diff --git a/Latex/Presentation/presentation2.tex b/Latex/Presentation/presentation2.tex new file mode 100644 index 0000000..7bf2892 --- /dev/null +++ b/Latex/Presentation/presentation2.tex @@ -0,0 +1,916 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Beamer Presentation +% LaTeX Template +% Version 1.0 (10/11/12) +% +% This template has been downloaded from: +% http://www.LaTeXTemplates.com +% +% License: +% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/) +% +% Changed theme to WSU by William King +% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%---------------------------------------------------------------------------------------- +% PACKAGES AND THEMES +%---------------------------------------------------------------------------------------- + +\documentclass[xcolor=dvipsnames,aspectratio=169]{beamer} + + +%Import Preamble bits +\input{../assets/preambles/FormattingPreamble.tex} +\input{../assets/preambles/TikzitPreamble.tex} +\input{../assets/preambles/MathPreamble.tex} +\input{../assets/preambles/BibPreamble.tex} +\input{../assets/preambles/GeneralPreamble.tex} + + + + +%---------------------------------------------------------------------------------------- +% TITLE PAGE +%---------------------------------------------------------------------------------------- + +\title[Clinical Trials]{The Effects of Market Conditions on Recruitment and Completion of Clinical Trials} + +\author{Will King} % Your name +\institute[WSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space +{ +Washington State University \\ % Your institution for the title page +\medskip +\textit{william.f.king@wsu.edu} % Your email address +} +\date{\today} % Date, can be changed to a custom date + + + + + +\begin{document} +\begin{frame} +\titlepage % Print the title page as the first slide +\end{frame} +%---------------------------------- +\begin{frame} %Allow frame breaks +\frametitle{Clinical Trials} % Table of contents slide, comment this out to remove it + % - Intro and hook (Clinical Trials are key part of pharmacological pipeline) + Pharmaceuticals are a frequently discussed aspect of health care cost management. + Their development is dictated by scientific and regulatory hurdles + including passing clinical trials + (\cite{noauthor_fda_nodate}), + while their market is characterized by strategic competition and ambiguous + patent protection + (\cite{van_der_gronde_addressing_2017}). + + \vspace{12pt} + + This research investigates the ways by which market conditions + affect clinical trial completion. +\end{frame} +%-------------------------------- +\begin{frame}[allowframebreaks] %Allow frame breaks +\frametitle{Overview} % Table of contents slide, comment this out to remove it +\tableofcontents +% - Intro and hook +% - Literature review +% - Causal Identification +% - Data +% - Econometric model +% - Results +% - Improvements +\end{frame} +%------------------------------- + +%------------------------------------------------------------------------------------- +%%%%%%%%%%%%%%%%%%%% Introduction and Background %%%%%%%%%%%%%%%%%%%%%%%% +\section{Background} +% TOC +% - Background on drug process +% - Literature on clinical trials +% - My questions +% add info about trials +% - Requirements (pre registered design [2007], updated "regularly" on clinicaltrials.gov) +% - Phases (1,2,3,4, mixed) +% - Safety and Ethicas (oversight boards, restrictions on payments) +% - Approval processes (biologics vs small-molecule) +% add info about drugs +%------------------------------------------------------------------------------------- + +%------------------------------- +\begin{frame} + \frametitle{Clinical Trials and Drug develoment} + + The FDA requires clinical trials before approving new drug compounds + \begin{itemize} + \item Pre-registered design + \item Updated regularly on clinicaltrials.gov + \item Often requires an oversight board. + \item Goal is to prove efficacy and safety of a compound/dosage/route. + \item A new drug candidate (NDC) must complete 3 phases of clinical trials before approval. + \item Phases are reviewed with FDA. + \item Not all clinical trials are for new drugs. + \end{itemize} +\end{frame} +%----------------------------- +\begin{frame} + \frametitle{Literature Highlights} + \begin{itemize} + \item \cite{van_der_gronde_addressing_2017}: + High level synthesis of overall discussion regarding drug costs. + Both academic and non-academic sources. + \item \cite{hwang_failure_2016}: + Answered the question "Why do late-stage (phase III) trials fail?" + Found that efficacy, safety, and competition reasons accounted