# Pearson Chi-squared Conditional Randomization Test

arXiv: https://arxiv.org/abs/2111.00027 · HTML: https://mlsanity.com/paper/2111.00027/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2021.

## Built on by (most cited first, 4 of 4)

- [Model-X Sequential Testing for Conditional Independence via Testing by Betting](https://mlsanity.com/paper/2210.00354/) (2022)
- [Score-based Generative Modeling for Conditional Independence Testing](https://mlsanity.com/paper/2505.23309/) (2025)
- [GRASP: A Goodness-of-Fit Test for Classification Learning](https://mlsanity.com/paper/2209.02064/) (2022)
- [A Model-free Closeness-of-influence Test for Features in Supervised Learning](https://mlsanity.com/paper/2306.11855/) (2023)

## Abstract (arXiv)

Conditional independence (CI) testing arises naturally in many scientific problems and applications domains. The goal of this problem is to investigate the conditional independence between a response variable Y and another variable X, while controlling for the effect of a high-dimensional confounding variable Z. In this paper, we introduce a novel test, called `Pearson Chi-squared Conditional Randomization' (PCR) test, which uses the distributional information on covariates X,Z and constructs randomizations to test conditional independence. PCR leverages the i.i.d-ness property of the observations to obtain high-resolution p-values with a very small number of conditional randomizations. We also provide a power analysis of the PCR test, which captures the effect of various parameters of the test, the sample size and the distance of the alternative from the set of null distributions, measured in terms of a notion called `conditional relative density'. In addition, we propose two extensions of the PCR test, with important practical implications: (i) parameter-free PCR, which uses Bonferroni's correction to decide on a tuning parameter in the test; (ii) robust PCR, which avoids inflations in the size of the test when there is slight error in estimating the conditional law PX|Z.