Propensity Score — Se Yoon Lee, Ph.D. (Mathematically Audited)

Mathematical formulation

Definition, balancing-score argument, causal identification, positivity, inverse weighting, and diagnostic implications.

NATIVE MATHML

Measured baseline covariates → treatment probability

X ⟶ e (X) = P (T = 1 | X) ⟶ \hat{e} (X)

Working treatment model

logit {e_{γ} (X)} = Z^{T} (X) γ

The model describes treatment assignment. The outcome is not used to fit it.

Estimated score

{\hat{e}}_{i} = expit {Z^{T} (X_{i}) \hat{γ}}

A fitted value of 0.70 is a modeled treatment probability for that measured profile, not a biological fraction of a person.

Dimension reduction

X \in R^{p} ⟶ e (X) \in [0, 1]

Different covariate profiles can have the same treatment probability. The score is a deterministic function of the measured baseline vector.

Balancing-score property of the true propensity score

T ⊥ X | e (X)

Iterated-expectation step

E {T | e (X)} = E [E {T | X} | e (X)] = e (X)

The equality holds almost surely because E(T|X)=e(X).

Conditioning step

P {T = 1 | X, e (X)} = e (X) = P {T = 1 | e (X)}

Thus, given the true score, the full measured vector provides no additional information about treatment.

Measure-theoretic qualification: these are regular conditional probabilities and the statement holds almost surely. It avoids treating the event e(X)=r as having positive probability when the score is continuous.

Estimated score: exact balance is a theorem for the true score. A fitted score and a finite-width band provide only an empirical approximation that must be checked.

Consistency

Y = T Y (1) + (1 - T) Y (0)

The observed outcome equals the potential outcome under the treatment actually received.

Conditional exchangeability

{Y (1), Y (0)} ⊥ T | X

The measured baseline vector must be sufficient for the no-unmeasured-confounding assumption.

Positivity in the target population

0 < e (X) < 1 almost surely

Both treatment levels must be possible for covariate profiles represented in the target population.

Dimension-reduced causal identification

{Y (1), Y (0)} ⊥ T | X \Rightarrow {Y (1), Y (0)} ⊥ T | e (X)

E {Y (t)} = E [E {Y | T = t, e (X)}]

Logical distinction: exchangeability is inherited by the true balancing score; positivity is separately required to identify both treatment-specific means. Neither condition is verified by a good propensity-model fit.

Unstabilized ATE weights

w_{i} = \frac{T_{i}}{{\hat{e}}_{i}} + \frac{1 - T_{i}}{1 - {\hat{e}}_{i}}

E {\frac{T f (X)}{e (X)}} = E {f (X)} = E {\frac{(1 - T) f (X)}{1 - e (X)}}

Population identity

The equality holds for every integrable f and the true propensity score. Estimated weights satisfy it only approximately in a finite sample.

Kish effective sample size

{ESS}_{t} = \frac{{\sum 𝟙 (T_{i} = t) w_{i}}^{2}}{\sum 𝟙 (T_{i} = t) w_{i}^{2}}

ESS summarizes weight concentration; it is not an actual count of independent subjects.

Practical positivity: fitted scores close to 0 or 1 can create large, unstable weights even when formal positivity is not literally violated.

Standardized mean difference

\begin{matrix} {SMD}_{j}^{(w)} = \frac{{\hat{μ}}_{1 j}^{(w)} - {\hat{μ}}_{0 j}^{(w)}}{{\hat{σ}}_{j, ref}} \\ {\hat{σ}}_{j, ref}^{2} = \frac{1}{2} ({\hat{σ}}_{1 j}^{2} + {\hat{σ}}_{0 j}^{2}) \\ {\hat{σ}}_{t j}^{2} = n_{t}^{- 1} \sum_{i : T_{i} = t} {X_{i j} - {\hat{μ}}_{t j}}^{2} \end{matrix}

The dashboard fixes this unweighted pre-weighting empirical reference scale before weighting, making the before–after mean differences directly comparable. The 0.10 line is a descriptive heuristic, not a causal-identification theorem.

