Last updated: 2019-07-03
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Here we investigate “selective inference” in the toy example of Wang et al (2018).
Now simulate some data with \(x_1 = x_2\) and \(x_3 = x_4\), and with effects at variables 1 and 3. (We simulate \(p = 100\) variables rather than \(p = 1000\) so that the example runs faster.)
set.seed(15)
n <- 500
p <- 100
x <- matrix(rnorm(n*p),n,p)
x[,2] <- x[,1]
x[,4] <- x[,3]
b <- rep(0,p)
b[1] <- 1
b[4] <- 1
y <- drop(x %*% b + rnorm(n))Unfortunately, the selective inference methods won’t allow duplicate columns.
try(fsfit <- fs(x,y))
try(larfit <- lar(x,y))
# Error in checkargs.xy(x = x, y = y) : x cannot have duplicate columns
# Error in checkargs.xy(x = x, y = y) : x cannot have duplicate columnsSo we modify x so that the identical columns aren’t quite identical.
x[,2] <- x[,1] + rnorm(n,0,0.1)
x[,4] <- x[,3] + rnorm(n,0,0.1)
cor(x[,1],x[,2])
cor(x[,3],x[,4])
# [1] 0.9955977
# [1] 0.9952243Now run the forward selection again, computing sequential p-values and confidence intervals.
fsfit <- fs(x,y)
out <- fsInf(fsfit)
out
#
# Call:
# fsInf(obj = fsfit)
#
# Standard deviation of noise (specified or estimated) sigma = 0.995
#
# Sequential testing results with alpha = 0.100
# Step Var Coef Z-score P-value LowConfPt UpConfPt LowTailArea UpTailArea
# 1 1 0.900 20.573 0.003 0.441 0.953 0.050 0.049
# 2 3 0.956 21.494 0.000 0.716 1.397 0.050 0.050
# 3 37 0.099 2.133 0.539 -0.439 0.156 0.050 0.050
# 4 45 0.083 1.817 0.911 -Inf 0.204 0.000 0.050
# 5 4 -0.838 -1.850 0.215 -Inf 10.125 0.000 0.050
# 6 46 0.077 1.747 0.715 -3.695 1.192 0.050 0.050
# 7 64 -0.077 -1.718 0.409 -3.906 2.666 0.050 0.050
# 8 79 0.076 1.678 0.542 -3.007 2.405 0.050 0.050
# 9 43 0.080 1.721 0.366 -1.354 2.373 0.050 0.050
# 10 68 -0.075 -1.669 0.456 -Inf Inf 0.000 0.000
# 11 13 0.081 1.738 0.434 -Inf Inf 0.000 0.000
# 12 61 -0.075 -1.588 0.102 -Inf 0.580 0.000 0.050
# 13 56 -0.064 -1.452 0.487 -3.159 3.055 0.050 0.050
# 14 84 0.058 1.332 0.901 -Inf 0.601 0.000 0.050
# 15 90 0.060 1.324 0.548 -3.950 3.120 0.050 0.050
# 16 65 -0.057 -1.284 0.298 -Inf Inf 0.000 0.000
# 17 69 -0.056 -1.243 0.071 -Inf 0.613 0.000 0.050
# 18 74 -0.053 -1.177 0.998 4.469 Inf 0.010 0.000
# 19 18 0.052 1.194 0.084 -3.562 Inf 0.050 0.000
# 20 70 -0.047 -1.097 0.229 -Inf Inf 0.000 0.000
# 21 49 -0.049 -1.075 0.092 -Inf 1.785 0.000 0.050
# 22 88 0.055 1.118 0.918 -Inf 1.226 0.000 0.050
# 23 25 0.049 1.055 0.060 -0.614 Inf 0.050 0.000
# 24 27 -0.047 -1.055 0.504 -Inf Inf 0.000 0.000
# 25 96 0.051 1.120 0.861 -Inf Inf 0.000 0.000
