# 15Missing Not at Random Models

## 15.1 Introduction

The examples and methods in Chapters 7–14 were based on the ignorable likelihood:

regarded as a function of the parameter θ for fixed observed data *Y*_{(0)}; in (15.1), *X* represents fully observed covariates, and *f*(*Y*_{(0)}|*X*, θ) is obtained by integrating the missing data *Y*_{(1)} out of the density *f*(*Y*|*X*, θ) = *f*(*Y*_{(0)}, *Y*_{(1)}|*X*, θ). In Chapter 6, we showed that sufficient conditions for basing inference about θ on (15.1), rather than the full likelihood from a model for *Y* and *M* given *X*, are that (i) the missing data are missing at random (MAR) and (ii) the parameters θ and *ψ* are distinct, as defined in Section 6.2. In this chapter, we consider situations where the missingness mechanism is missing not at random (MNAR), and valid ML, Bayesian, and multiple-imputation (MI) inferences generally need to be based on the full likelihood:

regarded as a function of the parameters θ, ψ for fixed observed data *Y*_{(0)} and missingness pattern *M*; here *f*(*Y*_{(0)}, *M*|*X*, θ, ψ) is obtained by integrating *Y*_{(1)} out of the joint density *f*(*Y*, *M*|*X*, θ, ψ) based on a joint model for *Y* and *M* given *X*.

Two main approaches for formulating MNAR models can be distinguished. We consider them for situations where the units' values of *M* and *Y* are modeled as independent ...

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