for + 57\%, 17\%, and 22\% respectively. + \item \cite{abrantes-metz_pharmaceutical_2004}: + Described how drugs progress through the 3 phases of clinical trials + and correlations between various trial characteristics and the + clinical trial failures. + \item \cite{khmelnitskaya_competition_2021}: + Modeled clinical trial life-cycle of drugs, found method to separate + scientific from competitive reasons for failure to progress to the + next phase. +% \item \cite{}: + + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{This research, in context} + + In contrast to previous work looking at multiple phases of trials, + I seek to figure out what causes individual trials to fail. + +% \vspace{12pt} +% +% Instead of focusing on the drug development pipeline, I attempt to +% investigate the population of drug-based, phase III trials. +% + \vspace{12pt} + + Questions + \begin{itemize} + \item How do the competitors on the market affect clinical trial completion? + \item How is this effect moderated by the enrollment of participants? + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Audience Questions} + + \center{What can I clarify?} +\end{frame} +%------------------------------------------------------------------------------------- +%%%%%%%%%%%%%%%%%%%% Causality and Data %%%%%%%%%%%%%%%%%%%%%%%% +\section{Causal Story and Data} +% TOC +% - Causal Story (no subsection) +% - Clinical trials: targets specific drug/condition combination. +% - Enrollment process: patients counsel with providers +% - Trials terminate if unsafe, ineffective, unprofitable, or cannot enroll patients +% - Ethical concerns exist throughout. +% - This is complicated by the fact that the experiment reveals information over time. +% - Formalization +% - Data Sources +% Data Generating process +% - Agents and their decisions +% - Factors that influence each decision +%------------------------------------------------------------------------------------- + +%------------------------------- +\begin{frame}[shrink=10] %evil option is helpful here. + \frametitle{How do clinical trials proceed?} + \begin{columns}[T] + \begin{column}{0.5\textwidth} + What does a \textif{Completed} trial look like? + \begin{enumerate} + \item Study sponsor comes up with design + \item Apply for NCT ID from ClinicalTrials.gov + \item Begin enrolling participants + \item Update ClinicalTrials.gov to recruit + \item Close Enrollment + \item Update ClinicalTrials.gov as not recruiting* + \item Reach primary objectives + \item Update ClinicalTrials.gov as complete + \item Reach secondary objectives + \item Update ClinicalTrials.gov with more information + \end{enumerate} + \end{column} + \begin{column}{0.5\textwidth} + What does an \textif{Terminated} trial look like? + \begin{enumerate} + \item Study sponsor comes up with design + \item Apply for NCT ID from ClinicalTrials.gov + \item Begin enrolling participants + \item Update ClinicalTrials.gov to advertise + \item Run into issues: + \begin{itemize} + \item Safety + \item Efficacy + \item Profitability + \item Feasiblity (enrollment, PI leaves, etc.) + \end{itemize} + \item Close Enrollment* + \item Decide to terminate clinical trial. + \item Update ClinicalTrials.gov as terminated. + \end{enumerate} + \end{column} + \end{columns} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{ClinicalTrials.gov} + Thus ClinicalTrials.gov becomes an (append only) repository of + the ``current'' status of clincal trials. + + As it is designed to help faciltate enrollment in clinical trials, + the record includes important information such as + + \begin{itemize} + \item drugs + \item study arms + \item conditions + \item expected and final enrollment figures + \item current status + \end{itemize} + + ClinicalTrials.gov also reports the history from previous + updates. +\end{frame} + +%------------------------------- +\begin{frame} + \frametitle{Decision-Making Process} + % study sponsors + Study Sponsors Decide to start a Phase 3 trial and whether to terminate it. + \\ + They ask themselves: + \begin{itemize} + \item Do safety incidents require terminating a trial? + \item Do efficacy results indicate the trial is worth continuing? + \item Is recruiting sufficient to achieve our results and contain costs? + \item Do expectations about future returns justify our expenditures? + \end{itemize} + + They plan their enrollment considering + \begin{itemize} + \item Total population affected + \item Likely participant response rates + \item Their network of clinicians + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Decision-Making Process} + % participants + Participants decide to enroll (and dis-enroll) themselves in a trial based on + \begin{itemize} + \item Doctor Recommendations + \item Disease severity + \item Relative safety/efficacy compared to other treatments + \end{itemize} + +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Questions?