Differentially Private Hypothesis Testing for Linear Regression

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" id="MathJax-Element-1-Frame" role="presentation" style="position:relative;" tabindex="0">F-statistic for the multivariate linear regression model framework. We also present other differentially private tests---not based on the F" id="MathJax-Element-2-Frame" role="presentation" style="position:relative;" tabindex="0">F-statistic---for these problems. We show that the differentially private F" id="MathJax-Element-3-Frame" role="presentation" style="position:relative;" tabindex="0">F-statistic converges to the asymptotic distribution of its non-private counterpart. As a corollary, the statistical power of the differentially private F" id="MathJax-Element-4-Frame" role="presentation" style="position:relative;" tabindex="0">F-statistic converges to the statistical power of the non-private F" id="MathJax-Element-5-Frame" role="presentation" style="position:relative;" tabindex="0">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.