Differentially Private Hypothesis Testing for Linear Regression
Daniel G. Alabi, Salil P. Vadhan.
Year: 2023, Volume: 24, Issue: 361, Pages: 1−50
Abstract
In this work, we design differentially private hypothesis tests for the following problems in the multivariate linear regression model: testing a linear relationship and testing for the presence of mixtures. The majority of our hypothesis tests are based on differentially private versions of the $F$-statistic for the multivariate linear regression model framework. We also present other differentially private tests---not based on the $F$-statistic---for these problems. We show that the differentially private $F$-statistic converges to the asymptotic distribution of its non-private counterpart. As a corollary, the statistical power of the differentially private $F$-statistic converges to the statistical power of the non-private $F$-statistic. Through a suite of Monte Carlo based experiments, we show that our tests achieve desired significance levels and have a high power that approaches the power of the non-private tests as we increase sample sizes or the privacy-loss parameter. We also show when our tests outperform existing methods in the literature.