Differentially Private One-Shot Federated Inference for Linear Mixed Models via Lossless Likelihood Reconstruction

Abstract

One-shot federated learning enables multi-site inference with minimal communication. However, sharing summary statistics can still leak sensitive individual-level information when sites have only a small number of patients. In particular, shared cross-product summaries can reveal patient-level covariate patterns under discrete covariates. Motivated by this concern, this study proposes a differentially private one-shot federated inference framework for linear mixed models with a random-intercept working covariance. The method reconstructs the pooled likelihood from site-level summary statistics and applies a Gaussian mechanism to perturb these summaries, ensuring a site-level differential privacy. Cluster-robust variance estimators are developed that are computed directly from the privatized summaries. Robust variance provides valid uncertainty quantification even under covariance mis-specification. Under a multi-site asymptotic regime, the consistency and asymptotic normality of the proposed estimator are established and the leading-order statistical cost of privacy is characterized. Simulation studies show that moderate privacy noise substantially reduces reconstruction risk while maintaining competitive estimation accuracy as the number of sites increases. However, very strong privacy settings can lead to unstable standard errors when the number of sites is limited. An application using multi-site COVID-19 testing data demonstrates that meaningful privacy protection can be achieved with a modest loss of efficiency.