Statistical Analysis with Missing Data., 3rd Edition

15Missing Not at Random Models

15.1 Introduction

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

(15.1)

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:

(15.2)

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 ...