# 26 39 0.048 1.029 0.908 -Inf 3.127 0.000 0.050
# 27 40 -0.050 -1.041 0.005 -Inf -4.787 0.000 0.011
# 28 17 -0.045 -0.968 0.190 -Inf 3.100 0.000 0.050
# 29 66 -0.041 -0.901 0.879 -2.066 Inf 0.050 0.000
# 30 85 0.042 0.916 0.874 -Inf Inf 0.000 0.000
# 31 8 -0.044 -0.932 0.391 -Inf Inf 0.000 0.000
# 32 22 -0.044 -0.933 0.144 -Inf 3.515 0.000 0.050
# 33 78 0.041 0.933 0.821 -Inf Inf 0.000 0.000
# 34 28 -0.041 -0.882 0.789 -4.262 Inf 0.050 0.000
# 35 7 0.043 0.896 0.894 -Inf 2.807 0.000 0.050
# 36 26 0.044 0.905 0.620 -Inf Inf 0.000 0.000
# 37 21 -0.041 -0.869 0.590 -Inf Inf 0.000 0.000
# 38 71 0.043 0.940 0.194 -Inf Inf 0.000 0.000
# 39 73 0.040 0.858 0.247 -Inf Inf 0.000 0.000
# 40 50 -0.041 -0.867 0.963 4.773 Inf 0.048 0.000
# 41 99 0.041 0.876 0.022 4.638 Inf 0.033 0.000
# 42 59 0.036 0.790 0.079 -1.322 Inf 0.050 0.000
# 43 76 0.034 0.738 0.060 -0.458 Inf 0.050 0.000
# 44 62 -0.037 -0.781 0.760 -3.209 Inf 0.050 0.000
# 45 19 0.031 0.717 0.638 -Inf Inf 0.000 0.000
# 46 36 -0.033 -0.711 0.705 -Inf Inf 0.000 0.000
# 47 6 0.030 0.684 0.828 -Inf Inf 0.000 0.000
# 48 34 0.034 0.725 0.041 1.281 Inf 0.050 0.000
# 49 57 -0.035 -0.740 0.915 -2.026 Inf 0.050 0.000
# 50 42 -0.034 -0.754 0.865 -4.424 Inf 0.050 0.000
# 51 14 -0.032 -0.692 0.212 -Inf 3.813 0.000 0.050
# 52 53 0.030 0.665 0.301 -Inf Inf 0.000 0.000
# 53 83 -0.028 -0.627 0.704 -Inf Inf 0.000 0.000
# 54 35 0.029 0.620 0.680 -Inf Inf 0.000 0.000
# 55 12 0.027 0.594 0.850 -Inf 2.456 0.000 0.050
# 56 2 -0.285 -0.587 0.877 -Inf Inf 0.000 0.000
# 57 38 0.031 0.617 0.632 -Inf Inf 0.000 0.000
# 58 41 -0.026 -0.562 0.987 0.322 Inf 0.000 0.000
# 59 51 0.027 0.583 0.023 -0.330 Inf 0.000 0.000
# 60 91 0.030 0.644 0.943 -Inf 0.654 0.000 0.050
# 61 58 0.027 0.574 0.773 -Inf Inf 0.000 0.000
# 62 97 0.029 0.558 0.002 5.116 Inf 0.004 0.000
# 63 87 0.025 0.523 0.552 -Inf Inf 0.000 0.000
# 64 5 0.025 0.509 0.108 -2.311 Inf 0.050 0.000
# 65 63 0.022 0.485 0.489 -Inf Inf 0.000 0.000
# 66 47 0.022 0.455 0.927 -Inf 2.634 0.000 0.050
# 67 9 0.021 0.459 0.642 -Inf Inf 0.000 0.000
# 68 31 0.021 0.448 0.064 -1.444 Inf 0.050 0.000
# 69 89 -0.021 -0.426 0.721 -Inf Inf 0.000 0.000
# 70 32 -0.022 -0.419 0.428 -Inf Inf 0.000 0.000
# 71 86 -0.021 -0.422 0.188 -Inf Inf 0.000 0.000
# 72 48 -0.020 -0.402 0.492 -Inf Inf 0.000 0.000
# 73 20 -0.017 -0.360 0.163 -Inf Inf 0.000 0.000
# 74 67 -0.018 -0.373 0.202 -Inf Inf 0.000 0.000
# 75 44 0.017 0.348 0.281 -Inf Inf 0.000 0.000
# 76 52 0.016 0.329 0.407 -3.741 Inf 0.050 0.000
# 77 11 0.015 0.309 0.697 -Inf Inf 0.000 0.000
# 78 23 0.014 0.308 0.357 -Inf Inf 0.000 