Statistical Analysis with Missing Data., 3rd Edition

7Factored Likelihood Methods When the Missingness Mechanism Is Ignorable

7.1 Introduction

We now assume that the missingness mechanism is ignorable, and for simplicity, write ℓ(θ ∣ y₍₀₎) for the ignorable loglikelihood ℓ_ign(θ ∣ y₍₀₎) based on the observed data y₍₀₎. This can be a complicated function with no obvious maximum and an apparently complex form for the information matrix. Analogous issues arise with simulating from the resulting posterior distributions without simple structures. For certain models and incomplete data patterns, however, analyses based on ℓ(θ ∣ y₍₀₎) can employ standard complete-data techniques. The general idea will be described here in this section, and then, specific examples will be given in the remainder of this chapter for normal data and in Section 7.2 for multinomial (that is, cross-classified) data.

For a variety of models and missingness patterns, an alternative parameterization φ = φ(θ), where φ is a one-to-one function of θ, can be found such that the loglikelihood decomposes into J terms

(7.1)

where

1. φ₁, φ₂,…, φ_J are distinct parameters, in the sense that the joint parameter space of φ = (φ₁, φ₂,…, φ_J) is the product of the individual parameter spaces for φ_j, j = 1,…, J; and
2. each component ℓ_j(φ_j ∣ y₍₀₎) corresponds to a loglikelihood for a complete-data problem, or more generally, for an incomplete-data problem that is easier to analyze ...