  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 re-analysis of the SUPPORT study confirms that
PMKS generally outperforms zero-imputation, 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.
