On the multiple imputation variance estimator for control-based and delta-adjusted pattern mixture models

Control-based pattern mixture models (PMM) and delta-adjusted PMMs are commonly used as sensitivity analyses in clinical trials with non-ignorable dropout. These PMMs assume that the statistical behavior of outcomes varies by pattern in the experimental arm in the imputation procedure, but the imputed data are typically analyzed by a standard method such as the primary analysis model. In the multiple imputation (MI) inference, Rubin's variance estimator is generally biased when the imputation and analysis models are uncongenial. One objective of the article is to quantify the bias of Rubin's variance estimator in the control-based and delta-adjusted PMMs for longitudinal continuous outcomes. These PMMs assume the same observed data distribution as the mixed effects model for repeated measures (MMRM). We derive analytic expressions for the MI treatment effect estimator and the associated Rubin's variance in these PMMs and MMRM as functions of the maximum likelihood estimator from the MMRM analysis and the observed proportion of subjects in each dropout pattern when the number of imputations is infinite. The asymptotic bias is generally small or negligible in the delta-adjusted PMM, but can be sizable in the control-based PMM. This indicates that the inference based on Rubin's rule is approximately valid in the delta-adjusted PMM. A simple variance estimator is proposed to ensure asymptotically valid MI inferences in these PMMs, and compared with the bootstrap variance. The proposed method is illustrated by the analysis of an antidepressant trial, and its performance is further evaluated via a simulation study.