0.000
# 79 94 0.014 0.313 0.402 -Inf Inf 0.000 0.000
# 80 30 0.014 0.287 0.339 -Inf Inf 0.000 0.000
# 81 95 0.013 0.264 0.269 -Inf Inf 0.000 0.000
# 82 33 0.012 0.251 0.553 -Inf Inf 0.000 0.000
# 83 15 -0.011 -0.226 0.534 -Inf Inf 0.000 0.000
# 84 98 0.011 0.235 0.305 -Inf Inf 0.000 0.000
# 85 72 -0.010 -0.205 0.119 -Inf 4.010 0.000 0.050
# 86 100 0.008 0.161 0.797 -Inf Inf 0.000 0.000
# 87 81 -0.008 -0.161 0.107 -Inf 2.010 0.000 0.050
# 88 24 0.007 0.135 0.771 -Inf Inf 0.000 0.000
# 89 10 0.006 0.123 0.440 -Inf Inf 0.000 0.000
# 90 60 0.006 0.120 0.592 -Inf Inf 0.000 0.000
# 91 75 -0.005 -0.113 0.153 -Inf 4.187 0.000 0.050
# 92 16 -0.004 -0.086 0.629 -Inf Inf 0.000 0.000
# 93 54 0.004 0.078 0.996 -Inf -5.340 0.000 0.004
# 94 29 0.004 0.077 0.842 -Inf Inf 0.000 0.000
# 95 82 0.004 0.078 0.004 4.839 Inf 0.005 0.000
# 96 80 0.000 -0.008 0.880 -Inf Inf 0.000 0.000
# 97 93 0.000 0.008 0.642 -Inf Inf 0.000 0.000
# 98 77 0.000 -0.007 0.185 -Inf Inf 0.000 0.000
# 99 55 0.000 -0.004 0.716 -Inf Inf 0.000 0.000
# 100 92 0.000 -0.003 0.443 -Inf Inf 0.000 0.000
#
# Estimated stopping point from ForwardStop rule = 2In this example, selective inference selected variables 1 and 3, each with very small \(p\) values. Of course we know that variable 3 is a false selection, so it might seem bad that the p value is small. However you have to remember that the p value does not measure significance of the variable selection: it measures the significance of the coefficient of the selected variable, conditional on the selection event.
Put another way, selective inference is not really trying to assess uncertainty in which variables should be selected, and is certainly not trying to produce inferences of the form \[(b_1 \neq 0 \text{ OR } b_2 \neq 0) \text{ AND } (b_3 \neq 0 \text{ OR } b_4 \neq 0)\] which was the goal in Wang et al.
sessionInfo()
# R version 3.4.3 (2017-11-30)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS High Sierra 10.13.6
#
# 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] selectiveInference_1.2.4 survival_2.41-3
# [3] intervals_0.15.1 glmnet_2.0-16
# [5] foreach_1.4.4 Matrix_1.2-12
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.1 knitr_1.23 whisker_0.3-2
# [4] magrittr_1.5 workflowr_1.4.0 splines_3.4.3
# [7] lattice_0.20-35 stringr_1.4.0 tools_3.4.3
# [10] grid_3.4.3 xfun_0.7 git2r_0.25.2.9008
# [13] htmltools_0.3.6 iterators_1.0.9 yaml_2.2.0
# [16] rprojroot_1.3-2 digest_0.6.18 fs_1.2.7
# [19] codetools_0.2-15 glue_1.3.1 evaluate_0.13
# [22] rmarkdown_1.13 stringi_1.4.3 compiler_3.4.3
# [25] backports_1.1.2