The pattern-mixture modeling framework

Since Yi and Riare independent, valid likelihood-based inference can obtained by fitting Pr (Yoi | θθθ)

and Pr (Ri | Yoi, ψψψ) separately (Rubin, 1976, 1996). For Bayesian inferences, a prior distribution is

assumed for each of the parameters in the selection model (Daniels and Hogan, 2008, pp. 168), and valid inferences can be obtained if besides the separability condition (when θθθ and ψψψ are distinct), the priors are independent (Little and Rubin, 1987; Daniels and Hogan, 2008, pp. 168). A posterior distribution is then constructed (Daniels and Hogan, 2008, see pp. 180-181), using the data likelihood and the prior distributions of the parameters, to sample/obtain parameter estimates in the selection model. Complexity often arise in the integration of likelihood with respect to the missing response and Bayesian MCMC approaches avoid direct evaluation of this integral through data augmentation (Daniels and Hogan, 2008, pp. 178).

We note that when the separability condition is satisfied, within the likelihood framework, ignor- ability is equivalent to the MAR or MCAR. This implies that non-ignorability and MNAR are equivalent in this context. A formal derivation given by Rubin (1976), and Little and Rubin (1987), showed that the same requirements hold for Bayesian inferences. The implication of the ignora- bility is that a software module with likelihood estimation facilities and with the ability to handle incompletely-observed subjects, manipulates the correct likelihood and thus provides valid param- eter estimates, standard errors if based on the observed information matrix, and likelihood ratio values (Molenberghs et al., 1998). This result makes direct-likelihood analyses viable candidates for the status of primary analysis in clinical trials (Molenberghs et al., 2004).

3.1.2

The pattern-mixture modeling framework

The pattern-mixture model (PMM) framework is a reverse factorization of the selection model (Heckman, 1976; Rubin, 1976; Diggle and Kenward, 1994; Little and Yau, 1996). The PMM ap- proach can be specified as

Pr (Yi, Ri | Xi, θθθ, ψψψ) = Pr (Yi | Xi, Ri, θθθ)

× Pr (Ri | Xi, ψψψ) .

(3.12)

The PMM has desirable properties especially when NMAR (probability that a response will be missing depends on the Ri and Yi) situations are examined in the analysis (Wu and Bailey, 1988).

For instance, where it is not substantively reasonable to consider non-responses as missing data, it may be desirable to limit the inferences to the subpopulation of patients whose responses are

observed. Thus, it is more meaningful to consider the distribution of Yij given Rij = 1 (Rij = 1

if subject is observed and 0 otherwise), rather than the marginal distribution of Yij (Rubin, 1976).

Contrary to the selection model, Pr (Ym

i | Yoi, Xi, Ri) is modeled directly from the pattern-mixture

model.

One important feature of the pattern-mixture model (3.12) is that it fits different response model for each pattern of missingness such that the observed data is a mixture of patterns weighted by their respective probabilities of missing patterns. That is, the first component in the PMM (3.12), Pr (Yi | Xi, Ri, θθθ) fits a response model for each pattern of missingness and Pr (Ri | Xi, ψψψ)

represents drop out probability for each pattern. It follows that if there are U number of missingness patterns in a data set, then the marginal distribution of Yi is a mixture of Pr (Yi | Xi, θθθ) =

U

P

u=1

Pr (Yi | Ri = Rui, Xi, θθθu) πu, where πu = Pr (Ri = u | Xi, ψψψ) and Ri counts the number of U

patterns, θθθu represents the parameters of marginal density Pr (Yi) in the uth pattern. It can be

observed that in the pattern-mixture model, parameters {θθθ1_{, . . . , θθθ}U_{} can have different dimensions.}

A logistic model is often assumed for dropout probabilities and a linear mixed effected model (Laird and Ware, 1982) for the measurement process.

The pattern-mixture model (3.12) is well understood using the second MAR assumption. The second MAR assumption states that observations that would have been recorded for a patient in the future, given that the observed history of such a patient has the same statistical behavior irrespective of whether such patient dropout or does not dropout in the future. This feature of the pattern-mixture model makes it possible for multiple imputation to provide a practical approach to estimation and inferences. In addition, this feature provides a framework for the formulation of the pattern-mixture model with multiple imputation (Carpenter et al., 2013).

The pattern-mixture modeling framework was proposed by Wu and Bailey (1988); Wu and Carroll (1988); Wu and Bailey (1989) as approximate methods for making inferences about the time course of the continuous responses. Their approach was aimed at avoiding dependence on the complex model fitting in the selection modelling approach, earlier proposed by Wu and Bailey (1989). Their approach was based on method-of-moments type fitting of a linear model to the least squares slopes, given the dropout time and then averaging over the distribution of the dropout time. Apart from assuming that dropouts occur at discrete times, their approach does not provide any distributional assumptions on the event times. This approach was latter extended by Hogan and Laird (1997) to allow censored dropout times, which might occur when there is a group of patients who joined the trial at the latter period and temporary analyses can be carried out. Following later modification to the pattern-mixture model proposed by Wu and Bailey (1989), Follmann and Wu (1995a) gen- eralized their conditional linear model to allow generalized linear models without any parametric assumption on the random effects. Some research related to pattern-mixture models can be found in Rubin (1977); Fitzmaurice and Laird (2000). Thijs et al. (2002) outlined a hierarchy for the

different ways of handling and fitting pattern-mixture models.

The pattern mixture models are rarely used for an arbitrary pattern of missingness because of the increase in the number of potential patterns. However, this is unlikely to occur when missingness is confined to dropout. The pattern-mixture model implies different distributions for each pattern of missingness or dropout. This implies that each deviation pattern will have its associated mean vector µ

µ

µu and covariance matrix ΣΣΣu, where u represents deviation or dropout pattern. By construction,

pattern-mixture models are under-identified or over-specified such that the data do not contain information on all the parameters specified by the model (Mallinckrodt et al. 2013; Molenberghs et al. 2014, pp.68).

The issue with estimating the inestimable parameters from the pattern-mixture model can be solved by using identifying restriction. Under such restriction, the parameters that cannot be estimated from the incomplete patterns are set equal to function of the parameters describing the distribution of other patterns. There are two main approaches to identifying pattern-mixture models. The first approach involves the use of outcome models that are sufficiently constrained so that such models can be identified within the different dropout patterns. However, the constraints that are required to make the model estimable in all the different dropout pattern implies the use of polynomial extrapolation which may be difficult to justify from substantive perspective (Molenberghs et al. 2014, pp.71).

The second approach, which is widely used to identifying pattern-mixture models relies on the idea of identifying restrictions. The basic idea behind identifying restriction is that one can either “borrow” the unidentifiable distributional information from the completers (Little, 1993), know as complete case missing values (CCMV) or from the nearest identified pattern (Mallinckrodt, Lin and Molenberghs, 2013), known as neighbouring case missing values (NCMV). An alternative identifying restriction approach is the available case missing value (ACMV); where the unavailable information is borrowed from the observed data.

A recent identifying restriction approach, which has been proven to be useful is to equate conditional distributions from the different treatment groups with the objective of representing patients in the study who deviate from the study protocol. However, whether such restriction will be intuitive is subject to the context of the study, particularly the objectives of the analysis, the nature of the outcome measurement and the action of the interventions (Albert et al., 2002; Carpenter et al., 2013). Analysis based on this identifying restriction has been developed by Little and Yau (1996) and then recently developed by Carpenter et al. (2013) for handling protocol deviations leading to missing data.