A Two-Sample Conditional Distribution Test Using Conformal Prediction and Weighted Rank Sum

Abstract

We consider the problem of testing the equality of conditional distributions of a response variable given a vector of covariates between two populations. Such a hypothesis testing problem can be motivated from various machine learning and statistical inference scenarios, including transfer learning and causal predictive inference. We develop a nonparametric test procedure inspired from the conformal prediction framework. The construction of our test statistic combines recent developments in conformal prediction with a novel choice of conformity score, resulting in a weighted rank-sum test statistic that is valid and powerful under general settings. To our knowledge, this is the first successful attempt of using conformal prediction for testing statistical hypotheses beyond exchangeability. Our method is suitable for modern machine learning scenarios where the data has high dimensionality and large sample sizes, and can be effectively combined with existing classification algorithms to find good conformity score functions. The performance of the proposed method is demonstrated in various numerical examples. Supplementary materials for this article are available online.

KEYWORD:

• Conformal prediction
• Covariate shift
• Distribution-free
• Model misspecification
• Rank-sum test

Supplementary Materials

The supplementary materials contain data and code to conduct the experiments in the article.

Additional information

Funding

Xiaoyu Hu’s research is partially supported by a scholarship from the China Scholarship Council. Jing Lei’s research is partially supported by NSF grant DMS-2015492.