FDR / \(s\) value on Correlated Null: betahat = betahat, sebetahat = betahat / zscore

Introduction

Apply two FDR-controlling procedures, Benjamini–Hochberg 1995 (“BH”) and Benjamini-Yekutieli 2001 (“BY”), and two \(s\) value models, ash and truncash (with the threshold \(T = 1.96\)) to the simulated, correlated null data. The data are obtained from 5 vs 5 GTEx/Liver samples and 10K top expressed genes, and \(1000\) independent simulation trials.

Compare the numbers of false discoveries (by definition, all discoveries should be false) obtained by these four methods, using FDR \(\leq 0.05\) and \(s\)-value \(\leq 0.05\) as cutoffs.

Simulation: \(p\) values for BH and BY procedures; \(\hat\beta = \hat\beta\), \(\hat s = \hat\beta / \hat z\), for ash and truncash.

\(\hat\beta\) obtained from the limma::lmFit step of the pipeline, \(\hat z\) obtained from the last step of the pipeline.

library(ashr)
source("../code/truncash.R")

p = read.table("../output/p_null_liver_777.txt")
z = read.table("../output/z_null_liver_777.txt")
betahat = read.table("../output/betahat_null_liver_777.txt")

m = dim(p)[1]
n = dim(p)[2]
fd.bh = fd.by = fd.ash = fd.truncash = c()

for (i in 1:m) {
  p_BH = p.adjust(p[i, ], method = "BH")
  fd.bh[i] = sum(p_BH <= 0.05)
  p_BY = p.adjust(p[i, ], method = "BY")
  fd.by[i] = sum(p_BY <= 0.05)
  betahat_trial = as.numeric(betahat[i, ])
  sebetahat_trial = - betahat_trial / as.numeric(z[i, ])
  fit.ash = ashr::ash(betahat_trial, sebetahat_trial, method = "fdr", mixcompdist = "normal")
  fd.ash[i] = sum(ashr::get_svalue(fit.ash) <= 0.05)
  fit.truncash = truncash(betahat_trial, sebetahat_trial, t = qnorm(0.975))
  fd.truncash[i] = sum(get_svalue(fit.truncash) <= 0.05)
}

Result

Simulated under the global null, FWER \(=\) FDR.

fdr.bh = mean(fd.bh >= 1)
fdr.bh

[1] 0.046

fdr.by = mean(fd.by >= 1)
fdr.by

[1] 0.006

fdr.ash = mean(fd.ash >= 1)
fdr.ash

[1] 0.096

fdr.truncash = mean(fd.truncash >= 1)
fdr.truncash

[1] 0.062

maxcount = max(c(fd.bh, fd.by, fd.ash, fd.truncash))
xlim = c(0, maxcount)
maxfreq = max(c(max(table(fd.bh)), max(table(fd.by)), max(table(fd.ash)), max(table(fd.truncash))))
ylim = c(0, maxfreq)
plot(table(fd.bh), xlab = "Number of False Discoveries / 10K", ylab = "Frequency", main = "Benjamini - Hochberg 1995", xlim = xlim, ylim = ylim)

plot(table(fd.by), xlab = "Number of False Discoveries / 10K", ylab = "Frequency", main = "Benjamini - Yekutieli 2001", xlim = xlim, ylim = ylim)

plot(table(fd.ash), xlab = "Number of False Discoveries / 10K", ylab = "Frequency", main = "ash", xlim = xlim, ylim = ylim)

plot(table(fd.truncash), xlab = "Number of False Discoveries / 10K", ylab = "Frequency", main = "truncash", xlim = xlim, ylim = ylim)

m = length(fd.bh)
fd.ind = (1:m)[!((fd.bh == 0) & (fd.by == 0) & (fd.ash == 0) & (fd.truncash == 0))]
plot(1:length(fd.ind), fd.bh[fd.ind], pch = 4, ylim = xlim, xlab = "Trials with False Discoveries", ylab = "Number of False Discoveries / 10K")
points(1:length(fd.ind), fd.by[fd.ind], pch = 4, col = 2)
points(1:length(fd.ind), fd.ash[fd.ind], pch = 4, col = 3)
points(1:length(fd.ind), fd.truncash[fd.ind], pch = 4, col = 4)
legend("topright", c("BH", "BY", "ash", "truncash"), col = 1:4, pch = 4)

Session Information

sessionInfo()

R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.2

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] SQUAREM_2017.10-1 ashr_2.2-2       

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.14      knitr_1.17        magrittr_1.5     
 [4] workflowr_0.8.0   MASS_7.3-47       doParallel_1.0.11
 [7] pscl_1.5.2        lattice_0.20-35   foreach_1.4.4    
[10] stringr_1.2.0     tools_3.4.3       parallel_3.4.3   
[13] grid_3.4.3        git2r_0.20.0      htmltools_0.3.6  
[16] iterators_1.0.9   yaml_2.1.16       rprojroot_1.3-1  
[19] digest_0.6.13     Matrix_1.2-12     codetools_0.2-15 
[22] evaluate_0.10.1   rmarkdown_1.8     stringi_1.1.6    
[25] compiler_3.4.3    backports_1.1.2   truncnorm_1.0-7