Missing data are a common problem for both the construction and implementation of a prediction algorithm. Pattern submodels (PS)-a set of submodels for every missing data pattern that are fit using only data from that pattern-are a computationally efficient remedy for handling missing data at both stages. Here, we show that PS (i) retain their predictive accuracy even when the missing data mechanism is not missing at random (MAR) and (ii) yield an algorithm that is the most predictive 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 PS 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 MAR, MI can yield comparable forecasting performance but generally requires a larger computational cost. We also show that predictions from the PS approach are equivalent to the limiting predictions for a MI procedure that is dependent on missingness indicators (the MIMI model). The focus of this article is on out-of-sample prediction; implications for model inference are only briefly explored.

中文翻译：

---

数据丢失和预测：模式子模型。

对于预测算法的构造和实现而言，数据丢失是一个普遍的问题。模式子模型（PS）-每个丢失数据模式的一组子模型，这些子模型仅使用来自该模式的数据就可以拟合-是一种在两个阶段都可以处理丢失数据的有效计算方法。在这里，我们表明PS（i）即使丢失的数据机制并非随机丢失（MAR）仍保持其预测准确性，并且（ii）产生一种在所有标准丢失数据策略中最具预测性的算法。具体来说，我们显示出当每种模式特定的损失最小时，预测算法的预期损失也最小。仿真和对SUPPORT研究的重新分析证实，PS通常优于零输入，均值输入，完整案例分析，完整案例子模型，甚至多重插补（MI）。改善的程度高度取决于缺失机制和缺失预测变量的影响大小。当数据为MAR时，MI可以产生可比的预测性能，但通常需要更大的计算成本。我们还表明，来自PS方法的预测等同于依赖于缺失指标（MIMI模型）的MI过程的极限预测。本文的重点是样本外预测。仅简要探讨了模型推断的含义。我们还表明，来自PS方法的预测等同于依赖于缺失指标（MIMI模型）的MI过程的极限预测。本文的重点是样本外预测。仅简要探讨了模型推断的含义。我们还表明，来自PS方法的预测等同于依赖于缺失指标（MIMI模型）的MI过程的极限预测。本文的重点是样本外预测。仅简要探讨了模型推断的含义。