Covariate Selection Strategy for the Extended Propensity Score to Adjust for Missing Not at Random Data

Missing data can introduce biases in the estimation of the indicator of interest if appropriate adjustments are not made. The case of Missing Not at Random (MNAR), a missing mechanism in which the missingness also depends on the missing values themselves, has been under-explored. When an outcome has MNAR data, one method to estimate the population mean of the outcome is using the extended propensity score. This method first estimates the extended propensity score, which is the missing probability conditional on the outcome and covariates. Then, the population mean of the outcome is estimated using these estimates. In this paper, we discuss which variables should be included in or excluded from the extended propensity score model to obtain an unbiased estimate of the population mean with small standard errors. First, we show which covariates, at a minimum, should be included in the model of missing probability so that the population mean estimator of the outcome is consistent. Next, we show that the inclusion of some covariates in the missing probability model results in a large variance of the population mean estimates even if they explain the missing probability well. Then, we verify these arguments using simulation experiments and argue that to obtain unbiased, small-variance estimates of the population mean, it is desirable to include only those covariates necessary for consistency. This study allows us to obtain such estimates when the outcome is MNAR and adjusted by the extended propensity score.