Last updated: 2020-06-23
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Knit directory: mr-ash/analysis/
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To our surprise, we discovered that the Lasso (or Elastic Net) sometimes provides more considerably more accurate predictions than mr.ash in settings where many predictors have a small effect on the outcome. Here I expand on an example from Matthew’s brief investigation of this phenomenon to better understand the (mis) behaviour of mr.ash.
These are the packages used in the analysis.
library(glmnet)
library(varbvs)
library(mr.ash.alpha)
library(ggplot2)
library(cowplot)I simulate the data just as Matthew did, except that I split the data into a training and a test set.
These are the data simulation settings: the number of samples in training set, “n”; number of simulated variables, “p”; the number of variables affecting the outcome (“p1”); and the proportion of variance in the outcome explained by the variables (“pve”).
n <- 500
p <- 1000
p1 <- 467
pve <- 0.95Simulate a \(2n x p\) design matrix; the first \(n\) rows is the training set data, and the remaining \(n\) rows are the test data.
X <- matrix(rnorm(2*n*p),2*n,p)
X <- scale(X,center = TRUE,scale = TRUE)Simulate the \(p\) regression coefficients; only \(p_1 < p\) of the coefficients are nonzero.
b <- rep(0,p)
j <- sample(p,p1)
b[j] <- rnorm(p1)Simulate the responses so that the target PVE is met.
y <- drop(X %*% b)
se <- sqrt((1 - pve)/pve) * sd(y)
y <- y + rnorm(n,sd = se)Split the data 50-50 into a training set and a test set.
test <- 1:n
Xtest <- X[test,]
ytest <- y[test]
X <- X[-test,]
y <- y[-test]
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.5
#
# Matrix products: default
# BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] cowplot_1.0.0 ggplot2_3.3.0 mr.ash.alpha_0.1-35
# [4] varbvs_2.6-5 glmnet_4.0-2 Matrix_1.2-18
#
# loaded via a namespace (and not attached):
# [1] shape_1.4.4 tidyselect_0.2.5 xfun_0.11
# [4] purrr_0.3.3 splines_3.6.2 lattice_0.20-38
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