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120 lines
4.4 KiB
TeX
120 lines
4.4 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\begin{document}
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%\subsection{Data Exploration} %TODO: fill this out later.
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%look at trial
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\subsection{Model Fitting}
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In this section we examine the results from fitting the econometric model using
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mc-stan (\cite{mc-stan}) through the rstan (\cite{rstan}) interface.
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%describe
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The model was based on the hierarchal logistic regression model
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presented in the Stan Users Guide (\cite{mc-stan}),
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and was run with 2,500 warmup iterations and
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2,500 sampling iterations in six chains.
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There were various issues, including 160 divergent transitions and the R-hat
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measure was 1.49.
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Overall these suggest that the econometric model is incorrect as
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written or requires reparameterization.
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%TODO: and info about how I learned about these diagnostics
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% \subsubsection{Diagnostics}
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% %Examine trank plots
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% To identify which parameters were problematic, I first looked at trace rank
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% histograms.
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% Under idea circumstances, each line (representing a chain) should exchange
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% places with the other lines frequently.
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% In both \cref{fig:mu_trank} and \cref{fig:sigma_trank}, most parameters seem
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% to mix well but there are a couple of exceptions.
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% This warrants further investigation.
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%
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% \begin{figure}[H]
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% \includegraphics[width=\textwidth]{../assets/img/mu_trank.png}
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% \caption{Trace Rank Histogram: Mu values}
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% \label{fig:mu_trank}
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% \end{figure}
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%
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% \begin{figure}[H]
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% \includegraphics[width=\textwidth]{../assets/img/sigma_trank.png}
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% \caption{Trace Rank Histogram: Sigma values}
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% \label{fig:sigma_trank}
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% \end{figure}
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%
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% %Take a look at batman and points for mu
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% In the case of the Mu values, a parallel coordinates plot
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% doesn't seem to indicate any parameters as likely candidates
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% for causing the issues with divergent transitions.
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% \begin{figure}[H]
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% \includegraphics[width=\textwidth]{../assets/img/mu_batman.png}
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% \caption{Parallel Coordinate Plot: Mu values}
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% \label{fig:mu_batman}
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% \end{figure}
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% Note that at each parameter, there is some level of dispersion between
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% values that diverged.
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%
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% On the other hand, in the parallel coordinates plot for sigma values,
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% it appears that most divergent transitions occur with values of
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% sigma[1], sigma[3], sigma[6], and sigma[7] close to zero.
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% \begin{figure}[H]
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% \includegraphics[width=\textwidth]{../assets/img/sigma_batman.png}
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% \caption{Parallel Coordinate Plot: Sigma values}
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% \label{fig:sigma_batman}
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% \end{figure}
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% Overall this suggests that there is an issue with the specification
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% of the covariance structures of the hyperparameters.
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%
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% Additional evidence that the covariance structure is incorrect comes from
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% plotting pairs of parameter values and examining the chains with divergent
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% transitions.
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%
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% \begin{figure}[H]
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% \includegraphics[width=\textwidth]{../assets/img/sigma_pairs_5-9.png}
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% \caption{Parameter Pairs plots: Sigma[5] through Sigma[9]}
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% \label{fig:sigma_pairs_5-9.png}
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% \end{figure}
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% From this we can see that divergent pairs are highly correlated with the cases
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% where sigma[6] or sigma[7] are equal to zero.
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% This has an impact on the shape of both of those estimated parameters, causing
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% both to be bimodal.
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\subsection{Interpretation}
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The key results so far are related to the distribution of differences in $p$.
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In figure \ref{fig:pred_dist_dif_delay} we see that there while most trials do not see any increased risk
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from a delay in closing enrollment, there is a small group that does experience this.
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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-delay}
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\caption{}
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\label{fig:pred_dist_diff_delay}
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\end{figure}
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Figure \ref{fig:pred_dist_dif_delay2} shows how this varies across disease categories
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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-delay-group}
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\caption{}
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\label{fig:pred_dist_dif_delay2}
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\end{figure}
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We can also examine the direct effect from adding a single generic competitior drug.
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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-generic}
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\caption{}
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\label{fig:pred_dist_diff_generic}
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\end{figure}
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Figure \ref{fig:pred_dist_dif_generic2} shows how this varies across disease categories
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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-generic-group}
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\caption{}
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\label{fig:pred_dist_dif_generic2}
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\end{figure}
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\end{document}
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