Differentially Private Inference for Extrema of Parameters in Clinical Studies

Abstract In clinical studies, inference on extrema parameters is crucial for identifying and confirming critical health outcomes and promising subgroups. Directly applying the usual statistical inference to the extrema of parameters often suffers from selection bias, and the problem becomes more acute as in many practical scenarios of clinical studies, we often need to protect the privacy of the data, which might further hinder the correction of the bias and the improvement of statistical accuracy and efficiency. In this paper, we construct a valid and efficient lower confidence bound for the extrema of parameters under privacy protection in clinical studies. We focus on the exponential family of distributions and propose a privatized parametric bootstrap method to address selection bias in the extrema of parameters problem under the scheme of differential privacy. While the usual privatized parametric bootstrap does not address selection bias appropriately, we show that with a privatized bias correction term, the proposed parametric bootstrap can lead to a valid, efficient, and privatized lower confidence limit for the extrema of parameters. Examples of the exponential family of distributions used in clinical studies, such as multivariate Gaussian and linear regression models, are discussed, with an additional focus on extensions for partial privacy protection. We demonstrate the merits of the proposed method by revisiting the AIDS Clinical Trials Group Study 175 (ACTG175) program with the proposed method.