## Shared Random Effects Models and Informative Missingness

In longitudinal clinical trials, repeated measures are often highly variable over time. Rather than modeling the missing data mechanism directly as a function of Yt, as in the selection models discussed in the previous section, Wu and Carroll (1988) exploit the fact that we have repeated responses on each individual to model the missing data mechanism in terms of features of an individual's underlying response process. They propose methodology in which informative dropout is accounted for by introducing random effects that are shared between the model for the longitudinal observations and the dropout mechanism. Thus, the missing data mechanism depends on an individual's underlying response process as opposed to the actual responses. Let d, denote the dropout time for the ith subject. The joint distribution of (Yt, d,, Bi | Xj) can be factored as

/(Yi, di, Bi | Xj) = g(Yi | Xi; Bi)k(Bi | Xi)h(di | Xi; Yj, Bi) (6)

where h(d,|Xi, Yj, Bi) is assumed to depend on Yi only through Bi, i.e., h(d, |Xi, Yi, Bi) = h(d, |Xi, Bi).

These models were first proposed by Wu and Carroll (1988) for modeling nonignorable dropout and have been referred to as random coefficient selection models (Little, 1995). Wu and Carroll (1988) propose their methodology in the setting of a two group longitudinal clinical trial where interest focuses on comparing change over time in the presence of informative dropout. They proposed a random effects model with a random intercept and slope (e.g., a random vector Bi = (b1i, b2i)' for the ¿th subject). The dropout mechanism was modeled as a multinomial with shared random effects incorporated through a probit link. Specifically, the probability of the ¿th subject dropping out within the first j intervals is parameterized as P(d, V j) = pj = /(a'Bi + a0j), where a = (a1, a2). Large positive values of a1 or a2 correspond to the situation where those individuals with large intercept or slopes tend to drop out of the study sooner. They propose jointly estimating the parameters of both probability mechanisms using weighted least squares. Others have proposed shared random effects models of this type. Schluchter (1992) proposed an E-M algorithm for maximum-likelihood estimation for a model where an individual's slope and log survival are assumed multivariate normal. Mori et al. (1994) proposed a model where the slope of continuous repeated data is related to the number of observations on each subject though a shared random effect. Similarly, Follmann and Wu (1995), Ten Have et al. (1998), and Pulkstenis et al. (1998) have proposed shared random effects models for binary longitudinal data subject to missingness. Albert and Follmann (2000) proposed a shared random effects model for repeated count data with informative dropout.

Wu and Bailey (1989) proposed an alternative approach to account for informative dropout which conditions on the dropout times. In this approach, they model the joint distribution of Yi and Bi given dt and Xj. Specifically,

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