The shared-parameter modeling framework

In the shared-parameter modelling framework, the measurement and the missingness models share a common random effect or latent variable. The random effect captures the association between the measurements and missingness processes and also accounts for the correlation between the repeated measurement. This approach is often used to jointly modeled longitudinal measurements and time-to-event analysis of dropout.

The shared-parameter model (SPM) is defined as

Pr (Ri, Yi, bi | Xi, θθθ, ψψψ) = Pr (Yi | Xi, bi, θθθ) Pr (Ri | Xi, bi, ψψψ) Pr (bi | Xi, G) , (3.13)

where Pr (Yi | Xi, bi, θθθ) is density function of the measurement process. The missingness process

model is Pr (Ri | Xi, bi, ψψψ) and Pr (bi | Xi, G) is the model for the random effects. Yi and Ri are

assumed to be conditionally independent, given the random effects bi. The equation (3.13) can

alternatively be written as

Pr (Ri, Yi | Xi, θθθ, ψψψ) =

Z

Pr (Yi | Xi, bi, θθθ) Pr (Ri | bi, ψψψ) Pr (bi | Gi) dbi. (3.14)

Tsonaka et al. (2009) have demonstrated that shared-parameter model, by construction, implies a missing not at random (MNAR) mechanism. Thus we can write the conditional distribution

Pr (Ri | Yoi, Y m i , ψψψ) = Z Pr (Ri | bi, ψψψ) Pr (bi | Yoi, Y m i ) dY m i , (3.15)

where the random effects for the ith patient is assumed to be bi ∼ N (000, G). It can be observed

that the response indicators depend on the missing response Ym

i through the posterior distribution

Pr (bi | Yoi, Yim) of the random effects. This shows that the SPM is a joint model for non-ignorable

missingness.

A shared-parameter model is defined by specifying a model Pr (Yi | bi) for the measurement process

and a model Pr (Ri | bi) for the missingness process. The most commonly used measurement model

is the linear mixed effect model for the continuous longitudinal outcomes and a logistic regression model describing the probabilities of dropout.

Since the measurement and the dropout models (3.13) share common random effects, setting up a model that assumes that these models do not share common random effects (no linkage) is analogous to MAR mechanism and hence standard methods of analyses can be use to produce valid inferences. On the other hand, we may also assume that the measurement and the dropout models share a common random effect (linkage). This allows us to make alternative and plausible assumptions about the manner in which these random effects are shared between the measurement and the dropout models. This provides a formal framework for assessing sensitivity of the no linkage analysis results to the different assumptions about the linkage between the measurement and the

dropout models. This implies that the dependence of the measurement and the dropout models on the random effects forms the basis for sensitivity analyses.

Several authors have proposed the SPM for incomplete longitudinal data. The use of the shared- parameter model (Wu and Bailey, 1988; Wu and Carroll, 1988; Creemers et al., 2010) has been one approach to accounting for nonrandom missingness. In Wu and Carroll (1988) study of repeated measurements of lung function, they proposed the SPM. Their study assumed the linear mixed effect model (LMM) (Laird and Ware, 1982) with a random intercept and slope terms. The dropout process was modeled using a probit model for the censoring process. The LMM was linked with the censoring process model by including a patient’s random slope as a covariate in the probit model for the censoring process. In this way, when the probit regression coefficient for the random slope is non-zero, it gives an indication that there is dependence between the measurement and the missingness processes.

The random effects in the SPM reflect patient’s deviation from the mean estimates of the fixed effects. Each patient draws a random slope from a Gaussian distribution. This slope governs both the patient’s expected rate of decline in the response and the probability of dropping out (see Albert and Follmann in Fitzmaurice et al., 2008, Ch. 19). They also pointed out that the shared-parameter models make assumptions which can only in part be justified, using the data. For instance, one can determine whether, for the observed data, change in response follows a linear function of time or a quadratic function of the log time, using standard diagnostics tools such as examining the residual correlation. On the contrary, changes in the response for the unobserved data, which are imposed by the SPM, cannot be verified.

The choice of measurement model depends on the type longitudinal data being analyzed. For normally distributed longitudinal data, the linear mixed model is often assumed. For discrete or dichotomous longitudinal data, the response model can be formulated as a generalized linear mixed effect model GLMM (Follmann and Wu, 1995a; Thomas et al., 1998; Albert and Follmann, 2000). The type of model for the missingness process depends on the type of missing data being considered. For instance, when the data are a discrete time to dropout then a geometric distribution can be used to model the missingness process (Mori et al., 1994). Mori et al. (1994) geometric model assumes that the number of observations per patient in the study is determined by his or her true rate of change. Several authors have proposed the SPM for the case where dropout is a continuous event time (Schluchter, 1992; De Gruttola and Tu, 1994; Tsiatis et al., 1995; Tsiatis and Davidian, 2001; Vonesh et al., 2002) and the logistic regression model is assumed for the probability of dropout.