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%\title{Significance}
%\author{Brian}
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Our statistical approach is based on a mixed-membership framework. We use the data to determine a small number, $K$, of {\em canonical trajectories} that correspond to {\em idealized progressions} of patient outcomes as a function of age. We model individual outcomes by assuming that individuals borrow characteristics from the canonical profiles in different individual degrees. Our outcome of interest, $Y$, is the cognitive status of each individual at each time point in the study, where $Y=1$ for {\em Normal}, $Y=2$ for {\em Mildly Impaired}, and $Y=3$ for {\em Severely Impaired}. 

The basic MMTM implemented in \cite{molsberry-etalReview}, which we will describe in the remainder of this subsection, begins by modeling the canonical trajectories toward various levels of impairment in each profile $K$, $k = 1, \ldots, K$, as 
%
%The goal of the basic MMTM is to identify canonical trajectories based solely on one time varying predictor, age. The basic model is described by a polytomous logistic regression in each canonical profile $k$, %, and contains three \help{it was six. Error?} parameters: $\boldsymbol{\beta} =(\beta_{0k}, \beta_{1k},c_k)$ for each canonical profile.
%
%Our statistical approach is to model patient outcomes in terms of a single variable Y that codes the cognitive status of each individual at each time point in the study, where Y=1 (Normal), =2 (Mildly Impaired), or =3 (Severely Impaired). The goal of the basic MMTM is to identify canonical trajectories based solely on one time varying predictor, age. The basic model is described by a polytomous logistic regression in each canonical profile $k$, %, and contains three \help{it was six. Error?} parameters: $\boldsymbol{\beta} =(\beta_{0k}, \beta_{1k},c_k)$ for each canonical profile.
%$$
%\begin{array}{rcl}
%P(Y = 1|\rm{ Age}) & = & 1 - P(Y > 1 | \rm{ Age})\\
%P(Y = 2|\rm{ Age}) & = & P(Y > 1| \rm{ Age}) - P(Y> 2|\rm{Age})\\
%P(Y= 3|\rm{ Age}) & = & P(Y > 2|\rm{Age}).
%\end{array}
%$$
\begin{align*}
\mbox{logit} \Pr(Y>1|Age, \boldsymbol{\theta_k}) &= \beta_{0k} + \beta_{1k}Age\\
\mbox{logit} \Pr(Y >2|Age, \boldsymbol{\theta_k}) &= \beta_{0k} + \beta_{1k}Age - c_k,
\end{align*}
where each trajectory $k=1, \ldots,K$ has its own set of trajectory parameters $\boldsymbol{\theta_k} =(\beta_{0k}, \beta_{1k},c_k)$.
The probabilities of each outcome, $d_k(y|Age) = \Pr(Y=y|Age, \boldsymbol{\theta_k})$, can be computed from these two equations by subtraction. In our work to date, $K=3$ profiles best fit the data. In the proposed project we will empirically test models with various numbers of canonical profiles, to determine the number of profiles that best reflects heterogeneity in the data. 


The MMTM allows each individual, indexed by $i$, to have partial  membership in all of the canonical profiles simultaneously. We specify the degree of membership through a vector $\mathbf{g_i} = (g_{i1},...,g_{iK})$, where component $g_{ik}$ specifies the degree of membership of individual $i$ in profile $k$. 

For example, with $K=3$, an individual who is perfectly aligned with the first canonical trajectory would have membership weights of (1, 0 , 0), whereas an individual who shared components of all three trajectories might, for example, have membership weights (.5, .2, .3). We construct individual trajectories toward impairment from the canonical ones by weighting them according to membership,
%For each individual $i$ we estimate the degree $g_{ik}$ to which that individual is a member of the $k^{th}$ canonical profile.  The trajectory for that individual is then calculated as 
$$
\Pr(Y_i = y|Age, \boldsymbol{\theta}, \mathbf{g_i}) = \sum^K_{k=1} g_{ik} d_k(y|Age).
$$
%Each individual has a set of three membership parameters $g_i$, which express the extent to which that individual shares property with each of the individual trajectories. 
%In an effort to describe characteristics of individuals relative to each of the canonical trajectories, we divided the sample into three groups as a function of which trajectory had the highest membership weight. Individuals listed under the ``normal aging'' profile were closest to the normal aging profile compared to either the premature aging or unhealthy profiles (e.g., (.6, .3, .1)). Table 2 shows the percent of individuals within each of these broadly defined groups who had the condition represented by each covariate. That is, 34.9 percent of the individuals closest to the normal aging profile were HIV-infected. We cannot test whether there is a difference in the distribution of the covariates across the canonical profiles because the variables used in the table occur twice: once in the MMTM (as column variables), and the second time in the table itself (as row variables).



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
\cite{manrique-vallier-fienberg2010b}. Following Chapter 6 of \cite{manrique-vallier-fienberg2010b}, and \cite{lecci2014}, 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],%\left[\sum^3_{k=1}g_{ik}w(s_i|\theta_k, \delta_k)\right]
\end{equation}
%\help{Brian: Here we're not considering the joint distribution of all measurements of $y_{it}$. Doing so would complicate the exposition. Do you think it would be worth doing it?}
where $h_k(s_i)$ is the density of the survival distribution for the $k$-th canonical profile, and the outcome is censored if $Age > s_i$. %Following (ME, fabrizzio) we plan to use the Weibull distribution for mathematical tractability. 
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 \cite{holland-rosenbaum}.  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.

Finally, the basic MMTM implemented \cite{molsberry-etalReview} incorporates recruitment cohort and other time-invariant covariates as predictors for membership scores. %Since we keep the extreme trajectories the same for the whole population, this arrangement allows us to read differences in the individual impairment trajectories as differences in the underlying distribution of membership, conditional on cohort.
%\help{Here I've changed the detailed specification of the logit model to something more generic. I think it flows better, but would like your opinion}
%The final MMTM includes critical time invariant covariates by treating the mixed membership parameters, 
Specifically, we treat the mixed membership scores, $\mathbf{g}_{i}$, as random effects sampled from a distribution specified conditional on the individual covariates, $\mathbf{X}_i$,
\begin{equation}
\mathbf{g_i} | \mathbf{X}_i \sim F_{\boldsymbol{\alpha}}(\mathbf{X}_i).
\label{e:baseline-covariates}
\end{equation}
%with means determined by a logit model of the form:
%\begin{equation}\label{eq:2}
%\log \frac{\mathds{E}(g_{.1})}{\mathds{E}(g_{.2})} = (\alpha_{01} - \alpha_{02}) + (\alpha_{11} - \alpha_{12}) X_1 + \ldots + (\alpha_{121} - \alpha_{122})X_{12}.
%\end{equation}
%where each  represents the weighted impact of each covariate on membership in each canonical profile (See \help{(12)} in Appendix). 
We have implemented this specification through a multinomial logit model, and tested time invariant covariates including recruitment cohort, race, the presence of confounding conditions, infection with hepatitis C virus, a history of depression, HIV Disease, and AIDS. %We have found that there are significant differences between recruitment cohorts in the NP substudy \help{(5)}.

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