} + \center{What clarifying questions do you have?} +\end{frame} +%------------------------------- +%-------------------------------- +%%%%%%%%%%%%%%%%%%%% Causal Formalization +\subsection{Formalization} +% - Introduce basic triangle +% - discuss total vs direct effects +% - +% - Add confounders and controls +% - Introduce backdoor criterion +%-------------------------------- +%------------------------------- +\begin{frame} %Allow frame breaks +\frametitle{Why this approach?} + \begin{figure} + \includegraphics[height=0.8\textheight]{../assets/img/methodology_trial.png} + \label{FIG:xkcd2726} + \caption{``If you think THAT'S unethical, you should see the stuff we approved via our Placebo IRB.'' + - \url{https://xkcd.com/2726} + } + \end{figure} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Framing my Questions} + \begin{columns}[T] + \begin{column}{0.5\textwidth} + Two potential causes of trial termination include + \begin{enumerate} + \item Alternative (competitor) treatments exist + \begin{itemize} + \item reduces future profitability. + \item reduces incentives to enroll as participants. + \end{itemize} + \item It can be difficult to recruit patients + \begin{itemize} + \item Are there few patients? + \item Are potential participants choosing other alternatives? + \end{itemize} + \end{enumerate} + \end{column} + \begin{column}{0.5\textwidth} + Overall this can be described graphically as: + + \begin{figure} + \scalebox{0.8}{ + \tikzfig{../assets/tikzit/4Node} + } + \label{FIG:4Node} + \caption{Total Effect} + \end{figure} + \end{column} + \end{columns} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Causal Effects} + %Discuss the two different effects: total effect, direct effects + + \begin{columns} + \begin{column}{0.5\textwidth} + Total Effect of Competitors + + \begin{figure} + \scalebox{0.8}{ + \tikzfig{../assets/tikzit/4Node_total} + } + \label{FIG:4Node} + \caption{Total Effect} + \end{figure} + \end{column} + \begin{column}{0.5\textwidth} + Direct Effects of Competitors and Enrollment + + \begin{figure} + \scalebox{0.8}{ + \tikzfig{../assets/tikzit/4Node_direct} + } + \label{FIG:4Node} + \caption{Direct Effect} + \end{figure} + \end{column} + \end{columns} +\end{frame} + +%------------------------------- +\begin{frame} + \frametitle{Rephrasing Questions} + To rephrase my questions + \begin{enumerate} + \item How large is the total effect of increasing the number + of competing drugs on completing clinical trials? + \item How large is the direct effect of increasing the number + of competing drugs on completing clincial trials? + \end{enumerate} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Additional Concerns} + %Confounders + Of course, there are other confounding relationships + \begin{enumerate} + \item Population Effects + \item Fundamental Safety and Efficacy of compound/dosage/route + \item How long it is taking + \item + \end{enumerate} + %TODO: Fill out with more details from graph +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Complete graph} + %introduce backdoor criterion + + \begin{figure} + \scalebox{0.6}{ + \tikzfig{../assets/tikzit/CausalGraph} + } + \label{FIG:CausalGraph} + \caption{Full Causal Graph} + \end{figure} + Discuss concerns about Elapsed Duration and Enrollment + +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Causal Diagram: Backdoor Criterion} + \small + \begin{block}{$d$-separation} + A set $S$ of nodes blocks a path $p$ on a DAG if either + \begin{enumerate} + \item $p$ contains at least one arrow-emitting node in $S$ + \item $p$ contains at least one collision node $c$ that is outside $S$ + and has no descendants in $S$. + \end{enumerate} + If $S$ blocks all paths from X to Y, then it is said to ``$d$-separate'' + $X$ and $Y$, and then $X \perp Y | S$. + \end{block} + \begin{block}{Back-Door Criterion} + A set $S$ of covariates is admissible as controls on the + causal relationship $X \rightarrow Y$ if: + \begin{enumerate} + \item No element of $S$ is a descendant of $X$ + \item The elements of $S$ d-separate all paths from $X$ to $Y$ that include + parents of $X$. + \end{enumerate} + \end{block} + \cite{pearl_causality_2000} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Sufficent Adjustment Set} + %introduce backdoor criterion + + Thus the required