Propensity Scores

The propensity score is the probability of receiving treatment given observed pretreatment covariates:

\[\boxed{e(X)=P(W=1\mid X)}.\]

Estimation

A common specification is logistic regression:

\[\hat e(X)= \frac{1} {1+\exp[-(\hat\beta_0+X^\top\hat\beta)]}.\]

The goal is covariate balance for causal adjustment, not treatment prediction for its own sake. A model with excellent classification may produce extreme scores and poor overlap.

Balancing property

For the true propensity score,

\[W\perp X\mid e(X).\]

Within a propensity-score stratum, treated and control units have the same distribution of observed covariates in expectation.

If treatment is unconfounded given $X$, then it is also unconfounded given the true propensity score:

\[(Y(0),Y(1))\perp W\mid e(X).\]

The propensity score reduces a potentially high-dimensional adjustment problem to a scalar balancing score. It does not solve unmeasured confounding.

Workflow

  1. Select pretreatment covariates using causal and subject-matter knowledge.
  2. Estimate $e(X)$ without using outcomes to chase a favorable effect estimate.
  3. Inspect treatment-control overlap in the score distributions.
  4. Apply stratification, weighting, or matching.
  5. Check covariate balance after adjustment.
  6. Revise the assignment model if balance remains poor.

Diagnostics

Useful checks include standardized mean differences, variance ratios, score overlap plots, effective sample size, and the presence of extreme weights. Balance on $\hat e(X)$ alone is insufficient; examine balance on the original covariates and important transformations.

See