Anytime Valid Tests of Conditional Independence Under Model-X

TL;DR

This paper introduces a sequential, anytime-valid testing procedure for conditional independence based on e-statistics and martingales, applicable under the model-X setting, with strong theoretical guarantees and competitive empirical performance.

Contribution

It develops a general method for constructing e-statistics for conditional independence testing within the model-X framework, including growth-rate optimal tests and asymptotic power guarantees.

Findings

The proposed test is valid at arbitrary stopping times.

It achieves asymptotic power one in logistic regression scenarios.

Simulation results show competitive power and robustness.

Abstract

We propose a sequential, anytime-valid method to test the conditional independence of a response $Y$ and a predictor $X$ given a random vector $Z$. The proposed test is based on e-statistics and test martingales, which generalize likelihood ratios and allow valid inference at arbitrary stopping times. In accordance with the recently introduced model-X setting, our test depends on the availability of the conditional distribution of $X$ given $Z$, or at least a sufficiently sharp approximation thereof. Within this setting, we derive a general method for constructing e-statistics for testing conditional independence, show that it leads to growth-rate optimal e-statistics for simple alternatives, and prove that our method yields tests with asymptotic power one in the special case of a logistic regression model. A simulation study is done to demonstrate that the approach is competitive in terms of power when compared to established sequential and nonsequential testing methods, and robust with respect to violations of the model-X assumption.