Covariance Measure Tests (COMETs) in R

The Generalised [1] and Projected [2] Covariance Measure tests (GCM, PCM) can be used to test conditional independence between a real-valued response Y and features/modalities X given additional features/modalities Z using any sufficiently predictive supervised learning algorithms. The comets R package implements these covariance measure tests (COMETs) with a user-friendly interface which allows the user to use any sufficiently predictive supervised learning algorithm of their choosing. The default is to use random forests implemented in ranger for all regressions. A Python version of this package is available here.

Here, we showcase how to use comets with a simple example in which Y is not independent of X given Z. More elaborate examples including conditional variable significance testing and modality selection on real-world data can be found in [3].

set.seed(1)
n <- 300
X <- matrix(rnorm(2 * n), ncol = 2)
colnames(X) <- c("X1", "X2")
Z <- matrix(rnorm(2 * n), ncol = 2)
colnames(Z) <- c("Z1", "Z2")
Y <- X[, 1]^2 + Z[, 2] + rnorm(n)
GCM <- gcm(Y, X, Z) # plot(GCM)

The output for the GCM test, which fails to reject the null hypothesis of conditional independence in this example, is shown below. The residuals for the Y on Z and X on Z regressions can be investigated by calling plot(GCM) (not shown here).

## 
##  Generalized covariance measure test
## 
## data:  gcm(Y = Y, X = X, Z = Z)
## X-squared = 3.6131, df = 2, p-value = 0.1642
## alternative hypothesis: true E[cov(Y, X | Z)] is not equal to 0

The PCM test can be run likewise.

PCM <- pcm(Y, X, Z) # plot(PCM)

The output is shown below: The PCM test correctly rejects the null hypothesis of conditional independence in this example.

## 
##  Projected covariance measure test
## 
## data:  pcm(Y = Y, X = X, Z = Z)
## Z = 4.7107, p-value = 1.235e-06
## alternative hypothesis: true E[Y | X, Z] is not equal to E[Y | Z]

The comets package contains an alternative formula-based interface, in which $H_0 : Y \perp\hspace{-5pt}\perp X \mid Z$ can be supplied as Y ~ X | Z with a corresponding data argument. This interface is implemented in comet() and shown below.

dat <- data.frame(Y = Y, X, Z)
comet(Y ~ X1 + X2 | Z1 + Z2, data = dat, test = "gcm")

## 
##  Generalized covariance measure test
## 
## data:  comet(formula = Y ~ X1 + X2 | Z1 + Z2, data = dat, test = "gcm")
## X-squared = 3.5, df = 2, p-value = 0.1738
## alternative hypothesis: true E[cov(Y, X | Z)] is not equal to 0

Different regression methods can supplied for both GCM and PCM tests using the reg_* arguments (for instance, reg_YonZ in gcm() for the regression of Y on Z). Pre-implemented regressions are "rf" for random forests and "lasso" for cross-validated L₁-penalized regression. Custom regression functions can be supplied as character strings or functions, require a residual() (GCM and PCM) or predict() (PCM only) method and the following structure:

my_regression <- function(y, x, ...) {
  ret <- <run the regression>
  class(ret) <- "my_regression"
  ret
}

predict.my_regression <- function(object, data, ...) {
  <run the prediction routine>
}

residuals.my_regression <- function(object, response, data, ...) {
  <run the routine for computing residuals>
}

The input y and x and data are vector and matrix-valued. The output of predict.my_regression() should be a vector of length NROW(data).

Installation

The development version of comets can be installed using:

# install.packages("remotes")
remotes::install_github("LucasKook/comets")

A stable version of comets can be installed from CRAN via:

install.packages("comets")

Replication materials

All results in [3] can be reproduced by running make all in ./inst after downloading all required data from the zenodo repository. The scripts for reproducing the results manually can be found in ./inst/code/ for the CCLE data (ccle.R), TCGA data (multiomics.R) and MIMIC data (mimic.R).

References

[1] Rajen D. Shah, Jonas Peters “The hardness of conditional independence testing and the generalised covariance measure,” The Annals of Statistics, 48(3), 1514-1538. doi:10.1214/19-aos1857

[2] Lundborg, A. R., Kim, I., Shah, R. D., & Samworth, R. J. (2024). The Projected Covariance Measure for assumption-lean variable significance testing. The Annals of Statistics. To appear. doi:10.48550/arXiv.2211.02039

[3] Kook, L. & Lundborg A. R. (2024). Algorithm-agnostic significance testing in supervised learning with multimodal data. Briefings in Bioinformatics 25(6) 2024. doi:10.1093/bib/bbae475