Last updated: 2020-04-04
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Knit directory: causal-TWAS/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | f43fd46 | simingz | 2020-03-24 | fix results display bug |
| html | f43fd46 | simingz | 2020-03-24 | fix results display bug |
| Rmd | 84ad96a | simingz | 2019-11-25 | X tilde |
| html | 84ad96a | simingz | 2019-11-25 | X tilde |
| Rmd | eee8d19 | simingz | 2019-11-22 | simu20191111 |
| html | eee8d19 | simingz | 2019-11-22 | simu20191111 |
| 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 expression based on the following model:
\[ X = G\alpha + \xi\] 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.
We simulate phenotype following
\[Y = G\alpha\gamma + G\theta + \epsilon\].
For \(\alpha\), we used the same as used in simulating expression. \(\gamma \sim N(0,\sigma_\gamma^2), \theta \sim N(0, \sigma_\theta^2), \epsilon \sim N (0,1)\). To choose proper \(\sigma_\gamma\) and \(\sigma_\theta\). We used the following formula:
\[\begin{align} PVE_{SNP} = &\frac{var(G\theta)}{var(G\theta) + var(\tilde{X}\gamma) + \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + \Sigma_jvar(\tilde{X_j})\gamma_j^2+ \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}\\ \end{align}\]
\[ PVE_{expr} \approx \frac{Jvar(\tilde{X})\sigma_\gamma^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}\]
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/"
load("/home/simingz/causalTWAS/WTCCC/bd.toy.RData")
Gvar <- mean(apply(X,2,var))
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/Gvar , 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 0.4844631 0.576745 0.1373830
sigma_theta^2 0.4365506 0.206319 0.2265750
sigma_e^2 1.0000000 1.048360 0.0723783
est est.se
PVE.expr 0.1150480 0.0274049
PVE.snp 0.0527935 0.0579769
PVE.total 0.1678420 0.0574520show_res("S1.2") truth est est.se
sigma_gamma^2 0.7751409 1.02284 0.229406
sigma_theta^2 2.7939238 1.50579 0.393121
sigma_e^2 1.0000000 1.27095 0.128254
est est.se
PVE.expr 0.127666 0.0286335
PVE.snp 0.241090 0.0629421
PVE.total 0.368756 0.0637001show_res("S2.1") truth est est.se
sigma_gamma^2 0.4844631 0.423657 0.129243
sigma_theta^2 0.4365506 0.138791 0.205555
sigma_e^2 1.0000000 1.061920 0.064932
est est.se
PVE.expr 0.0877619 0.0267731
PVE.snp 0.0368805 0.0546217
PVE.total 0.1246420 0.0535243show_res("S2.2") truth est est.se
sigma_gamma^2 0.7751409 0.644789 0.209481
sigma_theta^2 2.7939238 1.510700 0.363672
sigma_e^2 1.0000000 1.307530 0.115167
est est.se
PVE.expr 0.0828177 0.0269061
PVE.snp 0.2489020 0.0599185
PVE.total 0.3317200 0.0588617show_res("S3.1") truth est est.se
sigma_gamma^2 0.4844631 0.4286100 0.1308600
sigma_theta^2 0.4365506 0.0860036 0.2247160
sigma_e^2 1.0000000 1.1093200 0.0704246
est est.se
PVE.expr 0.0865320 0.0264194
PVE.snp 0.0222729 0.0581962
PVE.total 0.1088050 0.0565770show_res("S3.2") truth est est.se
sigma_gamma^2 0.7751409 0.710255 0.220093
sigma_theta^2 2.7939238 1.592890 0.386914
sigma_e^2 1.0000000 1.339000 0.122368
est est.se
PVE.expr 0.0878836 0.0272333
PVE.snp 0.2528280 0.0614121
PVE.total 0.3407110 0.0602505show_res("S4.1") truth est est.se
sigma_gamma^2 0.4844631 0.4104640 0.1367470
sigma_theta^2 0.4365506 0.0964066 0.2334440
sigma_e^2 1.0000000 1.1115300 0.0738127
est est.se
PVE.expr 0.0828015 0.0275855
PVE.snp 0.0249469 0.0604076
PVE.total 0.1077480 0.0592509show_res("S4.2") truth est est.se
sigma_gamma^2 0.7751409 0.653871 0.224143
sigma_theta^2 2.7939238 1.958680 0.438850
sigma_e^2 1.0000000 1.191170 0.142630
est est.se
PVE.expr 0.0827021 0.0283498
PVE.snp 0.3177850 0.0712011
PVE.total 0.4004870 0.0717854
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_1.6.0 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