Multivariate Mean Comparison under Differential Privacy

TL;DR

This paper develops a differentially private hypothesis test for multivariate mean comparison using a privatized Hotelling's $t^2$-statistic and a bootstrap method, ensuring privacy without sacrificing test reliability.

Contribution

It introduces a novel private testing procedure for multivariate means that combines Hotelling's statistic with a bootstrap algorithm to control error rates.

Findings

The proposed test maintains type-1 error control under differential privacy.

Empirical results demonstrate the test's practical applicability.

The method effectively balances privacy guarantees with statistical power.

Abstract

The comparison of multivariate population means is a central task of statistical inference. While statistical theory provides a variety of analysis tools, they usually do not protect individuals' privacy. This knowledge can create incentives for participants in a study to conceal their true data (especially for outliers), which might result in a distorted analysis. In this paper we address this problem by developing a hypothesis test for multivariate mean comparisons that guarantees differential privacy to users. The test statistic is based on the popular Hotelling's $t^2$-statistic, which has a natural interpretation in terms of the Mahalanobis distance. In order to control the type-1-error, we present a bootstrap algorithm under differential privacy that provably yields a reliable test decision. In an empirical study we demonstrate the applicability of this approach.