Last updated: 2018-05-15

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Introduction

Applying CASH to linear regression variable selection, compared with other popular methods including BH, Knockoff.

Simulation Setting

The simulation setting is very similar to what’s used in Knockoff’s vignette. The only notable difference is that the non-zero signals are normally distributed centered at zero, rather than constant.

Data are simulated by \[ y_n = X_{n \times p}\beta_p + e_n \] where \[ \begin{array}{c} n = 2000 \\ p = 1000 \\ e_n \sim N(0, 1) \\ \beta \sim \eta\delta_0 + (1 - \eta)N(0, \sigma /\sqrt{n}) \end{array} \] Each row of \(X\) is generated independently from a \(N(0, \Sigma_\rho)\) distribution, where \(\left(\Sigma_\rho\right)_{j, k} = \rho^{|j - k|}\), a Toplitz matrix.

Every method selects the variables with respect to a nominal false discovery rate \(q = 0.1\).

In simulations, we are changing the values of the sparsity level \(\eta\), the signal strength \(\sigma\), the feature correlation \(\rho\).

Methods

  • BH: First run multiple linear regression, then apply BH to obtained \(p\)-values.

  • qvalue: First run multiple linear regression, then apply qvalue::qvalue to obtained \(p\)-values.

  • Knockoff: Directly apply knockoff::knockoff on \(X\), \(y\).

  • ASH: First run multiple linear regression, then apply ashr::ash on obtained \(\hat\beta\) and \(\hat{\text{se}}\left(\hat \beta\right)\), using normal mixture and normal likelihood.

  • CASH: First run multiple linear regression, then apply cash on obtained \(\hat\beta\) and \(\hat{\text{se}}\left(\hat \beta\right)\), using normal mixture and normal likelihood, with default penalty on Gaussian derivative coefficients.

  • CASH+: CASH with perfect knowledge, using real noise level \(\text{se}\left(\hat{\beta}\right)\).

Observations

  • BH is very robust, very fast.

  • Knockoff is way too slow and way too conservative with signals being unimodal at zero. Perhaps it needs strong signals distinctly different from the “bulk.” Unimodal setting is really adversary to this method.

  • CASH works fine, but not better than the basic ASH.

\(\eta = 0.9\), \(\sigma = 5\), \(\rho = 0.5\)

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\(\eta = 0.8\), \(\sigma = 4\), \(\rho = 0.5\)

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\(\eta = 0.5\), \(\sigma = 4\), \(\rho = 0.5\)

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\(\eta \in \left\{0.5, 0.6, 0.7, 0.8, 0.9\right\}\), \(\sigma = 4\), \(\rho = 0.5\)

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\(\eta = 0.9\), \(\sigma = 4\), \(\rho = 0.5\)

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\(\eta \in \left\{0.5, 0.6, 0.7, 0.8, 0.9\right\}\), \(\sigma = 3\), \(\rho = 0.7\)

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\(\eta = 0.5\), \(\sigma = 3\), \(\rho = 0.7\)

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\(\eta = 0.9\), \(\sigma = 3\), \(\rho = 0.7\)

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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] ggplot2_2.2.1  reshape2_1.4.3

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   munsell_0.4.3    
 [7] colorspace_1.3-2  rlang_0.1.6       stringr_1.3.0    
[10] plyr_1.8.4        tools_3.4.3       grid_3.4.3       
[13] gtable_0.2.0      R.oo_1.21.0       git2r_0.21.0     
[16] htmltools_0.3.6   lazyeval_0.2.1    yaml_2.1.18      
[19] rprojroot_1.3-2   digest_0.6.15     tibble_1.4.1     
[22] R.utils_2.6.0     evaluate_0.10.1   rmarkdown_1.9    
[25] labeling_0.3      stringi_1.1.6     pillar_1.0.1     
[28] compiler_3.4.3    scales_0.5.0      backports_1.1.2  
[31] R.methodsS3_1.7.1

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