Mixture models of missing data

This paper proposes a general framework for the analysis of survey data with missing observations. The approach presented here treats missing data as an unavoidable feature of any survey of the human population and aims at incorporating the unobserved part of the data into the analysis rather than trying to avoid it or make up for it. To handle coverage error and unit non-response, the true distribution is modeled as a mixture of an observable and of an unobservable component. Generally, for the unobserved component, its relative size (the no-observation rate) and its distribution are not known. It is assumed that the goal of the analysis is to assess the fit of a statistical model, and for this purpose the mixture index of fit is used. The mixture index of fit does not postulate that the statistical model of interest is able to account for the entire population rather, that it may only describe a fraction of it. This leads to another mixture representation of the true distribution, with one component from the statistical model of interest and another unrestricted one. Inference with respect to the fit of the model, with missing data taken into account, is obtained by equating these two mixtures and asking, for different no-observation rates, what is the largest fraction of the population where the statistical model may hold. A statistical model is deemed relevant for the population, if it may account for a large enough fraction of the population, assuming the true (if known) or a sufficiently small or a realistic no-observation rate.