Prediction with Missing Data: Target Probabilities and Missingness Mechanisms
Conditions ensuring optimal parameter estimation in the presence of missing data are well established in inference, typically relying on the Missing-at-Random (MAR) assumption. In prediction, similar principles are often assumed to apply. However, methods considered biased in inference, such as pattern sub-modelling or unconditional imputation, have been shown to achieve optimal predictive performance under any missingness mechanism, including non-MAR (MNAR). To explain this apparent contradiction, we introduce a new formal framework for describing missingness in prediction. Central to this framework is a distinction between two prediction targets, defined according to whether or not the indicator of observation of the predictors is exploited to predict the outcome. This distinction leads to a classification of the missingness mechanisms describing the conditions under which these targets are equal, and when consistent prediction of each is achievable. A key result is that both targets may be consistently predicted under conditions weaker than MAR. We discuss the implications of this paradigm for handling missing data in prediction, distinguishing between missingness at development, validation and deployment of a forecaster. The findings are illustrated using simulated data and a real-world application with the prediction of significant injury after trauma upon arrival at the emergency department.