12.3. Data Setting and Modeling Framework

Assume that for subject i = 1,...,N in the study a sequence of responses Y_ij is designed to be measured at occasions j = 1,...,n. The outcomes are grouped into a vector Y_i = (Y_i1,...,Y_in)′. In addition, define a dropout indicator D_i for the occasion at which dropout occurs and make the convention that D_i = n + 1 for a complete sequence. It is often necessary to split the vector Y_i into observed (Y_i^o) and missing (Y_i^m) components respectively. Note that dropout is a particular case of monotone missingness. In order to have a monotone pattern of missingness, there has to exist a permutation of the measurement components such that a measurement earlier in the permuted sequence is observed for at least those subjects that are observed at later measurements. For this definition to be meaningful, we need to have a balanced design in the sense of a common set of measurement occasions. Other patterns are called nonmonotone or intermittent missingness. When intermittent missingness occurs, it is best to use a vector of binary indicators R_i = (R_i1,...,R_in)′ rather than the dropout indicator D_i.

In principle, we would like to consider the density of the full data f(Y_i,d_i|θ,ψ), where the parameter vectors θ and ψ describe the measurement and missingness processes, respectively. Covariates are assumed to be measured but, for notational simplicity, they are suppressed from notation.

The taxonomy, constructed by Rubin (1976), further developed in Little and Rubin (1987), and informally sketched in Section 12.1, is based on the factorization

where the first factor is the marginal density of the measurement process and the second one is the density of the missingness process, conditional on the outcomes. Factorization (12.3.1) forms the basis of selection modeling as the second factor corresponds to the (self-)selection of individuals into observed and missing groups. An alternative taxonomy can be built based on so-called pattern-mixture models (Little, 1993; Little, 1994a). These are based on the factorization

Indeed, (12.3.2) can be seen as a mixture of different populations, characterized by the observed pattern of missingness.

Rubin (1976) and Little and Rubin (1987) have shown that, under MAR and mild regularity conditions (parameters θ and ψ are functionally independent), likelihood-based and Bayesian inferences are valid when the missing data mechanism is ignored (see also Verbeke and Molenberghs, 2000; Molenberghs and Verbeke, 2005). Practically speaking, the likelihood of interest is then based upon the factor f(Y_i^o|θ). This is called ignorability.

The practical implication is that a software module with likelihood estimation facilities and with the ability to handle incompletely observed subjects manipulates the correct likelihood,providing valid parameter estimates and likelihood ratio values.Dmitrienko et al. (2005, Chapter 5) provide detailed guidelines on how to implement such analyses. They also issue a number of cautionary remarks. An important one is that the flexibility and ease of MAR, (and hence ignorable) analyses, do not rule out the option of an MNAR mechanism to operate. These authors also focused primarily on continuous outcomes. In this text, both discrete outcomes, as well as modeling approaches under MNAR are of primary interest.