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The basic MMTM extends this impairment model, through joint
modeling, to address the critical issue of survivorship.  It is known 
that the presence of HAND, at least prior to the use of cART, is strongly
associated with the risk of death.  Joint
modeling allows us to incorporate the information provided by survival
times to better reconstruct impairment trajectories---and
vice-versa---and to address the complications derived from attrition
due to death [3]. Following Chapter 6 of [3], and [2], we introduce a 
survival time $s_i$ for each individual $i$, and
expand the trajectory model for that individual to be
\begin{equation}\label{eq:one}
p(y_i,s_i|Age, \mathbf{g_i}) = \left[\sum^K_{k=1}
  g_{ik}d_k(y_i|Age)\right] \left[\sum^K_{k=1}g_{ik}h_k(s_i)\right],
\end{equation}
where $h_k(s_i)$ is the density of the survival distribution for the
$k$-th canonical profile, and the outcome is censored if $Age > s_i$.
This model builds critically on two assumptions. First, we assume that 
each canonical profile comprises both an idealized trajectory to severe 
impairment, and an idealized mortality distribution; i.e. canonical 
profiles describe both impairment and survival.  Second, given an 
individual's membership vector, $\mathbf{g_i}$,
survival times and cognitive classification are independent.  This is 
last assumption, sometimes referred to as local independence, is common 
and well-studied in factor analysis models, latent class models, and
latent variable modeling generally [1].    Impairment and survival are
dependent in the observed data, but the assumption of local
independence forces $\mathbf{g_i}$ to be rich enough to fully explain
this dependence when we condtion on it.  We deal with the problem of 
right censoring---people who do not
survive long enough to determine their final cognitive
disposition---through a data-augmentation scheme at the time of
estimation via Markov Chain Monte Carlo Simulation.


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