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

TL;DR

This paper develops differentially private chi-squared hypothesis tests for goodness of fit and independence, ensuring privacy while maintaining statistical validity and power.

Contribution

It introduces new private chi-squared tests that account for noise distribution, enabling accurate significance levels and comparable power to classical tests.

Findings

Tests achieve desired significance levels despite privacy noise

Require modest increase in sample size for similar power

Can be applied to sensitive categorical data with privacy guarantees

Abstract

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.