Estimators based on unconventional likelihoods with nonignorable missing data and its application to a children's mental health study

Nonignorable missing data is common in studies where the outcome is relevant to the subject's behaviour. Ibrahim, Lipsitz, and Horton [(2001), ‘Using Auxiliary Data for Parameter Estimation with Non-ignorably Missing Outcomes’, Journal of the Royal Statistical Society: Series C (Applied Statistics), 50, 361–373] fitted a logistic regression for a binary outcome subject to nonignorable missing data, and they proposed to replace the outcome in the mechanism model with an auxiliary variable that is completely observed. They had to correctly specify a model for the auxiliary variable; unfortunately the outcome variable subject to nonignorable missingness is still involved. The correct specification of this model is mysterious. Instead, we propose two unconventional likelihood-based estimation procedures where the nonignorable missingness mechanism model could be completely bypassed. We apply our proposed methods to the children's mental health study and compare their performance with existing methods. The large sample properties of the proposed estimators are rigorously justified, and their finite sample behaviours are examined via comprehensive simulation studies.