One number that stands in for every confounder. Rosenbaum & Rubin proved that e(x)=P(T=1 | X=x) is a balancing score: condition on that scalar and treatment becomes independent of the whole covariate vector. Adjust on the one number, and the bias from all of X falls away.
source Rosenbaum, P.R. & Rubin, D.B. (1983), “The Central Role of the Propensity Score in Observational Studies for Causal Effects,” Biometrika 70(1), 41–55. Rendered, not quoted — the engine below recomputes the balancing property and the IPW identity live.
A confounder X∈{0,1,2} drives both assignment and outcome — the classic observational trap. Structural equations, coefficients fixed:
Everything downstream is exact: population expectations are enumerated over the three strata — no Monte-Carlo noise in the checks. The true causal effect is read from a do-intervention that replaces T’s equation with a constant.
Neyman–Rubin gave us Y(1), Y(0) — but only one is ever observed. The the-backdoor-criterion says adjust on X to identify the effect. That can mean a high-dimensional X. This sphere is the compression: Rosenbaum & Rubin showed a scalar summary e(X) already carries all the adjustment X does. The neighbour’s potential outcomes become computable through one dimension.
Recomputes the live IPW estimate against the true ATE and re-tests the balancing identity every render. Confirms green while the propensity is correct; flips red the instant window 6 mis-specifies it.
The true propensity e(x) and the inverse-propensity weights it induces (treated: 1/e, control: 1/(1−e)):
| x | P(x) | e(x) | w·treated | w·control |
|---|
Note x=0 and x=1 share e=0.3 though they differ in X — the level set the balancing property lives on.
True effect from the do-intervention: —
Live IPW estimate (uses the current propensity): —
Balancing on shared-e group {0,1}: P(X=0 | e=.3, T=1) = —, P(X=0 | e=.3, T=0) = —
Scalar-e adjustment = — ≡ full-X adjustment = — (one number does the work of all of X)
Fixed-seed Monte-Carlo check (N=200000): —
With the correct propensity, the scalar e(X) balances X and IPW recovers the true effect exactly:
The propensity is only a balancing score for the covariates you put in it. An unmeasured confounder U — a hidden common cause of T and Y — is invisible to e(X); no weighting fixes it. And when e(x)→0 or 1 for some stratum, positivity fails, weights explode, and the estimator’s variance blows up even though it stays “unbiased” on paper.
“Matching on the propensity score removes all confounding.”
→ Only measured confounding. Unobserved U survives untouched. assumes ignorability
“A more flexible propensity model is always safer.”
→ Over-fit e(x) pushes estimated scores toward 0/1, wrecking positivity and inflating variance.
“e(X) being scalar means you lose information.”
→ No — it is sufficient for adjustment. Window 0 shows scalar-e = full-X, exactly.
Swap e(X) for a constant 0.5 that ignores X. The weights stop balancing; confounding leaks back in and IPW drifts off the true ATE. The witness (7) catches it.