Statistical Analysis with Missing Data., 3rd Edition

8Maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse

8.1 Alternative Computational Strategies

Patterns of incomplete data in practice often do not have the particular forms that allow explicit maximum likelihood (ML) estimates to be calculated by exploiting factorizations of the likelihood. Furthermore, for some models a factorization exists, but the parameters in the factorization are not distinct, and thus maximizing the factors separately does not maximize the likelihood. In this chapter, we consider iterative methods of computation for situations without explicit ML estimates. In some cases, these methods can be applied to incomplete-data factors discussed in Section 7.5.

Suppose, as before, that we have a model for the complete data Y, with associated density f (Y ∣ θ) indexed by unknown parameter θ, generally a vector. We write Y = (Y₍₀₎, Y₍₁₎), where Y₍₀₎ represents the observed part of Y and Y₍₁₎ denotes the missing part. In this chapter, we assume for simplicity that the data are missing at random (MAR) and that the objective is to maximize the likelihood

(8.1)

with respect to θ. Similar considerations apply to the more general case when the data are not MAR, and consequently a factor representing the missingness mechanism is included in the model; such cases are considered in Chapter 15.

When the likelihood ...