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Pearson Chi-squared Conditional Randomization Test

Adel Javanmard, Mohammad Mehrabi

TL;DR

The paper tackles conditional independence testing in the model-X framework by introducing the PCR test, which leverages distributional information on covariates to construct counterfactuals and perform a high-resolution multinomial Pearson $\chi^2$ test on grouped scores. PCR achieves valid finite- and asymptotic-size control with a small number of randomizations, and its power is analyzed through the conditional relative density, revealing regimes where PCR outperforms existing CRT-based methods. It introduces practical extensions including parameter-free PCR via Bonferroni aggregation and robust PCR that remains valid under misspecification of $P_{X|Z}$. Empirically, PCR demonstrates superior power and computational efficiency in simulations and real data (Capital Bikeshare), underscoring its usefulness for scalable CI testing in complex, high-dimensional problems.

Abstract

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 $P_{X|Z}$.

Pearson Chi-squared Conditional Randomization Test

TL;DR

The paper tackles conditional independence testing in the model-X framework by introducing the PCR test, which leverages distributional information on covariates to construct counterfactuals and perform a high-resolution multinomial Pearson test on grouped scores. PCR achieves valid finite- and asymptotic-size control with a small number of randomizations, and its power is analyzed through the conditional relative density, revealing regimes where PCR outperforms existing CRT-based methods. It introduces practical extensions including parameter-free PCR via Bonferroni aggregation and robust PCR that remains valid under misspecification of . Empirically, PCR demonstrates superior power and computational efficiency in simulations and real data (Capital Bikeshare), underscoring its usefulness for scalable CI testing in complex, high-dimensional problems.

Abstract

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 and another variable , while controlling for the effect of a high-dimensional confounding variable . In this paper, we introduce a novel test, called `Pearson Chi-squared Conditional Randomization' (PCR) test, which uses the distributional information on covariates 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: parameter-free PCR, which uses Bonferroni's correction to decide on a tuning parameter in the test; robust PCR, which avoids inflations in the size of the test when there is slight error in estimating the conditional law .

Paper Structure

This paper contains 41 sections, 23 theorems, 174 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation and summary of contributions
  3. Few randomizations: high-resolution p-values and speed-up
  4. High statistical power with simple score functions
  5. Standard and Non-Standard Setups: High Flexibility
  6. Related literature on conditional randomization tests
  7. Pearson Chi-squared randomization (PCR) test
  8. PCR test statistic
  9. Decision rule
  10. Size of the PCR test
  11. A power analysis of the PCR test
  12. Analytical assessment of the power advantage of PCR over CRT
  13. Parameter-free PCR test
  14. Robustness of the PCR test
  15. Numerical Experiments
  16. ...and 26 more sections

Key Result

Theorem 2.2

Under the null hypothesis eq: CI hypothesis and Assumption assum: mu_mapping , the statistic $U_{n_g,L}$ constructed in Algorithm algorithm: model-xz converges uniformly to the $\chi^2$ distribution with $L-1$ degrees of freedom, for $L\ge 2$. Concretely, let $V\sim \chi^2_{L-1}$. Then, for an absolute positive constant $C$. In addition, uniformly across $L,\alpha, n_g$, we have

Figures (8)

  • Figure 1: ODC function and dependency power for the regression setting of Corollary \ref{['coro: 1-sqrt']}.
  • Figure 2: Size of PCR test with $\theta^{\mathsf{finite}}_{L,\alpha}$ for dataset of size $n=100$ drawn iid from \ref{['eq:example-size']}. Three significance levels $\alpha=0.05,0.1,$ and $0.15$ are considered. Reported numbers are averaged over $10,000$ trials.
  • Figure 3: Size of PCR test with $\theta^{\mathsf{asym}}_{L,\alpha}$ for data-generating law \ref{['eq:example-size']} with $n=100$. Three significance levels $\alpha=0.05,0.1,$ and $0.15$ are considered. Reported numbers are averaged over $10,000$ trials.
  • Figure 4: Power of PCR test for (left) $n=1000$ and (right) $n=2000$ data points. Data points are generated under the setup \ref{['eq:example-size']} and the conditional law \ref{['eq: power-example-conditional-law']}. We consider the decision rule \ref{['eq:decision rule']} with both of the rejection thresholds $\theta^{\mathsf{asym}}_{L,\alpha}$ and $\theta^{\mathsf{finite}}_{L,\alpha}$. Each reported power is obtained by averaging over $1000$ trials at significance level $\alpha=0.1$.
  • Figure 5: Comparison between statistical power of PCR and a group of model-X CI tests for the data generating law \ref{['eq: compare-generative']} for low-dimensional (left) and high-dimensional (right) settings. For the low-dimensional setting, we consider $n=8000$, $p=50$ with the ordinary least square as the score function. For the high-dimensional setting, we consider $n=5000$ and $p=6000$ with the cross-validated lasso as the score function.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Theorem 2.2
  • Definition 3.1
  • Definition 3.4
  • Remark 3.5
  • Proposition 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Theorem 3.9
  • Remark 3.10
  • Theorem 3.11
  • ...and 18 more