Empirical min–max common support

\begin{matrix} a_{t} = \min_{i : T_{i} = t} {\hat{e}}_{i}, b_{t} = \max_{i : T_{i} = t} {\hat{e}}_{i} \\ I_{CS} = [\max (a_{1}, a_{0}), \min (b_{1}, b_{0})] \end{matrix}

The interval is declared empty if its lower endpoint exceeds its upper endpoint. It is sample-dependent and is not proof of population positivity.

Calibration is not balance

E {T | e (X)} = e (X)

Calibration checks treatment probabilities. Balance checks measured covariate distributions. One does not imply the other for a fitted finite-sample model.

Treatment prediction is secondary

high AUC \neq good causal design

A highly discriminating score can signal separation and poor overlap. The primary diagnostics are balance, overlap, and weight stability.

1. Many measured covariates, one propensity score

Subject A is the selected participant. Subject B is the nearest participant from the opposite observed treatment group on the active score.

smokernon-smokercovariates compressed into score

Selected subject

Click a point or table row to inspect another participant.

2. Propensity-score overlap

At a score value, causal comparison is empirically supported only when both observed treatment groups are represented nearby.

smokersnon-smokerscurrent score bandoutside empirical common support

3. Treatment calibration on the score scale

Within score bins, compare the mean score with the observed smoking proportion.

identity linescore-bin estimate

4. Balancing within a narrow propensity-score band

Choose a covariate and inspect smokers and non-smokers with similar active scores.

Covariate

Exact balance is a property of conditioning on the true score. A finite-width band and an estimated score produce an approximation; the displayed within-band SMD is therefore a diagnostic, not a theorem.

5. Overlap in the original covariate space

Age and BMI are only a two-dimensional projection of the full measured covariate space.

smokernon-smokeroutside PS common support

A clean-looking age–BMI projection cannot establish multivariate positivity. The plot deliberately avoids drawing a rectangular “overlap region,” because overlap in separate age and BMI ranges does not imply joint multivariate overlap. Race, education, income, and their combinations still matter.

6. Positivity and inverse-weight consequences

Extreme scores generate large ATE weights: 1/e for smokers and 1/(1−e) for non-smokers.

smoker weight 1/enon-smoker weight 1/(1−e)

7. Covariate balance before and after ATE weighting

Absolute standardized mean differences summarize measured baseline imbalance.

before weightingafter weighting|SMD| = 0.10 reference

For before–after comparability, the dashboard standardizes both unweighted and weighted mean differences by the same unweighted pre-weighting pooled reference SD. Other software may use a different documented SMD convention.

8. Propensity-score stratification

Within each score quintile, smokers and non-smokers should have more similar measured covariate summaries.

Covariate

smoker mean/proportionnon-smoker mean/proportion

9. The propensity score is not outcome risk

The horizontal axis predicts smoking; the vertical axis predicts CVD under no smoking in the simulation.

10. What the propensity score is—and is not

Common interpretation errors from the lecture.

Not outcome riske(X) predicts treatment assignment, not CVD.

Not a recommendationA high score does not mean a participant should smoke.

Not proof of exchangeabilityA well-fitting treatment model cannot rule out unmeasured confounding.

Not for post-treatment variablesUse baseline covariates, not variables caused by treatment.

Diagnostics are centralBalance and overlap matter more than treatment-model p-values.

Extreme scores are substantive warnings about limited comparability, not merely inconvenient numbers to hide.

Inspect coefficients, participants, and export resolution-independent graphics

Treatment-model coefficient table. Coefficients and odds ratios are shown for transparency. Wald inference is displayed only for an unpenalized converged logistic maximum-likelihood fit and remains secondary to balance and overlap diagnostics.

Term	Coefficient	Odds ratio	SE	p-value

Participant-level data. Click a row to select that participant.

ID	T	Y	Age	Gender	Race	Education	Income	BMI	true e	ê	active score	ATE weight	PS quintile