The Exact Risks of Reference Panel-based Regularized Estimators

Reference panel-based estimators have become widely used in genetic prediction of complex traits due to their ability to address data privacy concerns and reduce computational and communication costs. These estimators estimate the covariance matrix of predictors using an external reference panel, instead of relying solely on the original training data. In this paper, we investigate the performance of reference panel-based $L_1$ and $L_2$ regularized estimators within a unified framework based on approximate message passing (AMP). We uncover several key factors that influence the accuracy of reference panel-based estimators, including the sample sizes of the training data and reference panels, the signal-to-noise ratio, the underlying sparsity of the signal, and the covariance matrix among predictors. Our findings reveal that, even when the sample size of the reference panel matches that of the training data, reference panel-based estimators tend to exhibit lower accuracy compared to traditional regularized estimators. Furthermore, we observe that this performance gap widens as the amount of training data increases, highlighting the importance of constructing large-scale reference panels to mitigate this issue. To support our theoretical analysis, we develop a novel non-separable matrix AMP framework capable of handling the complexities introduced by a general covariance matrix and the additional randomness associated with a reference panel. We validate our theoretical results through extensive simulation studies and real data analyses using the UK Biobank database.