Balancing Weights for Non-monotone Missing Data

Balancing weights have been widely applied to single or monotone missingness due to empirical advantages over likelihood-based methods and inverse probability weighting approaches. This paper considers non-monotone missing data under the complete-case missing variable condition (CCMV), a case of missing not at random (MNAR). Using relationships between each missing pattern and the complete-case subsample, we construct a weighted estimator for estimation, where the weight is a sum of ratios of the conditional probability of observing a particular missing pattern versus that of observing the complete-case, given the variables observed in the corresponding missing pattern. However, plug-in estimators of the propensity odds can be unbounded and lead to unstable estimation. Using further relations between propensity odds and balancing of moments across response patterns, we employ tailored loss functions, each encouraging empirical balance across patterns to estimate propensity odds flexibly using a functional basis expansion. We propose two penalizations to control propensity odds model smoothness and empirical imbalance. We study the asymptotic properties of the proposed estimators and show that they are consistent under mild smoothness assumptions. Asymptotic normality and efficiency are developed. Simulation results show the superior performance of the proposed method.