# Differentially Private Chi-Squared Hypothesis Testing: Goodness of Fit and Independence Testing

arXiv: https://arxiv.org/abs/1602.03090 · HTML: https://mlsanity.com/paper/1602.03090/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2016.

## Built on by (most cited first, 24 of 24)

- [Differentially private inference via noisy optimization](https://mlsanity.com/paper/2103.11003/) (2021)
- [Private High-Dimensional Hypothesis Testing](https://mlsanity.com/paper/2203.01537/) (2022)
- [The Test of Tests: A Framework For Differentially Private Hypothesis Testing](https://mlsanity.com/paper/2302.04260/) (2023)
- [Non-parametric Differentially Private Confidence Intervals for the Median](https://mlsanity.com/paper/2106.10333/) (2021)
- [Nonparametric extensions of randomized response for private confidence sets](https://mlsanity.com/paper/2202.08728/) (2022)
- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)
- [Does Differentially Private Synthetic Data Lead to Synthetic Discoveries?](https://mlsanity.com/paper/2403.13612/) (2024)
- [Private Sequential Hypothesis Testing for Statisticians: Privacy, Error Rates, and Sample Size](https://mlsanity.com/paper/2204.04597/) (2022)
- [Private independence testing across two parties](https://mlsanity.com/paper/2207.03652/) (2022)
- [ResidualPlanner+: a scalable matrix mechanism for marginals and beyond](https://mlsanity.com/paper/2305.08175/) (2023)
- [On Differentially Private U Statistics](https://mlsanity.com/paper/2407.04945/) (2024)
- [Optimal Algorithms for Augmented Testing of Discrete Distributions](https://mlsanity.com/paper/2412.00974/) (2024)
- [Private measurement of nonlinear correlations between data hosted across multiple parties](https://mlsanity.com/paper/2110.09670/) (2021)
- [On the Computational Complexity of Private High-dimensional Model Selection](https://mlsanity.com/paper/2310.07852/) (2023)
- [Private Statistical Estimation via Truncation](https://mlsanity.com/paper/2505.12541/) (2025)
- [Differentially private ratio statistics](https://mlsanity.com/paper/2505.20351/) (2025)
- [Fiducial Matching: Differentially Private Inference for Categorical Data](https://mlsanity.com/paper/2507.11762/) (2025)
- [DP-SPRT: Differentially Private Sequential Probability Ratio Tests](https://mlsanity.com/paper/2508.06377/) (2025)
- [The power of private likelihood-ratio tests for goodness-of-fit in frequency tables](https://mlsanity.com/paper/2109.09630/) (2021)
- [Differentially Private Conditional Independence Testing](https://mlsanity.com/paper/2306.06721/) (2023)
- [DPpack: An R Package for Differentially Private Statistical Analysis and Machine Learning](https://mlsanity.com/paper/2309.10965/) (2023)
- [Optimal Survey Design for Private Mean Estimation](https://mlsanity.com/paper/2501.18121/) (2025)
- [Near-Optimal Private Tests for Simple and MLR Hypotheses](https://mlsanity.com/paper/2601.21959/) (2026)
- [Equivalence Testing Under Privacy Constraints](https://mlsanity.com/paper/2604.06499/) (2026)

## Abstract (arXiv)

Hypothesis testing is a useful statistical tool in determining whether a given model should be rejected based on a sample from the population. Sample data may contain sensitive information about individuals, such as medical information. Thus it is important to design statistical tests that guarantee the privacy of subjects in the data. In this work, we study hypothesis testing subject to differential privacy, specifically chi-squared tests for goodness of fit for multinomial data and independence between two categorical variables. We propose new tests for goodness of fit and independence testing that like the classical versions can be used to determine whether a given model should be rejected or not, and that additionally can ensure differential privacy. We give both Monte Carlo based hypothesis tests as well as hypothesis tests that more closely follow the classical chi-squared goodness of fit test and the Pearson chi-squared test for independence. Crucially, our tests account for the distribution of the noise that is injected to ensure privacy in determining significance. We show that these tests can be used to achieve desired significance levels, in sharp contrast to direct applications of classical tests to differentially private contingency tables which can result in wildly varying significance levels. Moreover, we study the statistical power of these tests. We empirically show that to achieve the same level of power as the classical non-private tests our new tests need only a relatively modest increase in sample size.