The selection modelling framework - Summary and discussion

2.7 Summary and discussion

3.1.1 The selection modelling framework

The selection model can be viewed as a multivariate model where one variable represents the marginal density of the measurements process and the other variable represents the conditional density of the missingness process, given the outcomes.

Some of the earlier work on methods for handling non-ignorable missing data in longitudinal studies was conducted by Wu and colleagues (Wu and Carroll, 1988). Wu and Carroll (1988) proposed a selection modelling approach, which is now being used by many subsequent researchers. Their selection modelling approach assumes that the continuous responses follow a simple linear random- effects model (Laird and Ware, 1982) and that the dropout process depends upon an individual’s random intercept and slope. Models where the dropout probabilities depend indirectly upon the distribution of the unobserved responses, via the random effects, are often referred to as shared- parameter models (Fitzmaurice et al., 2008, Ch. 13 and Ch. 19). We discuss shared-parameter models in Section 3.1.3. An alternative selection modelling approach, based on earlier work in the univariate setting by econometricians Heckman (Heckman, 1976), was proposed by Diggle and Kenward (1994). These author allowed the probability of non-response to depend directly on the distribution of the unobserved outcome rather than on underlying random effects.

Heckman (1976) introduced a Tobit model for a potentially missing continuous outcome such that this model combines a marginal Gaussian regression model for the measurement process with a probit model for the missingness process. However, the structure of this model and its associated estimation approach was subjected to much debate. This debate focus was on the problem of

identifiability (data not enough to estimate all parameters specified by the model) and sensitivity (robustness of inferences to changing assumptions about the missing data). This basic structure of the Tobit model forms the basis of the simplest forms of the selection models that have been proposed in a wide range of disciplines (Molenberghs et al., 2014, pp.55). In these models, a suitable model such as a multivariate normal model for the measurement process is combined with a binary regression model for the missingness process. Often, logit of the probability of dropout at each time point is regressed on the previous and current outcomes.

The selection model (SeM) can be defined as (Rubin, 1976; Little and Yau, 1996)

Pr (Yij, Di | θθθ, ψψψ) = Pr (Yij | θθθ) Pr (Di | Yij, ψψψ) , (3.1)

where θθθ is a vector of parameters describing the measurement process, ψψψ is a vector of parameters describing the missing data mechanism and Diindicates a scheduled visit time where the ith patient

dropped out. The first component Pr (Yij | θθθ) of this selection model (3.1) is the marginal density

of the measurement processes and the second component Pr (Di | Yij, ψψψ) is the conditional density

of missingness and the measurement processes. Rubin (1976) showed that when the missing data mechanism is described (MAR) and θθθ and ψψψ are functionally independent, the likelihood-based and Bayesian methods yield valid inferences (Verbeke and Molenberghs, 2009; Fitzmaurice et al., 2008). The likelihood-based methods involve maximization of the full-data likelihood function

L∗ ≡ L∗(θθθ, ψψψ | Y, D) = N Y i=1 Pr (Yi, Di | θθθ, ψψψ) = N Y i=1 Pr (Y_io, Ym_i , Di | θθθ, ψψψ) . (3.2)

In the presence of missing values, however, inferences must be based on what is observed, and thus, the full-data likelihood L∗ must be replaced by the observed-data likelihood L, for which the individual likelihood contributions need to be integrated over the missing values defined as

L ≡ L (θθθ, ψψψ | Y, D) = N Y i=1 Z Pr (Y_io, Ym_i , Di | θθθ, ψψψ) dYmi . (3.3)

Restricting our focus on the observed-data likelihood, the contribution of the ith _{subject, under the}

SeM framework (3.1), the integral in (3.3) becomes

Li ≡ Li(θθθ, ψψψ | Yi, Di) = N Y i=1 Z Pr (Yo_i, Y_im | θθθ, ψψψ) Pr (Ri | Yio, Y m i | ψψψ) dY m i . (3.4)

Rubin (1976) classified the missing data mechanism according to the selection model (3.1) as follows. For MCAR, the missing data mechanism component is defined as Pr (Di | Yoi, Y

i , θθθ, ψψψ) =

Pr (Di | ψψψ). It follows that the selection model can now be written as

Pr (Yo_i, Di | θθθ, ψψψ) = Pr (Yoi | θ) Pr (Di | ψψψ) . (3.5)

produce valid inference. This is because, the dropout Di process is independent of the observed Yoi

and the unobserved Yo_i components of Yi.

For MAR, the missing data mechanism component is defined as Pr (Di | Yoi, Y m

i , θθθ, ψψψ) = Pr (Di | Yoi, ψψψ)

and the selection model can now be written as

Pr (Yo_i, Di | θθθ, ψψψ) = Pr (Yio| θθθ) Pr (Di | Yoi, ψψψ) . (3.6)

The implication of model (3.6) is that valid inferences cannot be obtained without accounting for the missing data mechanism. This is because, the dropout Di process depends on the observed Yoi

component of Yi. However, under the ignorable condition, the observed data and the missing data

models can be fitted separately to obtain valid inferences.

For NMAR, the missing data mechanism component is defined as Pr (Di | Yoi, Y m

i , θθθ, ψψψ) = Pr (Di | Yoi, Y m i ψψψ).

This defines the non-ignorable missingness, and the selection model can now be written as

Pr (Yo_i, Di | θθθ, ψψψ) = Z Pr (Yo_i, Ym_i | θθθ) Pr (Di | Yoi, Y m i , ψψψ) dY m i . (3.7)

The implication of model (3.7) is that valid inference cannot be obtained without accounting for the missing data mechanism. This is because the dropout Di process depends on the observed Yoi and

the unobserved Ym_i components of Yi. The model (3.7) is a joint model for non-ignorable missing

data. This is because the measurement and the missingness processes have to be modeled jointly to obtained valid inferences.

Rubin (1976) showed that when data are assumed to be MAR and ψψψ and θθθ are functionally independent, valid inference can be obtained by modeling the measurement and the missingness processes separately. This is called ignorability. This means that the likelihood-based inference remains valid when the missing data mechanism is ignored. In this case, the likelihood of interest is then based upon the observed data.

The ith _{patient contribution to the likelihood based on (3.6) is of the form}

L∗_i (θθθ, ψψψ | Yi, Ri) ∝ (Yi, Ri | θθθ, ψψψ) . (3.8)

Since inference has to be based on what is observed, the full data likelihood Li has to be replaced

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