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93 lines
3.4 KiB
TeX
93 lines
3.4 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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The model I use is a
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hierarchal logistic regression model where the
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hierarchies are based on disease categories.
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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
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terminated (true, 1) or completed (false, 0).
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\item $d_i$: indexes the ICD-10 disease category of the trial.
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\item $x_{i,n}$: represents the independent
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variables associated with the snapshot.
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\end{itemize}
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The goal is to take each snapshot and predict
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The actual specification of the model to measure
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the direct effect of enrollment is:
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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_i))
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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_i) \sim \text{Normal}(\mu_i,\sigma_i I)
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\end{align}
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With hyperpriors
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%Checked on 2024-11-27. Is corrrect. \todo{Double check that these are the priors I used.}
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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 actual spec}
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The independent variables include:
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\todo{Make sure data is described before this point.}
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\begin{subequations}
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\begin{align}
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x_{i,n}\beta(d_i)
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= & \bx{1}{\text{Elapsed Duration}} \\
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&+ \bx{2}{\arcsinh \left(\text{\# Generic compunds}\right)} \\
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&+ \bx{3}{\arcsinh \left(\text{\# Branded compunds}\right)} \\
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&+ \bx{4}{\text{\# DALYs in High SDI Countries}} \\
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&+ \bx{5}{\text{\# DALYs in High-Medium SDI Countries}} \\
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&+ \bx{6}{\text{\# DALYs in Medium SDI Countries}} \\
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&+ \bx{7}{\text{\# DALYs in Low-Medium SDI Countries}} \\
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&+ \bx{8}{\text{\# DALYs in Low SDI Countries}} \\
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&+ \bxi{9}{\text{Not yet Recruiting}}{\text{Trial Status}}\\
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&+ \bxi{10}{\text{Recruiting}}{\text{Trial Status}}\\
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&+ \bxi{11}{\text{Enrolling by Invitation Only}}{\text{Trial Status}}\\
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&+ \bxi{12}{\text{Active, not recruiting}}{\text{Trial Status}}
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\end{align}
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\end{subequations}
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The arcsinh transform is used because it is similar to a log transform but
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differentiably handles counts of zero since
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$\text{arcsinh}(0) = \ln (0 + \sqrt{0^2 + 1}) =0$.
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Note that in this is a heirarchal model, each IDC-10 disease category
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gets it's own set of parameters, and that is why the $\beta$s are parameterized
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by $d_i$.
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%%%% Not sure if space should go here. I think these work well together.
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Other variables are implicitly controlled for as they are used
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to select the trials of interest.
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These include:
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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.
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\todo{double check where this happened in the code.
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I may have only done it in the CBO analysis.}
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}
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\end{itemize}
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\end{document}
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