Recursive Equations For Imputation Of Missing Not At Random Data With Sparse Pattern Support

A common approach for handling missing values in data analysis pipelines is multiple imputation via software packages such as MICE (Van Buuren and Groothuis-Oudshoorn, 2011) and Amelia (Honaker et al., 2011). These packages typically assume the data are missing at random (MAR), and impose parametric or smoothing assumptions upon the imputing distributions in a way that allows imputation to proceed even if not all missingness patterns have support in the data. Such assumptions are unrealistic in practice, and induce model misspecification bias on any analysis performed after such imputation. In this paper, we provide a principled alternative. Specifically, we develop a new characterization for the full data law in graphical models of missing data. This characterization is constructive, is easily adapted for the calculation of imputation distributions for both MAR and MNAR (missing not at random) mechanisms, and is able to handle lack of support for certain patterns of missingness. We use this characterization to develop a new imputation algorithm -- Multivariate Imputation via Supported Pattern Recursion (MISPR) -- which uses Gibbs sampling, by analogy with the Multivariate Imputation with Chained Equations (MICE) algorithm, but which is consistent under both MAR and MNAR settings, and is able to handle missing data patterns with no support without imposing additional assumptions beyond those already imposed by the missing data model itself. In simulations, we show MISPR obtains comparable results to MICE when data are MAR, and superior, less biased results when data are MNAR. Our characterization and imputation algorithm based on it are a step towards making principled missing data methods more practical in applied settings, where the data are likely both MNAR and sufficiently high dimensional to yield missing data patterns with no support at available sample sizes.