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The aim here is to illustrate in a very simple example why the hierarchical inference approach, e.g. as implemented in hierinf, may select too many variables.
Suppose we have three highly-correlated variables \(x_1,x_2,x_3\), and that \(x_2,x_3\) are most highly correlated. Then the hierchical clustering will put \(x_2,x_3\) together first, and then join with \(x_1\). As a consequence the hierarchical approach cannot select just \(x_1\) and \(x_2\) say, even if the data would support that selection (e.g. because \(x_1\) is the effect variable and \(x_2\) is sufficiently correlated with \(x_1\) to be difficult to distinguish from it, whereas \(x_3\) is not). The hierarchical approach can only choose to select \(x_1\) alone or all 3. On the other hand SuSiE can select any pair of variables that is appropriate.
Here we provide a simple simulation that illustrates the idea. (Note that this is not deterministic.. so a different seed may produce a different result.)
library("hierinf")
library("mvtnorm")First simulate some data with \((x_2,x_3)\) most highly correlated, and with \(x_1\) being most correlated with \(x_2\). Note that \(x_1\) is the effect variable.
set.seed(4)
n = 10000 # chosen large to reduce the variation among simulations, although different seeds can give different answers
rho12 = 0.97
rho13 = 0.92
rho23 = 0.98
Sigma = cbind(c(1,rho12,rho13),c(rho12,1,rho23),c(rho13,rho23,1))
x = mvtnorm::rmvnorm(n, c(0,0,0),Sigma)
b = c(0.07,0,0) # this was chosen such that there is usually some uncertainty in the variable selection
y = drop(x %*% b + rnorm(n))Run susie:
s = susieR::susie(x,y)
susieR::susie_get_cs(s)$cs
$cs[[1]]
[1] 1 2
$coverage
[1] 0.95susieR::susie_get_pip(s)[1] 0.871336030 0.122517502 0.006146468Note that susie reports a CS with just variables \((x_1,x_2)\). Essentially this is because, although \(x_3\) is correlated with \(x_1\), it is sufficiently independent of \(x_1\) to not be confused with it (\(x_3\) has very small PIP).
This result is impossible for hierinf because it constrains itself by the hierarchical structure (which does not “really” exist here - it is imposed by the method.) Here hierinf reports a significant cluster containing all 3 variables:
colnames(x)<-1:3
h = hierinf::test_hierarchy(x,y,alpha=0.01,dendr=cluster_var(x))
h block p.value significant.cluster
1 NA 1.493e-07 1, 2, 3
sessionInfo()R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.4
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] mvtnorm_1.0-11 hierinf_1.3.1
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 MASS_7.3-51.4
[7] lattice_0.20-38 foreach_1.4.4 susieR_0.8.1.0525
[10] stringr_1.4.0 highr_0.8 tools_3.6.0
[13] glmnet_2.0-18 grid_3.6.0 xfun_0.8
[16] git2r_0.26.1 iterators_1.0.10 htmltools_0.3.6
[19] yaml_2.2.0 rprojroot_1.3-2 digest_0.6.20
[22] Matrix_1.2-17 codetools_0.2-16 fs_1.3.1
[25] glue_1.3.1 evaluate_0.14 rmarkdown_1.14
[28] wavethresh_4.6.8 stringi_1.4.3 compiler_3.6.0
[31] backports_1.1.4