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@ -19,97 +19,101 @@ written or requires reparameterization.
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%TODO: and info about how I learned about these diagnostics
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%TODO: and info about how I learned about these diagnostics
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\subsubsection{Diagnostics}
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% \subsubsection{Diagnostics}
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%Examine trank plots
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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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% To identify which parameters were problematic, I first looked at trace rank
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histograms.
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% histograms.
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Under idea circumstances, each line (representing a chain) should exchange
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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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% 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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% 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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% to mix well but there are a couple of exceptions.
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This warrants further investigation.
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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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\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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\begin{figure}[H]
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\subsection{Interpretation}
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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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%Take a look at batman and points for mu
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The key results so far are related to the distribution of differences in $p$.
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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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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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for causing the issues with divergent transitions.
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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/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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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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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/sigma_batman.png}
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-delay}
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\caption{Parallel Coordinate Plot: Sigma values}
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\caption{}
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\label{fig:sigma_batman}
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\label{fig:pred_dist_diff_delay}
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\end{figure}
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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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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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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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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/sigma_pairs_5-9.png}
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-delay-group}
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\caption{Parameter Pairs plots: Sigma[5] through Sigma[9]}
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\caption{}
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\label{fig:sigma_pairs_5-9.png}
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\label{fig:pred_dist_dif_delay2}
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\end{figure}
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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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Ignoring the diagnosed issues with the model, we do see some interesting
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preliminary results.
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%in mu, mu[5] shifted strongly
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In \cref{fig:mu_posterior} we see that mu[5], the parameter corresponding
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to enrollment appears to be strongly negative.
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This is consistent with the idea that enrollment close to planned enrollment
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decreases the probability of terminating the trial.
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In \cref{fig:sigma_posterior}, sigma[2] (corresponding to the number of brands
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selling the drug of interest) has a large variance covers some relatively
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high values.
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This suggests that the impact of how frequently the drug is sold varies greatly
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across different ICD-10 categories of disease.
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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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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/mu_posterior.png}
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-generic}
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\caption{Posterior Parameter Estimates: Mu}
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\caption{}
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\label{fig:mu_posterior}
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\label{fig:pred_dist_diff_generic}
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\end{figure}
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\end{figure}
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% Sigma[2] suggests there is a high variance in the impact that the number of drugs on the market has.
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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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\begin{figure}[H]
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\includegraphics[width=\textwidth]{../assets/img/sigma_posterior.png}
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\includegraphics[width=\textwidth]{../assets/img/current/pred_dist_diff-generic-group}
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\caption{Posterior Hyperparameter Estimates: Sigma}
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\caption{}
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\label{fig:sigma_posterior}
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\label{fig:pred_dist_dif_generic2}
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\end{figure}
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\end{figure}
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Due to the deficiencies in the data and model, this is the limit of the
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analysis I will perform at this time.
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\end{document}
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\end{document}
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Will King
2 years ago
