You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
116 lines
4.3 KiB
TeX
116 lines
4.3 KiB
TeX
\documentclass[../Main.tex]{subfiles}
|
|
|
|
\begin{document}
|
|
%\subsection{Data Exploration} %TODO: fill this out later.
|
|
%look at trial
|
|
\subsection{Model Fitting}
|
|
In this section we examine the results from fitting the econometric model using
|
|
mc-stan (\cite{mc-stan}) through the rstan (\cite{rstan}) interface.
|
|
|
|
%describe
|
|
The model was based on the hierarchal logistic regression model
|
|
presented in the Stan Users Guide (\cite{mc-stan}),
|
|
and was run with 2,500 warmup iterations and
|
|
2,500 sampling iterations in six chains.
|
|
There were various issues, including 160 divergent transitions and the R-hat
|
|
measure was 1.49.
|
|
Overall these suggest that the econometric model is incorrect as
|
|
written or requires reparameterization.
|
|
%TODO: and info about how I learned about these diagnostics
|
|
|
|
|
|
\subsubsection{Diagnostics}
|
|
%Examine trank plots
|
|
To identify which parameters were problematic, I first looked at trace rank
|
|
histograms.
|
|
Under idea circumstances, each line (representing a chain) should exchange
|
|
places with the other lines frequently.
|
|
In both \cref{fig:mu_trank} and \cref{fig:sigma_trank}, most parameters seem
|
|
to mix well but there are a couple of exceptions.
|
|
This warrants further investigation.
|
|
|
|
\begin{figure}[H]
|
|
\includegraphics[width=\textwidth]{../assets/img/mu_trank.png}
|
|
\caption{Trace Rank Histogram: Mu values}
|
|
\label{fig:mu_trank}
|
|
\end{figure}
|
|
|
|
\begin{figure}[H]
|
|
\includegraphics[width=\textwidth]{../assets/img/sigma_trank.png}
|
|
\caption{Trace Rank Histogram: Sigma values}
|
|
\label{fig:sigma_trank}
|
|
\end{figure}
|
|
|
|
%Take a look at batman and points for mu
|
|
In the case of the Mu values, a parallel coordinates plot
|
|
doesn't seem to indicate any parameters as likely candidates
|
|
for causing the issues with divergent transitions.
|
|
\begin{figure}[H]
|
|
\includegraphics[width=\textwidth]{../assets/img/mu_batman.png}
|
|
\caption{Parallel Coordinate Plot: Mu values}
|
|
\label{fig:mu_batman}
|
|
\end{figure}
|
|
Note that at each parameter, there is some level of dispersion between
|
|
values that diverged.
|
|
|
|
On the other hand, in the parallel coordinates plot for sigma values,
|
|
it appears that most divergent transitions occur with values of
|
|
sigma[1], sigma[3], sigma[6], and sigma[7] close to zero.
|
|
\begin{figure}[H]
|
|
\includegraphics[width=\textwidth]{../assets/img/sigma_batman.png}
|
|
\caption{Parallel Coordinate Plot: Sigma values}
|
|
\label{fig:sigma_batman}
|
|
\end{figure}
|
|
Overall this suggests that there is an issue with the specification
|
|
of the covariance structures of the hyperparameters.
|
|
|
|
Additional evidence that the covariance structure is incorrect comes from
|
|
plotting pairs of parameter values and examining the chains with divergent
|
|
transitions.
|
|
|
|
\begin{figure}[H]
|
|
\includegraphics[width=\textwidth]{../assets/img/sigma_pairs_5-9.png}
|
|
\caption{Parameter Pairs plots: Sigma[5] through Sigma[9]}
|
|
\label{fig:sigma_pairs_5-9.png}
|
|
\end{figure}
|
|
From this we can see that divergent pairs are highly correlated with the cases
|
|
where sigma[6] or sigma[7] are equal to zero.
|
|
This has an impact on the shape of both of those estimated parameters, causing
|
|
both to be bimodal.
|
|
|
|
|
|
\subsection{Interpretation}
|
|
|
|
Ignoring the diagnosed issues with the model, we do see some interesting
|
|
preliminary results.
|
|
|
|
%in mu, mu[5] shifted strongly
|
|
In \cref{fig:mu_posterior} we see that mu[5], the parameter corresponding
|
|
to enrollment appears to be strongly negative.
|
|
This is consistent with the idea that enrollment close to planned enrollment
|
|
decreases the probability of terminating the trial.
|
|
In \cref{fig:sigma_posterior}, sigma[2] (corresponding to the number of brands
|
|
selling the drug of interest) has a large variance covers some relatively
|
|
high values.
|
|
This suggests that the impact of how frequently the drug is sold varies greatly
|
|
across different ICD-10 categories of disease.
|
|
|
|
|
|
\begin{figure}[H]
|
|
\includegraphics[width=\textwidth]{../assets/img/mu_posterior.png}
|
|
\caption{Posterior Parameter Estimates: Mu}
|
|
\label{fig:mu_posterior}
|
|
\end{figure}
|
|
|
|
% Sigma[2] suggests there is a high variance in the impact that the number of drugs on the market has.
|
|
\begin{figure}[H]
|
|
\includegraphics[width=\textwidth]{../assets/img/sigma_posterior.png}
|
|
\caption{Posterior Hyperparameter Estimates: Sigma}
|
|
\label{fig:sigma_posterior}
|
|
\end{figure}
|
|
|
|
Due to the deficiencies in the data and model, this is the limit of the
|
|
analysis I will perform at this time.
|
|
|
|
\end{document}
|