tweaked econometrics presentation, added todos

Paper/Main.tex

@@ -68,10 +68,10 @@ Section \ref{SEC:Results} discusses the results of the analysis.
 \subfile{sections/10_CausalStory}
 \subfile{sections/02_data}
 
-%---------------------------------------------------------------
-\section{Causal Identification}\label{SEC:CausalIdentification}
-%---------------------------------------------------------------
-\subfile{sections/03_CausalIdentification}
+% %---------------------------------------------------------------
+% \section{Causal Identification}\label{SEC:CausalIdentification}
+% %---------------------------------------------------------------
+% \subfile{sections/03_CausalIdentification}
 
 %---------------------------------------------------------------
 \section{Econometric Model}\label{SEC:EconometricModel}

Paper/sections/04_EconometricModel.tex

@@ -8,61 +8,77 @@
 %   How do I propose estimating that?
 
 %%NOTATION
+% change notation
+% i indexes trials for y and d 
+% n indexes snapshots within the trial
 
 First, some notation:
 \begin{itemize}
+    \item $i$: indexes trials
     \item $n$: indexes trial snapshots.
-    \item $y_n$: whether each trial terminated (true) or completed (false).
-    \item $d$: indexes ICD-10 disease categories.
-    \item $d_n$: represents the disease category of the trial associated with the snapshot $n$.
-    \item $x_n$: represents the other dependent variables associated to the snapshot.
-        This includes\footnote{No trials in the current dataset are ever suspended.}:
-        \begin{enumerate} 
-            \item Elapsed duration
-            \item arcsinh of the number of brands
-            \item arcsinh of the DALYs from high SDI countries
-            \item arcsinh of the DALYs from high-medium SDI countries
-            \item Enrollment (no distinction between anticipated or actual) 
-            \item Dummy Status: Not yet recruiting
-            \item Dummy Status: Recruiting
-            \item Dummy Status: Active, not recruiting
-            \item Dummy Status: Enrolling by invitation 
-        \end{enumerate} 
+    \item $y_i$: whether each trial terminated (true) or completed (false).
+    \item $d_i$: indexes the ICD-10 disease categories per trial.
+    \item $x_{i,n}$: represents the other dependent 
+        variables associated with the snapshot.
+        % This includes\footnote{No trials in the current dataset are ever suspended.}:
+        % \begin{enumerate} 
+        %     \item Elapsed duration
+        %     \item arcsinh of the number of brands
+        %     \item arcsinh of the DALYs from high SDI countries
+        %     \item arcsinh of the DALYs from high-medium SDI countries
+        %     \item Enrollment (no distinction between anticipated or actual) 
+        %     \item Dummy Status: Not yet recruiting
+        %     \item Dummy Status: Recruiting
+        %     \item Dummy Status: Active, not recruiting
+        %     \item Dummy Status: Enrolling by invitation 
+        % \end{enumerate} 
 \end{itemize} 
-The arcsinh transform is used because it is similar to a log transform but
-maps $\text{arcsinh}(0)=0$.
+% The arcsinh transform is used because it is similar to a log transform but
+% maps $\text{arcsinh}(0)=0$.
 
 
-The bayesian model to measure the direct effects of enrollment and the number 
-of other brands is easily specified as a hierarchal logistic regression.
+The bayesian model to measure the direct effect of enrollment
+is specified as a hierarchal logistic regression.
 \begin{align}
-    y_n \sim \text{Bernoulli}(p_n) \\
-	p_n = \text{logit}(x_n \vec \beta(d_n))
+    y_i \sim \text{Bernoulli}(p_{i,n}) \\
+        p_{i,n} = \text{logit}(x_{i,n} \vec \beta(d_n))
 \end{align}
-Where beta is indexed by $k$ for each parameter in $x$, and by
-$d \in \{1,2,\dots,21,22\}$ for each general ICD-10 category.
+Where beta is indexed by 
+$d \in \{1,2,\dots,21,22\}$ 
+for each general ICD-10 category.
 The betas are distributed
 \begin{align}
-    \beta_k(d) \sim \text{Normal}(\mu_k,\sigma_k)
+    \beta(d) \sim \text{Normal}(\mu,\sigma I)
 \end{align}
-With hyperparameters
+With hyperpriors
 \begin{align}
     \mu_k \sim \text{Normal}(0,0.05) \\
     \sigma_k \sim \text{Gamma}(4,20)
 \end{align}
+\todo{Double check that these are the priors I used.}
 
 
-Other variables are implicitly conditioned on as they were used 
-to select trials of interest.
-These include:
+Other variables are implicitly conditioned-on as they are used 
+to select the trials of interest.
+I ensured that:
+        \todo{double check these in the code.}
 \begin{itemize}
-    \item Is the trial Phase 3?\footnote{
-       Conditioning on phase 3 is equivalent to asserting that previous trials 
-       occured and had acceptable safety and efficacy results.
+    \item The trial is Phase 3.
+    \item The trial has a Data Monitoring Committee.
+    \item The compounds are FDA regulated drug.
+    \item The trial was never suspended\footnote{
+        This was because I wasn't sure how to handle it in the model
+        when I started scraping the data. 
+        Later the website changed.
+        This is technically post selection in some cases.
     }
-    \item Does the trial have a Data Monitoring Committee?
-    \item Are the compounds an FDA regulated drug?
 \end{itemize}
-%TODO: double check the sql used to select trials of interest.
 
+
+\todo{Make sure data is described before this point.}
+\todo{Put in a standard econometrics model}
+\begin{equation}
+    x\beta = \beta_0 + \beta_1 \times \text{test}
+    \label{eq:test}
+\end{equation}
 \end{document}