Illustration of SuSiE in a “difficult” situation

Last updated: 2020-01-15

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Rmd	63a0e77	Peter Carbonetto	2020-01-15	wflow_publish(“susie_jrssb_example.Rmd”)
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html	f73d1cb	Peter Carbonetto	2020-01-14	First build of the susie example.
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Rmd	781b68e	Peter Carbonetto	2020-01-14	Added/modified workflowr files.
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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.

Load packages

The mvtnorm package is used to simulate the data.

library(mvtnorm)
library(varbvs)
library(susieR)

Overview

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,  0.8,  0.8, 0.9),
           c(0.99,    1,  0.8,  0.8, 0.9),
           c( 0.8,  0.8,    1, 0.99, 0.9),
           c( 0.8,  0.8, 0.99,    1, 0.9),
           c( 0.9,  0.9,  0.9,  0.9,   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:

\[\mbox{Among coefficients $\beta_1, \ldots, \beta_5$, two (or more) are not zero}\]

SuSiE is not designed to produce such inferential statements, but it could state instead:

\[(\beta_1 \neq 0 \mbox{ or } \beta_2 \neq 0 \mbox{ or } \beta_5 \neq 0) \mbox{ and } (\beta_3 \neq 0 \mbox{ or } \beta_4 \neq 0)\]

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.

Simulate data

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 <- drop(X %*% b + rnorm(n)/0.3)

Multiple regression using varbvs

First, to provide a counterexample that does not accomplish what we set out to achieve, we fit the “varbvs” model, and show that model, although very similar to SuSiE in many ways, does not adequately capture uncertainty in the selected variables.

fit1 <- varbvs(X,NULL,y,logodds = log10(0.5),sa = 1,verbose = FALSE)
print(summary(fit1)$top.vars)
#   index variable       prob         PVE          coef  Pr(coef.>0.95)
# 1     3       X3 0.99995825 0.128995948 -1.3311483474 [-1.836,-0.826]
# 2     1       X1 0.98264586 0.073088751  0.9266196082 [+0.460,+1.425]
# 3     4       X4 0.03756227 0.004674475  0.0005357177 [-0.495,+0.524]
# 4     5       X5 0.03643073 0.004849222 -0.0017998663 [-0.534,+0.435]
# 5     2       X2 0.03618984 0.004751830  0.0012398904 [-0.452,+0.521]

From this summary, we see that varbvs correctly determines that two variables are useful for predicting the outcome, but it incorrectly places all the weight on the first and third variables. This is expected behaviour due to the particular variational approximation used in varbvs, but it is undesirable when we want to better quantify uncertainty in the correlated variables.

Multiple regression using SuSiE

Next, we fit a SuSiE model. (For simplicity, we fit a SuSiE model with $L = 2$, but SuSiE will obtain similar results for $L > 2$.)

fit2 <- susie(X,y,L = 2,standardize = FALSE,estimate_prior_variance = FALSE,
              scaled_prior_variance = 1,min_abs_corr = 0,tol = 1e-8)
print(susie_get_cs(fit2,X,min_abs_corr = 0)$cs)
# $L2
# [1] 1 2 5
# 
# $L1
# [1] 3 4 5

SuSiE identified two Credible Sets, in which $x_5$ is included in both, so we have:

\[(\beta_1 \neq 0 \mbox{ or } \beta_2 \neq 0 \mbox{ or } \beta_5 \neq 0) \mbox{ and } (\beta_3 \neq 0 \mbox{ or } \beta_4 \neq 0 \mbox{ or } \beta_5 \neq 0) )\]

Looking more closely at the posterior inclusion probabilities, most of the weight is assigned to $x_3$ and $x_4$ in the first Credible Set, and to $x_1$ and $x_2$ in the second Credible Set, but, in light of the strong correlations with $x_5$, each Credible Set also allows for the possibility that $x_5$ affects the outcome.

round(fit2$alpha,digits = 3)
#       [,1]  [,2]  [,3]  [,4]  [,5]
# [1,] 0.001 0.001 0.629 0.318 0.052
# [2,] 0.466 0.478 0.010 0.012 0.035

Therefore, although this statement is not the same as, “among coefficients $\beta_1, \ldots, \beta_5$, two (or more) are not zero”—for example, it allows for only selecting one variable, $x_5$—it is does achieve the aim of capturing all relevant variables.