Correlated \(z\) scores

Because Efron said so!

Efron 2004 JASA, Efron 2007 JASA, Efron 2008 StatSci, Efron 2009 AoAS, Efron 2010 JASA (with discussion), Efron 2010 Book.

Intuition

Suppose all \(z\) scores are perfectly correlated.

Simulation

\(z = Lx\)

\(z_i = \sqrt{\rho}x + \sqrt{1-\rho}e_i\)

Real data

Efron’s method: locfdr

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     

loaded via a namespace (and not attached):
 [1] compiler_3.4.3  backports_1.1.2 magrittr_1.5    rprojroot_1.3-1
 [5] tools_3.4.3     htmltools_0.3.6 yaml_2.1.16     Rcpp_0.12.14   
 [9] splines_3.4.3   stringi_1.1.6   rmarkdown_1.8   knitr_1.17     
[13] git2r_0.20.0    stringr_1.2.0   digest_0.6.13   locfdr_1.1-8   
[17] evaluate_0.10.1