Last updated: 2020-01-15
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Untracked: R/demo.nmf.R
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Unstaged changes:
Modified: R/demo.susie.R
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These are the previous versions of the repository in which changes were made to the R Markdown (analysis/susie_jrssb_example.Rmd) and HTML (docs/susie_jrssb_example.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.
| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | e69de40 | Peter Carbonetto | 2020-01-15 | wflow_publish(“susie_jrssb_example.Rmd”) |
| Rmd | a70ba28 | Peter Carbonetto | 2020-01-14 | Removed old home page. |
| html | 040c081 | Peter Carbonetto | 2020-01-14 | Adjusted the knitr settings in the susie example. |
| Rmd | 88668a4 | Peter Carbonetto | 2020-01-14 | wflow_publish(“susie_jrssb_example.Rmd”) |
| html | f73d1cb | Peter Carbonetto | 2020-01-14 | First build of the susie example. |
| Rmd | fd08150 | Peter Carbonetto | 2020-01-14 | wflow_publish(“susie_jrssb_example.Rmd”) |
| Rmd | 781b68e | Peter Carbonetto | 2020-01-14 | Added/modified workflowr files. |
| Rmd | 43ad1c3 | Peter Carbonetto | 2020-01-14 | Added workflowr files. |
Here we explore the behaviour of SuSiE in a toy example where the correlations between the predictors are not so straightforward. We will see that SuSiE still achieves the goal of correctly inferring Credible Sets in this slightly more complicated example.
The mvtnorm package is used to simulate the data.
library(mvtnorm)
library(varbvs)
library(susieR)One of the referees was interested in understanding the behaviour of SuSiE in the following scenario in which the correlation structure is slightly more complex than the motivating example provided in the paper.
Consider 5 correlated predictors in a linear regression, \(x_1, \ldots, x_5\), in which only two variables, \(x1\) and \(x3\), have an effect on the outcome, \(y\); that is, their regression coefficients \(b_1\) and \(b_3\) are not zero. The first two predictors are very strongly correlated with each other, and the third and fourth predictors are also very strongly correlated. Therefore, it will be difficult to distinguish between \(x1\) and \(x2\), and the same for \(x3\) and \(x4\). This correlation structure is the same as the simple motivating example given in the paper.
To complicate the example, there is a fifth predictor that has no effect on the outcome, and is also strongly correlated with the other variables. This is the correlation matrix for the 5 predictors:
S <- rbind(c( 1, 0.99, 0.5, 0.5, 0.8),
c(0.99, 1, 0.5, 0.5, 0.8),
c( 0.5, 0.5, 1, 0.99, 0.8),
c( 0.5, 0.5, 0.99, 1, 0.8),
c( 0.8, 0.8, 0.8, 0.8, 1))Therefore, without an abundance of data, it will be difficult to distinguish between \((x_1, x_2, x_3)\) and \((x_3, x_4, x_5)\). The inferential statement that would best capture our uncertainty about which variables affect the outcome would look something like this:
\[\mbox{Among coefficients $\beta_1, \ldots, \beta_5$, two (or more) are not zero}\]
Since SuSiE is not designed to produce such inferential statements, the best we can do is instead to state:
\[(\beta_1 \neq 0 \mbox{ or } \beta_2 \neq 0 \mbox{ or } \beta_5 \neq 0) \mbox{ and } (\beta_3 \neq 0 \mbox{ or } \beta_4 \neq 0)\]
Another equally valid statement could be:
\[(\beta_1 \neq 0 \mbox{ or } \beta_2 \neq 0) \mbox{ and } (\beta_3 \neq 0 \mbox{ or } \beta_4 \neq 0 \mbox{ or } \beta_5 \neq 0)\]
Although neither of these statements completely capture our uncertainty about which variables affect the outcome, it does satisfy the following aim, which was given in the introduction to the paper:
*A key feature of our method, which distinguishes it from most existing BVSR methods, is that it produces “Credible Sets” of variables that quantify uncertainty in which variable should be selected when multiple, highly correlated variables compete with one another. These Credible Sets are designed to be as small as possible while still each capturing a relevant variable.
