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69 lines
2.5 KiB
TeX
69 lines
2.5 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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First, some notation:
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\begin{itemize}
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\item $n$: indexes trial snapshots.
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\item $y_n$: whether each trial terminated (true) or completed (false).
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\item $d$: indexes ICD-10 disease categories.
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\item $d_n$: represents the disease category of the trial associated with the snapshot $n$.
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\item $x_n$: represents the other dependent variables associated to 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 effects of enrollment and the number
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of other brands is easily specified as a hierarchal logistic regression.
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\begin{align}
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y_n \sim \text{Bernoulli}(p_n) \\
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p_n = \text{logit}(x_n \vec \beta(d_n))
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\end{align}
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Where beta is indexed by $k$ for each parameter in $x$, and by
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$d \in \{1,2,\dots,21,22\}$ 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_k(d) \sim \text{Normal}(\mu_k,\sigma_k)
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\end{align}
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With hyperparameters
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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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Other variables are implicitly conditioned on as they were used
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to select trials of interest.
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These include:
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\begin{itemize}
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\item Is the trial Phase 3?\footnote{
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Conditioning on phase 3 is equivalent to asserting that previous trials
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occured and had acceptable safety and efficacy results.
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}
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\item Does the trial have a Data Monitoring Committee?
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\item Are the compounds an FDA regulated drug?
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\end{itemize}
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%TODO: double check the sql used to select trials of interest.
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\end{document}
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