\documentclass[../Main.tex]{subfiles}
\graphicspath{{\subfix{Assets/img/}}}

\begin{document}
%% Describe goal
%   Estimate probability distribution of normalized durations and conclusion statuses.
%   Explain why this answers questions well.
%   How do I propose estimating that?

%%NOTATION

First, some notation:
\begin{itemize}
    \item $t,i$ index trial and snapshot of the trial respectively.
    \item $S_{it}$ : status (``Terminated'', ``Concluded'') 
        at the conclusion of the trial
    \item $\tau_{it}$ : elapsed percentage of planned duration, i.e. the normalized elapsed duration.
    \item $s_{it}$: current status variable(s) describing current trial status as one of:
        ``Recruiting'', ``Active, not recruiting'', or ``Suspended''
    \item $X_{it}$: data describing the trial at the snapshot level
        \begin{itemize}
            \item number of sponsor changes so far
            \item percentage currently enrolled of planned enrolled
        \end{itemize}
    \item $M_{t}$: data describing market conditions at the time of the snapshot.
        \begin{itemize}
            \item number of compounds approved for the same indication
            \item total number of marketers approved for the same indication
            \item number of marketers for this specific compound
        \end{itemize}
    \item $T_t$: Trial Level Data
        \begin{itemize}
            \item Indication or indication class 
            \item Sponsor Type 
        \end{itemize}
\end{itemize} 

Other variables used when selecting trials to include as observations are:
\begin{itemize}
    \item Trial Phase\footnote{
            Conditioning on previous trials can be as simple as selecting only 
            phase 2 or phase 3 trials.
        }.
    \item Does the trial have a Data Monitoring Committee?
    \item Are the compounds an FDA regulated drug?
\end{itemize}


The goal is to estimate the probability distribution of conclusion status and duration conditional on the active snapshot status.

\begin{align}
    \pr{ t, S | X,M,T} = \sum_{S\in\{\text{Terminated},\text{Concluded}\}} \pr{t|S,X,M,T} \pr{S|X,M,T}
\end{align}
Note how this has two component parts: 
\begin{itemize}
    \item The survival probability portion $\pr{t|S,X,M,T}$
    \item The conclusion status portion $\pr{S|X,M,T}$
\end{itemize}

\end{document}