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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 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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