Conditions for Ignoring the Missing-Data Mechanism in Likelihood Inferences for Parameter Subsets

ABSTRACT

For likelihood-based inferences from data with missing values, models are generally needed for both the data and the missing-data mechanism. However, modeling the mechanism can be challenging, and parameters are often poorly identified. Rubin in 1976 showed that for likelihood and Bayesian inference, sufficient conditions for ignoring the missing data mechanism are (a) the missing data are missing at random (MAR), in the sense that missingness does not depend on the missing values after conditioning on the observed data and (b) the parameters of the data model and the missingness mechanism are distinct, that is, there are no a priori ties, via parameter space restrictions or prior distributions, between these two sets of parameters. These conditions are sufficient but not always necessary, and they relate to the full vector of parameters of the data model. We propose definitions of partially MAR and ignorability for a subvector of the parameters of particular substantive interest, for direct likelihood/Bayesian and frequentist likelihood-based inference. We apply these definitions to a variety of examples. We also discuss conditioning on the pattern of missingness, as an alternative strategy for avoiding the need to model the missingness mechanism.

KEYWORDS:

• Bayesian inference
• Incomplete data
• Likelihood inference
• Missingness mechanism
• Partial likelihood
• Pattern-mixture models

Acknowledgments

The authors thank two referees and an associate editor for extensive and very helpful comments on earlier drafts. We also thank Peng Ding for pointing out an error in Example 2 in the “early view” version of the article, which is now corrected.