Anytime Valid Tests of Conditional Independence Under Model-X

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 full characterization of $E$-statistics for testing conditional independence, investigate growth-rate optimal $E$-statistics and their power properties, and show 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 robust with respect to violations of the model-X assumption and competitive in terms of power when compared to established sequential and non-sequential testing methods.