0.1246970
                 est    est.se
PVE.expr   0.3977290 0.0448574
PVE.snp   -0.0457914 0.0519251
PVE.total  0.3519380 0.0651690

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

3.1 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$

show_res("S3.1")

                  truth      est    est.se
sigma_gamma^2 0.1005991 0.107247 0.0382940
sigma_theta^2 0.1250000 0.626345 0.2070430
sigma_e^2     1.0000000 0.880455 0.0792357
               est    est.se
PVE.expr  0.110097 0.0393119
PVE.snp   0.166007 0.0548750
PVE.total 0.276105 0.0651462

3.2 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.4$

show_res("S3.2")

                  truth      est    est.se
sigma_gamma^2 0.6438345 0.702810 0.0720651
sigma_theta^2 0.2000000 0.968523 0.3470150
sigma_e^2     1.0000000 0.788533 0.1418140
               est    est.se
PVE.expr  0.443583 0.0454843
PVE.snp   0.157822 0.0565465
PVE.total 0.601405 0.0716856

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

4.1 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.1$

show_res("S4.1")

                  truth     est    est.se
sigma_gamma^2 0.1005991 0.12410 0.0409952
sigma_theta^2 0.1250000 0.33148 0.2124450
sigma_e^2     1.0000000 1.02347 0.0815844
                est    est.se
PVE.expr  0.1205590 0.0398255
PVE.snp   0.0831393 0.0532839
PVE.total 0.2036980 0.0634763

4.2 $PVE_{SNP} \approx 0.1$ , $PVE_{expr} \approx 0.4$

show_res("S4.2")

                  truth      est    est.se
sigma_gamma^2 0.6438345 0.703181 0.0755506
sigma_theta^2 0.2000000 0.617824 0.3450340
sigma_e^2     1.0000000 1.008390 0.1414000
               est    est.se
PVE.expr  0.420990 0.0452316
PVE.snp   0.095497 0.0533320
PVE.total 0.516487 0.0678001

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} \approx 0.1, PVEexpr≈0.1PVE_{expr} \approx 0.1

1.2 PVESNP≈0.1PVE_{SNP}\approx 0.1, PVEexpr≈0.4PVE_{expr}\approx 0.4

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

2.1 PVESNP≈0.1PVE_{SNP} \approx 0.1, PVEexpr≈0.1PVE_{expr} \approx 0.1

2.2 PVESNP≈0.1PVE_{SNP} \approx 0.1, PVEexpr≈0.4PVE_{expr} \approx 0.4

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

3.1 PVESNP≈0.1PVE_{SNP} \approx 0.1, PVEexpr≈0.1PVE_{expr} \approx 0.1

3.2 PVESNP≈0.1PVE_{SNP} \approx 0.1, PVEexpr≈0.4PVE_{expr} \approx 0.4

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

4.1 PVESNP≈0.1PVE_{SNP} \approx 0.1, PVEexpr≈0.1PVE_{expr} \approx 0.1

4.2 PVESNP≈0.1PVE_{SNP} \approx 0.1, PVEexpr≈0.4PVE_{expr} \approx 0.4