The key point here is that the Credible Sets are not necessarily intended to capture the full uncertainty in which variables should be selected, but does at least capture all relevant variables.
Below, we investigate these claims empirically.
We simulate 150 samples in this scenario.
set.seed(1)
n <- 200
b <- c(1, 0, -1, 0, 0)
X <- rmvnorm(n,sigma = S)
y <- c(X %*% b + rnorm(n)/0.3)First, to provide a counterexample, we fit the “varbvs” model, and show that model, although very similar to SuSiE in many ways, does not properly capture uncertainty in the selected variables.
fit1 <- varbvs(X,NULL,y,logodds = log10(0.5),sa = 1)
print(summary(fit1)$top.vars)
# Welcome to -- * *
# VARBVS version 2.6-5 -- | | |
# large-scale Bayesian -- || | | | || | | |
# variable selection -- | || | | | | || || |||| || | ||
# ****************************************************************************
# Copyright (C) 2012-2019 Peter Carbonetto.
# See http://www.gnu.org/licenses/gpl.html for the full license.
# Fitting variational approximation for Bayesian variable selection model.
# family: gaussian num. hyperparameter settings: 1
# samples: 200 convergence tolerance 1.0e-04
# variables: 5 iid variable selection prior: yes
# covariates: 0 fit prior var. of coefs (sa): no
# intercept: yes fit residual var. (sigma): yes
# variational max. incl variance params
# iter lower bound change vars sigma sa
# 00001 -5.356108e+02 5.5e-01 0001.9 1.2e+01 1.0e+00
# 00002 -5.333116e+02 3.9e-01 0002.1 1.1e+01 1.0e+00
# 00003 -5.331771e+02 1.3e-02 0002.1 1.1e+01 1.0e+00
# 00004 -5.331695e+02 6.6e-03 0002.1 1.1e+01 1.0e+00
# 00005 -5.331692e+02 1.9e-03 0002.1 1.1e+01 1.0e+00
# 00006 -5.331692e+02 3.3e-04 0002.1 1.1e+01 1.0e+00
# 00007 -5.331692e+02 5.0e-05 0002.1 1.1e+01 1.0e+00
#
# Estimating proportion of variance in Y explained by model.
# index variable prob PVE coef Pr(coef.>0.95)
# 1 3 X3 0.99996835 0.123298810 -1.3131491450 [-1.806,-0.820]
# 2 1 X1 0.97533345 0.065627231 0.8614275793 [+0.421,+1.345]
# 3 4 X4 0.03648400 0.004372519 0.0004970112 [-0.481,+0.508]
# 4 5 X5 0.03619460 0.004588351 -0.0020164184 [-0.534,+0.423]
# 5 2 X2 0.03479185 0.004470761 0.0013014624 [-0.428,+0.503]TO DO.
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.2
#
# 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] susieR_0.9.0.0575 varbvs_2.6-5 mvtnorm_1.0-11
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.3 knitr_1.26 whisker_0.4
# [4] magrittr_1.5 workflowr_1.6.0.9000 lattice_0.20-38
# [7] R6_2.4.1 jpeg_0.1-8.1 rlang_0.4.2
# [10] nor1mix_1.3-0 stringr_1.4.0 tools_3.6.2
# [13] grid_3.6.2 xfun_0.11 png_0.1-7
# [16] latticeExtra_0.6-29 git2r_0.26.1 htmltools_0.4.0
# [19] yaml_2.2.0 digest_0.6.23 rprojroot_1.3-2
# [22] Matrix_1.2-18 RColorBrewer_1.1-2 later_1.0.0
# [25] promises_1.1.0 fs_1.3.1 glue_1.3.1
# [28] evaluate_0.14 rmarkdown_2.0 stringi_1.4.3
# [31] compiler_3.6.2 backports_1.1.5 httpuv_1.5.2