Anytime-Valid Tests of Conditional Independence Under Model-X

Journal of the American Statistical Association, vol. 119, pp. 1554–1565

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. Supplementary materials for this article are available online.

Authors 3

1. Peter Grünwald corresponding Aachen Department of Statistics, Leiden University

   Leiden University · Centrum Wiskunde & Informatica

   Affiliation as printed

   Department of Machine Learning, Centrum Wiskunde & Informatica

   Department of Statistics, Leiden University

   Department of Machine Learning, Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

   Department of Statistics, Leiden University, Leiden, The Netherlands

2. Alexander Henzi

   ETH Zurich · Seminar for Statistics

   Affiliation as printed

   Seminar for Statistics, ETH Zürich

   Seminar for Statistics, ETH Zürich, Switzerland

3. Tyron Lardy Aachen Department of Statistics, Leiden University

   Leiden University · Centrum Wiskunde & Informatica

   Affiliation as printed

   Department of Machine Learning, Centrum Wiskunde & Informatica

   Department of Statistics, Leiden University

   Department of Machine Learning, Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

   Department of Statistics, Leiden University, Leiden, The Netherlands