Robust Optimal Designs when Missing Data Happen at Random

Abstract

In this article, we investigate the robust optimal design problem for the prediction of response when the fitted regression models are only approximately specified, and observations might be missing completely at random. The intuitive idea is as follows: We assume that data are missing at random, and the complete case analysis is applied. To account for the occurrence of missing data, the design criterion we choose is the mean, for the missing indicator, of the averaged (over the design space) mean squared errors of the predictions. To describe the uncertainty in the specification of the real underlying model, we impose a neighborhood structure on the deterministic part of the regression response and maximize, analytically, the \textbf{M}ean of the averaged \textbf{M}ean squared \textbf{P}rediction \textbf{E}rrors (MMPE), over the entire neighborhood. The maximized MMPE is the ``worst'' loss in the neighborhood of the fitted regression model. Minimizing the maximum MMPE over the class of designs, we obtain robust ``minimax'' designs. The robust designs constructed afford protection from increases in prediction errors resulting from model misspecifications.