On Optimal Prediction Rules with Prospective Missingness and Bagged Empirical Null Inference in Large-Scale Data

Abstract

Missing data are a common problem for both the construction and implementation of a prediction algorithm. Pattern mixture kernel submodels (PMKS) - a series of submodels for every missing data pattern that are fit using only data from that pattern - are a computationally efficient remedy for both stages. Here we show that PMKS yield the most predictive algorithm among all standard missing data strategies. Specifically, we show that the expected loss of a forecasting algorithm is minimized when each pattern-specific loss is minimized. Simulations and a reanalysis of the SUPPORT study confirms that PMKS generally outperforms zeroimputation, mean-imputation, complete-case analysis, complete-case submodels, and even multiple imputation (MI). The degree of improvement is highly dependent on the missingness mechanism and the effect size of missing predictors. When the data are Missing at Random (MAR) MI can yield comparable forecasting performance but generally requires a larger computational cost. We see that predictions from the PMKS are equivalent to the limiting predictions for a MI procedure that uses a mean model dependent on missingness indicators (the MIMI model). Consequently, the MIMI model can be used to assess the MAR assumption in practice. The focus of this paper is on out-of-sample prediction behavior; implications for model inference are only briefly explored.