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 | 5a4a588 | simingz | 2020-03-13 | v1.0 simulate expression using separate data 20191127 |
| html | 5a4a588 | simingz | 2020-03-13 | v1.0 simulate expression using separate data 20191127 |
Use WTCCC data obtained from Peter on CRI.
The difference between this analysis and previous run: used 500 samples from cad.RData from WTCCC to simulate expression, then train expression model using cv.glmnet. We impute expression in using bd.RData from WTCCC.
Simulation of phenotype: use the same weights as used in simulate expression using the 'cad.RData dataset, simulate phenotype according to:
\[Y = G\alpha\gamma + G\theta + \epsilon\]. For \(\alpha\), we used the same as used in simulating expression (Thus, not \(\tilde{X}\)). \(\gamma \sim N(0,\sigma_\gamma^2), \theta \sim N(0, \sigma_\theta^2), \epsilon \sim N (0,1)\).
The genotype data and other settings are the same.
We simulate expression based on the following model, using 500 samples from cad in WTCCC:
\[ X = G\alpha + \xi\] 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 using bd data from WTCCC following \[Y = G\alpha\gamma + G\theta + \epsilon\]. For \(\alpha\), we used the same as used in simulating expression. \(\gamma \sim N(0,\sigma_\gamma^2), \theta \sim N(0, \sigma_\theta^2), \epsilon \sim N (0,1)\).
\[\begin{align} PVE_{SNP} = &\frac{var(G\theta)}{var(G\theta) + var(\tilde{X}\gamma) + \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + \Sigma_jvar(\tilde{X_j})\gamma_j^2+ \sigma_e^2}\\ \approx &\frac{M\sigma_\theta^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}\\ \end{align}\]
\[ PVE_{expr} \approx \frac{Jvar(\tilde{X})\sigma_\gamma^2}{M\sigma_\theta^2 + Jvar(\tilde{X})\sigma_\gamma^2+ \sigma_e^2}\]
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_20191127/"
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 0.5451204 0.000225689 0.00021374
sigma_theta^2 0.4365506 0.872810000 0.23609300
sigma_e^2 1.0000000 1.018880000 0.07640700
est est.se
PVE.expr 0.00412807 0.00390952
PVE.snp 0.21549800 0.05829160
PVE.total 0.21962600 0.05852090show_res("S1.2") truth est est.se
sigma_gamma^2 0.8721927 0.000321793 0.000292108
sigma_theta^2 2.7939238 2.839750000 0.427679000
sigma_e^2 1.0000000 1.153150000 0.138334000
est est.se
PVE.expr 0.00370128 0.00335983
PVE.snp 0.44090200 0.06640170
PVE.total 0.44460300 0.06662640show_res("S2.1") truth est est.se
sigma_gamma^2 0.5451204 0.000111696 0.000139882
sigma_theta^2 0.4365506 0.528777000 0.208465000
sigma_e^2 1.0000000 1.060910000 0.067136400
est est.se
PVE.expr 0.00216157 0.00270702
PVE.snp 0.13813100 0.05445660
PVE.total 0.14029200 0.05440400show_res("S2.2") truth est est.se
sigma_gamma^2 0.8721927 0.000258743 0.000278473
sigma_theta^2 2.7939238 2.144150000 0.372914000
sigma_e^2 1.0000000 1.301840000 0.120221000
est est.se
PVE.expr 0.00309079 0.00332647
PVE.snp 0.34573400 0.06013060
PVE.total 0.34882500 0.06013430show_res("S3.1") truth est est.se
sigma_gamma^2 0.5451204 8.28456e-06 7.35267e-05
sigma_theta^2 0.4365506 5.06254e-01 2.20805e-01
sigma_e^2 1.0000000 1.10084e+00 7.12064e-02
est est.se
PVE.expr 0.000156494 0.00138891
PVE.snp 0.129088000 0.05630220
PVE.total 0.129244000 0.05632350show_res("S3.2") truth est est.se
sigma_gamma^2 0.8721927 0.000185945 0.000246806
sigma_theta^2 2.7939238 2.467070000 0.407404000
sigma_e^2 1.0000000 1.227570000 0.131413000
est est.se
PVE.expr 0.0021904 0.00290734
PVE.snp 0.3922910 0.06478150
PVE.total 0.3944810 0.06482130In this analysis, we see the PVE estimation for expression is much under estimated, this is because many of the gene expression training model has low \(R^2\), especially for low heritability ones. For a polygenic model, because we are estimating the total herability explained by expression, the ones with low \(R^2\) will affect this estimation. We are moving to model with a sparse prior for expression.
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