Every missingness not at random model has a missingness at random counterpart with equal fit

Over the last decade a variety of models to analyse incomplete multivariate and longitudinal data have been proposed, many of which allowing for the missingness to be not at random, in the sense that the unobserved measurements influence the process governing missingness, in addition to influences coming from observed measurements and/or covariates. The fundamental problems that are implied by such models, to which we refer as sensitivity to unverifiable modelling assumptions, has, in turn, sparked off various strands of research in what is now termed sensitivity analysis. The nature of sensitivity originates from the fact that a missingness not at random (MNAR) model is not fully verifiable from the data, rendering the empirical distinction between MNAR and missingness at random (MAR), where only covariates and observed outcomes influence missingness, difficult or even impossible, unless we are willing to accept the posited MNAR model in an unquestioning way. We show that the empirical distinction between MAR and MNAR is not possible, in the sense that each MNAR model fit to a set of observed data can be reproduced exactly by an MAR counterpart. Of course, such a pair of models will produce different predictions of the unobserved outcomes, given the observed outcomes. Theoretical considerations are supplemented with an illustration that is based on the Slovenian public opinion survey, which has been analysed before in the context of sensitivity analysis.