Last updated: 2018-05-12

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Introduction

For a fixed design matrix \(X\), \(\hat\beta \sim N(\beta, \sigma_e^2(X^TX)^{-1})\), and the empirical distribution of \(\hat z_j = \hat\beta_j / \hat{\text{SE}}(\hat\beta_j)\) under the null will depend on the average correlation in \((X^TX)^{-1}\).

Actually the most important quantity to determine the shape of the empirical distribution of \(\hat z_j\) is the square root of the mean squared correlation among \(\hat\beta_j\)’s, that is, \(\sqrt{\overline{\rho_{\hat\beta_i, \hat\beta_j}^2}}\).

Now we are taking a look at this quantity for some commonly used design matrix in linear regression simulations.

In all three settings, each row of \(X_{n \times p}\) is independently drawn from a \(N(0, \Sigma)\) distribution, where the diagonal elements of \(\Sigma\) are all one. Then the columns of \(X\) are normalized such that \(\|X_j\|_2^2 = 1\).

Independent and normalized columns

  • \(\Sigma = I\).
[1] 0.02234663

Toeplitz column correlation

  • \(\Sigma_{ij} = \rho^{|i - j|}\)

\(\text{SE}\left(\hat\beta_j\right)\)

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Average orrelation among \(X_j\) and \(\hat\beta_j\)

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Factor model column correlation

  • \(\Sigma_X = \texttt{cov2cor}(B_{p \times d}B_{d\times p}^T + I)\)

\(\text{SE}\left(\hat\beta_j\right)\)

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Average correlation among \(X_j\) and \(\hat\beta_j\).

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Factor model \(\hat\beta\) correlation

  • \(Cor(\hat\beta) = \texttt{cov2cor}(B_{p \times d}B_{d\times p}^T + I)\)

\(\text{SE}\left(\hat\beta_j\right)\)

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Average correlation among \(X_j\) and \(\hat\beta_j\).

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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     

loaded via a namespace (and not attached):
 [1] workflowr_1.0.1   Rcpp_0.12.16      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.21.0      magrittr_1.5      evaluate_0.10.1  
[10] stringi_1.1.6     whisker_0.3-2     R.oo_1.21.0      
[13] R.utils_2.6.0     rmarkdown_1.9     tools_3.4.3      
[16] stringr_1.3.0     yaml_2.1.18       compiler_3.4.3   
[19] htmltools_0.3.6   knitr_1.20

This reproducible R Markdown analysis was created with workflowr 1.0.1