The paradox is that
- treatment $B$ looks better in both of the subgroups,
- but treatment $A$ looks better when examining the total population ignoring the subgroups.
Lecture 1: Motivation
Simpson’s Paradox
New disease: COVID-27
- Treatment $T = \{A, B\}$.
- Condition $C = \{\text{mild}, \text{severe}\}$.
- Outcome $Y = \{\text{alive}, \text{dead}\}$
Currently receiving treatment $A$ vs. $B$: 73% / 27%
Goal: Choosing which treatment to use.
$$ \frac{50}{550} (0.10) + \frac{500}{550} (0.20) = 0.19 $$
The non-uniformity of allocation of people to the groups:
- 1400 of the 1500 people received treatment $A$ had mild condition,
- whereas 500 of the 550 people who received treatment B had severe condition.
people with mild condition are less likely to die $\Rightarrow$ total mortality rate for those with treatment A is lower.
Scenario 1
Doctors give treatment $A$ to most people with mild conditions, save the more expensive and limited treatment $B$ for ppl with severe conditions.
- $B$ assigned to dying people.
- $A$ assigned to health people.
$C$ confounds $T$. To correct for this confounding, examine the relationship of $T$ and $Y$ among patients with the same conditions $\Rightarrow$ consider subpopulations (“Mild” and “Severe” columns).
Scenario 2
Might have to wait a long time to take treatment $B$ (its scarce). In that time, the condition got worsen (mild $\to$ severe).
- 500/ 550 who wait for treament $B$ got worse condition.
$\Rightarrow$ Treatment $B$ is a cause of the Condition.
Even if treatment B
- is more effective than treatment A once administered (positive effect along $T \to Y$)
- because precription of treatment $B$ causes worse conditions (negative effect along $T \to C \to Y$)
$\Rightarrow$ $B$ is less effective in total.
Correlation Does Not Imply Causation
Correlation vs. Association
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correlation: measures linear relationships.
-
association: any kind of statistical dependence.
Association $\neq$ Causation
Just because two things are associated doesn’t mean one causes the other. The relationship could be partly causal, fully causal, or not causal at all.
Example - Shoes and Headaches Data might show that people who sleep with shoes on often wake up with headaches. Sleep with shoes on $\to$ wake up with headaches? Actually, both events are often caused by a third factor: drinking alcohol the night before.
- Drinking $\to$ passing out with shoes on.
- Drinking $\to$ headache.
The main problem motivating causal inference is that association is not causation.