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85 lines
2.9 KiB
TeX
85 lines
2.9 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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%% Describe goal
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% Estimate probability distribution of normalized durations and conclusion statuses.
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% Explain why this answers questions well.
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% How do I propose estimating that?
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%%NOTATION
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% change notation
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% i indexes trials for y and d
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% n indexes snapshots within the trial
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First, some notation:
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\begin{itemize}
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\item $i$: indexes trials
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\item $n$: indexes trial snapshots.
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\item $y_i$: whether each trial terminated (true) or completed (false).
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\item $d_i$: indexes the ICD-10 disease categories per trial.
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\item $x_{i,n}$: represents the other dependent
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variables associated with the snapshot.
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% This includes\footnote{No trials in the current dataset are ever suspended.}:
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% \begin{enumerate}
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% \item Elapsed duration
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% \item arcsinh of the number of brands
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% \item arcsinh of the DALYs from high SDI countries
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% \item arcsinh of the DALYs from high-medium SDI countries
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% \item Enrollment (no distinction between anticipated or actual)
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% \item Dummy Status: Not yet recruiting
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% \item Dummy Status: Recruiting
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% \item Dummy Status: Active, not recruiting
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% \item Dummy Status: Enrolling by invitation
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% \end{enumerate}
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\end{itemize}
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% The arcsinh transform is used because it is similar to a log transform but
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% maps $\text{arcsinh}(0)=0$.
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The bayesian model to measure the direct effect of enrollment
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is specified as a hierarchal logistic regression.
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\begin{align}
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y_i \sim \text{Bernoulli}(p_{i,n}) \\
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p_{i,n} = \text{logit}(x_{i,n} \vec \beta(d_n))
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\end{align}
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Where beta is indexed by
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$d \in \{1,2,\dots,21,22\}$
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for each general ICD-10 category.
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The betas are distributed
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\begin{align}
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\beta(d) \sim \text{Normal}(\mu,\sigma I)
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\end{align}
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With hyperpriors
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\begin{align}
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\mu_k \sim \text{Normal}(0,0.05) \\
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\sigma_k \sim \text{Gamma}(4,20)
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\end{align}
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\todo{Double check that these are the priors I used.}
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Other variables are implicitly conditioned-on as they are used
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to select the trials of interest.
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I ensured that:
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\todo{double check these in the code.}
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\begin{itemize}
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\item The trial is Phase 3.
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\item The trial has a Data Monitoring Committee.
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\item The compounds are FDA regulated drug.
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\item The trial was never suspended\footnote{
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This was because I wasn't sure how to handle it in the model
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when I started scraping the data.
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Later the website changed.
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This is technically post selection in some cases.
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}
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\end{itemize}
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\todo{Make sure data is described before this point.}
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\todo{Put in a standard econometrics model}
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\begin{equation}
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x\beta = \beta_0 + \beta_1 \times \text{test}
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\label{eq:test}
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\end{equation}
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\end{document}
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