Propensity scores

TL;DR

Propensity scores are likelihoods. The likelihood of a certain type of unit receiving the treatment.

Experiments are usually design balancing groups of treated \(T=1\) vs control people \(T=0\). And making sure that, if our population has some important features \(X\), those will be equally represented in both groups.

Imagine testing the effectiveness of a new dietary supplement. Given some characteristic \(X = \text{vegan}\), you'd want to see roughly the same number of these individuals in the treated and untreated groups.

In observational studies, groups are not balanced and propensity scores measure that unbalance. Formally,

\[e(x) \triangleq P(T = 1 | X = x)\]

If \(e(x) = 0.5\) for all \(x\), this means the treatment is actually independent of \(x\) and we've truly randomized it, guaranteeing no confounding!

TL;DR

Instead of controlling for \(X\), you can control for \(e(X)\) and that is great.

TL;DR

Inverse-probability weighting (IPW) is a method that uses propensity scores to estimate causal effects.