From 2397d093cfb28805049636a95e00442039dc5d09 Mon Sep 17 00:00:00 2001
From: Jennifer Yu <165851215+jennhyu26@users.noreply.github.com>
Date: Wed, 1 May 2024 22:43:06 -0700
Subject: [PATCH] Update logistic-regression.qmd

Using prior established notation (D, not D; E, not E) but specifying what exposures and disease are per example.
---
 logistic-regression.qmd | 38 +++++++++++++++++++-------------------
 1 file changed, 19 insertions(+), 19 deletions(-)

diff --git a/logistic-regression.qmd b/logistic-regression.qmd
index fb80680..6b1da02 100644
--- a/logistic-regression.qmd
+++ b/logistic-regression.qmd
@@ -769,28 +769,28 @@ In @exm-oc-mi, we have:
 
 $$
 \begin{aligned}
-\theta(MI; OC) 
+\theta(D_{MI}; E_{OC}) 
 &\eqdef
-\frac{\omega(MI|OC)}{\omega(MI|\neg OC)}\\
+\frac{\omega(D_{MI}|E_{OC})}{\omega(D_{MI}|\neg E_{OC})}\\
 &\eqdef \frac
-{\left(\frac{\Pr(MI|OC)}{\Pr(\neg MI|OC)}\right)}
-{\left(\frac{\Pr(MI|\neg OC)}{\Pr(\neg MI|\neg OC)}\right)}\\
+{\left(\frac{\Pr(D_{MI}|E_{OC})}{\Pr(\neg D_{MI}|E_{OC})}\right)}
+{\left(\frac{\Pr(D_{MI}|\neg E_{OC})}{\Pr(\neg D_{MI}|\neg E_{OC})}\right)}\\
 &= \frac
-{\left(\frac{\Pr(MI,OC)}{\Pr(\neg MI,OC)}\right)}
-{\left(\frac{\Pr(MI,\neg OC)}{\Pr(\neg MI,\neg OC)}\right)}\\
-&= \left(\frac{\Pr(MI,OC)}{\Pr(\neg MI,OC)}\right)
-\left(\frac{\Pr(\neg MI,\neg OC)}{\Pr(MI,\neg OC)}\right)\\
-&= \left(\frac{\Pr(MI,OC)}{\Pr(MI,\neg OC)}\right)
-\left(\frac{\Pr(\neg MI,\neg OC)}{\Pr(\neg MI,OC)}\right)\\
-&= \left(\frac{\Pr(OC,MI)}{\Pr(\neg OC,MI)}\right)
-\left(\frac{\Pr(\neg OC,\neg MI)}{\Pr(OC,\neg MI)}\right)\\
-&= \left(\frac{\Pr(OC|MI)}{\Pr(\neg OC|MI)}\right)
-\left(\frac{\Pr(\neg OC|\neg MI)}{\Pr(OC|\neg MI)}\right)\\
-&= \frac{\left(\frac{\Pr(OC|MI)}{\Pr(\neg OC|MI)}\right)}
-{\left(\frac{\Pr(OC|\neg MI)}{\Pr(\neg OC|\neg MI)}\right)}\\
-&\eqdef \frac{\omega(OC|MI)}
-{\omega(OC|\neg MI)}\\
-&\eqdef \theta(OC; MI)
+{\left(\frac{\Pr(D_{MI},E_{OC})}{\Pr(\neg D_{MI},E_{OC})}\right)}
+{\left(\frac{\Pr(D_{MI},\neg E_{OC})}{\Pr(\neg D_{MI},\neg E_{OC})}\right)}\\
+&= \left(\frac{\Pr(D_{MI},E_{OC})}{\Pr(\neg D_{MI},E_{OC})}\right)
+\left(\frac{\Pr(\neg D_{MI},\neg E_{OC})}{\Pr(D_{MI},\neg E_{OC})}\right)\\
+&= \left(\frac{\Pr(D_{MI},E_{OC})}{\Pr(D_{MI},\neg E_{OC})}\right)
+\left(\frac{\Pr(\neg D_{MI},\neg E_{OC})}{\Pr(\neg D_{MI},E_{OC})}\right)\\
+&= \left(\frac{\Pr(E_{OC},D_{MI})}{\Pr(\neg E_{OC},D_{MI})}\right)
+\left(\frac{\Pr(\neg E_{OC},\neg D_{MI})}{\Pr(E_{OC},\neg D_{MI})}\right)\\
+&= \left(\frac{\Pr(E_{OC}|D_{MI})}{\Pr(\neg E_{OC}|D_{MI})}\right)
+\left(\frac{\Pr(\neg E_{OC}|\neg D_{MI})}{\Pr(E_{OC}|\neg D_{MI})}\right)\\
+&= \frac{\left(\frac{\Pr(E_{OC}|D_{MI})}{\Pr(\neg E_{OC}|D_{MI})}\right)}
+{\left(\frac{\Pr(E_{OC}|\neg D_{MI})}{\Pr(\neg E_{OC}|\neg D_{MI})}\right)}\\
+&\eqdef \frac{\omega(E_{OC}|D_{MI})}
+{\omega(E_{OC}|\neg D_{MI})}\\
+&\eqdef \theta(E_{OC}; D_{MI})
 \end{aligned}
 $$
 :::