Last updated: 2019-11-22
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Knit directory: causal-TWAS/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | dd2b4ab | simingz | 2019-11-19 | simulate phenotype |
| html | dd2b4ab | simingz | 2019-11-19 | simulate phenotype |
| 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.
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.
\[\begin{align} PVE_{SNP} = &\frac{var(G\theta)}{var(G\theta) + var(\tilde{X}\gamma) + \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + \Sigma_jvar(\tilde{X_j})\gamma_j^2+ \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}\\ \end{align}\]
\[ PVE_{expr} \approx \frac{Jvar(\tilde{X})\sigma_\gamma^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}\]
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)
}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.
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 0.0643964show_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 0.0687856show_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 0.0617994show_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 0.0651690show_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 0.0651462show_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 0.0716856show_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 0.0634763show_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