Last updated: 2018-05-15

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    html 563b5a2 LSun 2017-02-16 FDR

Last updated: 2018-05-15

Code version: 388e65e

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.

Estimated FWER or FDR by BH

fdr.bh = mean(fd.bh >= 1)
fdr.bh
[1] 0.046

Estimated FWER or FDR by BY

fdr.by = mean(fd.by >= 1)
fdr.by
[1] 0.006

Estimated FWER or FDR by ash

fdr.ash = mean(fd.ash >= 1)
fdr.ash
[1] 0.096

Estimated FWER or FDR by truncash

fdr.truncash = mean(fd.truncash >= 1)
fdr.truncash
[1] 0.062

Happenstance of false discoveries by four approaches

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)

Expand here to see past versions of unnamed-chunk-7-1.png:
Version Author Date
0f36d99 LSun 2017-12-21
563b5a2 LSun 2017-02-16 
plot(table(fd.by), xlab = "Number of False Discoveries / 10K", ylab = "Frequency", main = "Benjamini - Yekutieli 2001", xlim = xlim, ylim = ylim)

Expand here to see past versions of unnamed-chunk-7-2.png:
Version Author Date
0f36d99 LSun 2017-12-21
563b5a2 LSun 2017-02-16 
plot(table(fd.ash), xlab = "Number of False Discoveries / 10K", ylab = "Frequency", main = "ash", xlim = xlim, ylim = ylim)

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Version Author Date
0f36d99 LSun 2017-12-21
563b5a2 LSun 2017-02-16 
plot(table(fd.truncash), xlab = "Number of False Discoveries / 10K", ylab = "Frequency", main = "truncash", xlim = xlim, ylim = ylim)

Expand here to see past versions of unnamed-chunk-7-4.png:
Version Author Date
0f36d99 LSun 2017-12-21
563b5a2 LSun 2017-02-16

Comparison of the numbers of false discoveries by four approaches

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)

Expand here to see past versions of unnamed-chunk-8-1.png:
Version Author Date
0f36d99 LSun 2017-12-21
563b5a2 LSun 2017-02-16

Session Information

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 10.13.4

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_0.12.16      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.0.1   MASS_7.3-47      
 [7] pscl_1.5.2        doParallel_1.0.11 lattice_0.20-35  
[10] foreach_1.4.4     stringr_1.3.0     tools_3.4.3      
[13] parallel_3.4.3    grid_3.4.3        R.oo_1.21.0      
[16] git2r_0.21.0      htmltools_0.3.6   iterators_1.0.9  
[19] yaml_2.1.18       rprojroot_1.3-2   digest_0.6.15    
[22] Matrix_1.2-12     codetools_0.2-15  R.utils_2.6.0    
[25] evaluate_0.10.1   rmarkdown_1.9     stringi_1.1.6    
[28] compiler_3.4.3    backports_1.1.2   R.methodsS3_1.7.1
[31] truncnorm_1.0-7

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