adjustment set depends on the effects of interest. + + For the total effect these are controls for: + \begin{itemize} + \item Proceed with Phase III + \item Condition + \item Population + \end{itemize} + Discuss Regime Switching + + For the direct effect these are controls for: + \begin{itemize} + \item Proceed with Phase III + \item Condition + \item Population (optional) + \item Enrollment + \end{itemize} + Not causally identified due to Regime Switching + + +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Other testable hypotheses} + One advantage of this approach tools can automatically + \begin{itemize} + \item verify causal identification + \item generate hypotheses to verify model + \end{itemize} + + Automatic hypotheses +% \begin{itemize} +% \item Condition $\perp$ Elapsed Duration +% \item Decision to continue Phase III $\perp$ Elapsed Duration +% \item Decision to continue Phase III $\perp$ Market Conditions | Condition +% \item Decision to continue Phase III $\perp$ Population | Condition +% \item Elapsed Duration $\perp$ Market Conditions +% \item Elapsed Duration $\perp$ Population +% \item Terminated $\perp$ Population | Condition, Decision to continue Phase III, Elapsed Duration, Enrollment Status, Market Conditions +% \end{itemize} + +% \href{Dagitty.net model}{http://dagitty.net/mLyFuc5} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Questions?} + +\end{frame} +%------------------------------- +%-------------------------------- +%%%%%%%%%%%%%%%%%%%% Data sources +\subsection{Data Sources} +% TOC +% - Main Data Sources +% - ClinicalTrials.gov and AACT +% - IHME Burden of Disease +% - Marketing Data +% - MeSH, RxNorm/RxNav +% - How did I Link Data Sources +% - Data Sizes +%-------------------------------- + +%------------------------------- +\begin{frame} %Allow frame breaks + \frametitle{Data Sources} + \begin{itemize} + \item ClinicalTrials.gov - AACT \& custom scripts + \begin{itemize} + \item Select trials of interest + \item Trial details: + \begin{itemize} + \item conditions + \item final status + \item drugs/interventions + \end{itemize} + \item Trial snapshots: + \begin{itemize} + \item elapsed duration + \item enrollment status (NYE,EBI,R,ANR) + \end{itemize} + \end{itemize} + \item Medical Subject Headings (MeSH) Thesaurus + \begin{itemize} + \item A standardized nomenclature used to classify interventions + and conditions in the clinical trials database. + \end{itemize} + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} %Allow frame breaks + \frametitle{Data Sources} + \begin{itemize} + \item USP Drug Classification (2023) + \begin{itemize} + \item Used to measure which drugs + \end{itemize} + \item NSDE Files (New drug code Structured product labels Data Element) + \begin{itemize} + \item Contains information about when a given drug was on the market. + \end{itemize} + \item RxNorm + \begin{itemize} + \item Links pharmaceuticals between MeSH standardized terms and + NSDE files. + \item Used to find brand names that share active ingredients with those from trial. + \end{itemize} + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} %Allow frame breaks + \frametitle{Data Sources} + \begin{itemize} + \item Global Disease Burden Survey (2019) + \begin{itemize} + \item Estimates of DALYs for categories of disease + \item Links of Categories to ICD-10 Codes + \end{itemize} + \item ICD-10 (2019) + \begin{itemize} + \item WHO version + \item CMS version (Clinical Management) + \item Used to group disease conditions in hierarchical model + \end{itemize} + \item Unified Medical Language System Thesaurus + \begin{itemize} + \item Used to link MeSH standardized terms and ICD-10 conditions + \item Manual matching process + \end{itemize} + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Linking data} + % + The following linking process was used: + \begin{enumerate} + \item AACT trials to snapshots (internal ID) + \item AACT trials to ICD-10 (hand match) + \item ICD-10 to IHME (IHME) + \item Snapshots to drug brands (RxNorm/RxNav/MeSh, SPL) + \item AACT to USP DC alternates (RxNorm, USP DC, hand match) + \end{enumerate} + +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Data used} + The following data points were used. + \begin{itemize} + \item elapsed duration + \item enrollment status + \item asinh(brands with identical ingredients) + \item asinh(brands in USP-DC category) + \item asinh(high sdi DALY estimate) + \item asinh(high-medium sdi DALY estimate) + \item asinh(medium sdi DALY estimate) + \item asinh(low-medium sdi DALY estimate) + \item asinh(low sdi DALY estimate) + \end{itemize} + The