March 22, 2019
A more general approach to the comparative method is to adjust using:
when we observe \(X\) and \(Y\) for multiple cases, we examine the correlation of \(X\) and \(Y\) within groups of cases that have the same values of confounding variables \(W, Z, \ldots\).
How does conditioning solve confounding?
Sometimes we think about "conditioning" like this:
But all else does not need to be equal:
We saw in toy example, an "unadjusted" correlation between \(Sanctuary\) and \(Crime\) led to confounding and bias
Conditioning on \(Urban\) solved the confounding, removing the bias
Examines crime rates in 2492 US counties. 608 are "Sanctuary" counties.
Researcher matched counties on:
Compared to similar non-sanctuary counties
Sanctuary policies do not cause an increase in crime
In order to infer \(X\) causes \(Y\) if \(X,Y\) correlated after adjustment
How do we know we have found and measured all confounding variables?
In order to infer \(X\) causes \(Y\) if \(X,Y\) correlated after adjustment
In addition to assumptions: adjustment has other limitations
We have shown adjustment that finds identical cases.
Adjustment: "adjusts" correlation of \(X\) (cause) and$ Y$ (outcome) by conditioning on specific, potentially confounding variables
We can infer causality ONLY IF:
design-based solutions to confounding:
Contrast to adjustment:
Designs using conditioning
Designs using random exposure to \(X\)
Choice of comparison holds many confounding variables constant
Holds constant many confounding variables