asinh operator was used because it parallels $\text{ln}(x)$ for + large values of $x$ but also handles $\text{asinh}(0)=0$. +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Measures of Causes and Effects} + + Here are the actual measures used for causes + \begin{itemize} + \item Final Status: Measured from AACT - status when trial is over. + \item Competitors on Market: Measured by the number of drugs + \begin{itemize} + \item with same active ingredients (at the time of the snapshot) + \item sharing the USP DC category and class (in 2023) + \end{itemize} + \end{itemize} + + Effects are measured in parameter values and changes in probability + +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Adjustment set} + Here are the actual measures of the adjustment set + \begin{itemize} + \item Enrollment: Measured by enrollment status at the snapshot level. + \item Elapsed Duration: Measured at snapshot level + by $\frac{\text{Current Date} - \text{Start Date}}{\text{Planned Completion Date} - \text{Start Date}}$ + \item Population Measures + \begin{itemize} + \item IHME Global Disease Burden: DALYs, spread over 5 levels of the Social Development Index + \end{itemize} + \item Beliefs about safety \& efficacy: Restricted to Phase 3 trials. + \item Disease Type: Hierarchal parameters in model + \end{itemize} + + Note the implicit conditioning on trials treating diseases with IHME data\footnote{ + IHME does not track data for W61.62XD: Struck by duck, subsequent encounter + }. + +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Other Details} + + Other Trial Selection Criteria + \begin{itemize} + \item Interventional Study + \item Involved an FDA Regulated Drug + \item Phase 3 trial + \item Started after 2010-01-01 + \item Ended before 2022-01-01 + \end{itemize} + +\end{frame} +%---------------------------------- +%%%%%%%%%%%%%%%%%%%% Summary +\subsection{Data Summary} +%---------------------------------- +%------------------------------- +\begin{frame} + \frametitle{Data Summaries} + %TODO: Update + \begin{itemize} + \item Number of Phase III, FDA monitored Drug Trials: 1,981 + \item Number of Trials matched to ICD-10: + \item Number of Trials matched to ICD-10 with population measures: + ( completed, terminated) + \item Number of Snapshots: + \end{itemize} +\end{frame} +%%---------------------------------- +%\begin{frame} +% \frametitle{Summaries: Trial Durations} +% \begin{figure} +% \includegraphics[height=0.8\textheight]{../assets/img/2023-04-12_durations_hist.png} +% \label{FIG:durations} +% \caption{Trial Durations (days)} +% \end{figure} +%\end{frame} +%%---------------------------------- +%\begin{frame} +% \frametitle{Summaries: snapshots} +% \begin{figure} +% \includegraphics[height=0.8\textheight]{../assets/img/2023-04-12_snapshots_hist.png} +% \label{FIG:snapshots} +% \caption{Number of Snapshots per matched trial} +% \end{figure} +%\end{frame} +%%---------------------------------- +%\begin{frame} +% \frametitle{Summaries: snapshots} +% \begin{figure} +% \includegraphics[height=0.8\textheight]{../assets/img/2023-04-12_status_duration_snapshots_points.png} +% \label{FIG:snapshot_duration_scatter} +% \caption{Scatterplot of snapshot count and durations} +% \end{figure} +%\end{frame} +%%------------------------------- +%\begin{frame} +% \frametitle{Questions?} +% +%\end{frame} +%------------------------------- +%------------------------------------------------------------------------------------- +%%%%%%%%%%%%%%%%%%%% Analysis %%%%%%%%%%%%%%%%%%%%%%%% +\section{Analysis} +% TOC +% - Review questions and datasets to use for each +% - +% - +%------------------------------------------------------------------------------------- + +%------------------------------- +\begin{frame} + \frametitle{General Approach} + + \begin{itemize} + \item Logistic model + \item Bayesian Hierarchal model + \begin{itemize} + \item Allows for transfer learning between groups + \end{itemize} + \item Distribution of Predicted Differences + \begin{itemize} + +\end{frame} + +%------------------------------- +\begin{frame} + \frametitle{Total Effects Model} + \begin{align} + y_i \sim \text{Bernoulli}(p_i) \\ + p_i = \text{logistic}(X_i \vec\beta_{c(i)}) \\ + \vec\beta_{c(i)} \sim \text{MvNormal}(\vec\mu,\vec\sigma I) + \end{align} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Questions?} + +\end{frame} +%-------------------------------- +%%%%%%%%%%%%%%%%%%%% Results +\subsection{Results} +%-------------------------------- +%-------------------------------- +\subsubsection{Total Effect} +% - Review Parameter Values +% - hyperparameters +% - Table of MLE +% - Distributions +% - betas +% - Table of MLE +% - Distributions +% - Review Posterior Prediction for interventions +%-------------------------------- +%------------------------------- +\begin{frame} + \frametitle{Results} + Because Bayesian estimation is typically done numerically, we will first + validate convergence. + + Then we will take a look at preliminary results. + + Sampling details + + %TODO: Update + \begin{itemize} + \item 6 chains + \item 2,500 warm-up, 2,500 sampling runs + \item seed = 11021585 + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Questions?} + +\end{frame} +%------------------------------- +%-------------------------------- +\subsubsection{Direct Effects} +% - Review Parameter Values +% - hyperparameters +% - Table of MLE +% - Distributions +% - betas +% - Table of MLE +% - Distributions +% - Review Posterior Prediction for interventions +%-------------------------------- +%------------------------------- +\begin{frame} + \frametitle{Convergence} + Sampling details + %TODO: UPDATE + \begin{itemize} + \item 6 chains + \item 2,500 warm-up, 2,500 sampling runs + \item seed = 11021585 + \end{itemize} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Questions?} + +\end{frame} +%------------------------------- +%------------------------------------------------------------------------------------- +%%%%%%%%%%%%%%%%%%%% Conclusion %%%%%%%%%%%%%%%%%%%%%%%% +\section{Conclusion} +%------------------------------------------------------------------------------------- + +%------------------------------- +\begin{frame} + \frametitle{Proposed improvements} + \begin{enumerate} + \item Match more trials to ICD-10 codes and Formularies + \item Add more formularies + \item Remove disease categories that don't exist in the data from the priors + \item Imputing Enrollment + \end{enumerate} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Summary} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Final Questions} + + \center{\huge{Time is yours to ask any remaining questions.}} +\end{frame} +%------------------------------------------------------------------------------------- +%%%%%%%%%%%%%%%%%%%% Appendicies %%%%%%%%%%%%%%%%%%%%%%%% +\section{Appendices} +%------------------------------------------------------------------------------------- +%---------------------------------- +%%%%%%%%%%%%%%%%%%%% Convergence Tests +\subsection{Convergence} +%---------------------------------- +%------------------------------- +\begin{frame} + \frametitle{Warnings} + %TODO: UPDATE + + \begin{itemize} + \item There were no diverging transitions. + \item There were 15,000 transitions that exceeded max treedepth. + Sampling efficiency is poor. + \item All chains had low Bayesian Fraction of Missing Information. + Some areas of the distribution were poorly explored. + \item R-hat = $1.23$, ideal is around 1, chains did not mix well. + \item Bulk and Tail Effective Sample sizes were low, + suggesting mean and variance/quantile estimates will be unreliable. + \end{itemize} + \cite{mc-stan} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Convergence: Mu} + \begin{figure} + %TODO: UPDATE + %\includegraphics[height=0.9\textheight]{../assets/img/2023-04-11_mu_points.png} + \label{FIG:caption} + \caption{Hyperparameter Points Plots: Mu} + \end{figure} +\end{frame} +%------------------------------- +\begin{frame} + \frametitle{Convergence: Sigma} + \begin{figure} + %TODO: UPDATE + %\includegraphics[height=0.9\textheight]{../assets/img/2023-04-11_mu_points.png} + \label{FIG:caption} + \caption{Hyperparameter Points Plots: Sigma} + \end{figure} +\end{frame} +%------------------------------- +\begin{frame}[allowframebreaks] + \frametitle{Bibliography} + \printbibliography +\end{frame} +%------------------------------- +\end{document} +%========================================= +%\begin{frame} +% \frametitle{MarginalRevenue} +% \begin{figure} +% \tikzfig{../Assets/owned/ch8_MarginalRevenue} +% \includegraphics[height=\textheight]{../Assets/copyrighted/KrugmanObsterfeldMeliz_fig8-7.jpg} +% \label{FIG:costs} +% \caption{Average Cost Curve as firms enter.} +% \end{figure} +%\end{frame} +%------------------------------- +%\begin{frame} +% \frametitle{Columns} +% \begin{columns} +% \begin{column}{0.5\textwidth} +% \end{column} +% \begin{column}{0.5\textwidth} +% \begin{figure} +% \tikzfig{../Assets/owned/ch7_EstablishedAdvantageExample2} +% \label{FIG:costs} +% \caption{Setting the Stage} +% \end{figure} +% \end{column} +% \end{columns} +%\end{frame} +% %---------------------------------------------------------------