Propensity Score Estimation for Augmented Estimators

Why Estimate a Propensity Score for AIPW?

The augmented inverse probability weighted (AIPW) estimator uses two fitted quantities: outcome predictions under each treatment and the probability of receiving treatment given measured covariates. The latter probability is the propensity score. Estimating it well matters, but “well” has more than one meaning. A model may predict treatment accurately while still being inadequate for causal estimation; conversely, prediction metrics alone do not determine whether the estimator’s causal assumptions hold.

This tutorial focuses on specifying and assessing propensity-score models when they are used in AIPW. It separates probability prediction diagnostics from the identification conditions that justify a causal interpretation. The earlier material on AIPW estimation and balance diagnostics supplies the broader estimator and balance concepts; here, the emphasis is on the propensity-score fit and what its diagnostics can and cannot establish.

Definition: Let \(A\) be a binary treatment, with \(A=1\) for treated and \(A=0\) for control, and let \(X\) denote measured pre-treatment covariates. The propensity score is \(e(X)=P(A=1\mid X)\). A fitted propensity score, \(\hat e(X_i)\), is an estimate of that conditional probability for individual \(i\).

For the average treatment effect, a common AIPW estimator is

\hat{\tau}_{\mathrm{AIPW}} = \frac{1}{n}\sum_{i=1}^{n} \left[ \hat m_1(X_i)-\hat m_0(X_i) +\frac{A_i}{\hat e(X_i)}\{Y_i-\hat m_1(X_i)\} -\frac{1-A_i}{1-\hat e(X_i)}\{Y_i-\hat m_0(X_i)\} \right].

Here \(Y_i\) is the observed outcome and \(\hat m_a(X_i)\) is the fitted outcome prediction under treatment level \(a\). The propensity score determines the residual weights in the two correction terms. Estimating \(\hat e\) is not, by itself, a demonstration that the causal effect is identified.

Specifying and Fitting the Assignment Model

A standard starting point for a binary treatment is logistic regression. Its covariates should be chosen for the causal analysis, not merely because they improve treatment classification. They are typically measured before treatment and include relevant causes or predictors of treatment and outcome. Treatment-affected variables should not be included as if they were baseline confounders.

\operatorname{logit}\{e(X_i)\} = \log\left\{\frac{e(X_i)}{1-e(X_i)}\right\} = \beta_0+\beta_1X_{i1}+\cdots+\beta_pX_{ip}.

The fitted model estimates the coefficients from the observed treatment assignments and covariates; substituting the fitted coefficients gives \(\hat e(X_i)\). The linear predictor need not be linear in the original covariates. Depending on the design, useful terms may include nonlinear functions, interactions, or indicators for categories. These terms should reflect plausible structure in treatment assignment and should be assessed with causal diagnostics as well as prediction diagnostics.

Worked Example: Calculating Fitted Scores From a Logistic Model

Suppose a hypothetical study fits a logistic treatment model using two binary pre-treatment covariates: \(X_1\), an indicator for a clinical risk factor, and \(X_2\), an indicator for prior service use. The fitted model is

\operatorname{logit}\{\hat e(X)\}=-1+0.7X_1+0.4X_2.

For an individual with \(X_1=1\) and \(X_2=0\), the fitted log-odds are \(-1+0.7=-0.3\). Converting log-odds to a probability gives

\hat e(X) = \frac{\exp(-0.3)}{1+\exp(-0.3)} = \frac{0.7408}{1.7408} \approx 0.4256.

For an individual with \(X_1=0\) and \(X_2=1\), the fitted log-odds are \(-1+0.4=-0.6\), so

\hat e(X) = \frac{\exp(-0.6)}{1+\exp(-0.6)} = \frac{0.5488}{1.5488} \approx 0.3543.

These values are estimated treatment probabilities for individuals with the stated covariates, not treatment effects. Whether the fitted model is useful for AIPW requires further assessment, including whether relevant covariates are adequately represented and whether the resulting analysis is supported by the causal assumptions.

The causal identification assumptions are distinct from the choice of regression model. For the usual ATE interpretation, the central conditions are consistency (the observed outcome equals the potential outcome under the treatment actually received), conditional exchangeability (potential outcomes are independent of treatment conditional on \(X\)), and positivity (both treatment levels have positive probability at covariate values in the target population). These conditions concern the design, measurement, and target population; a good fit statistic cannot verify them.

Why the Score Can Help: A Short Justification

The propensity score’s weighting role follows directly from its definition. For any function \(h(X)\) for which the expectation exists, and where \(e(X)>0\),

E\left[\left.\frac{A h(X)}{e(X)}\right|X\right] = \frac{h(X)}{e(X)}E(A\mid X) = \frac{h(X)}{e(X)}e(X) = h(X).

Taking expectations on both sides shows that weighting treated observations by \(1/e(X)\) reproduces the population expectation of \(h(X)\), in expectation. A parallel identity applies to controls using \(1/\{1-e(X)\}\). This explains why a suitable score can help balance measured covariates. It does not imply that unmeasured confounding disappears, that positivity holds in practice, or that every fitted score gives good finite-sample balance.

Worked Example: Comparing Probability Prediction Scores

Consider four hypothetical treatment assignments, \(A=(1,1,0,0)\), and two sets of fitted probabilities. Model C predicts \((0.9,0.6,0.4,0.1)\); Model D predicts \((0.7,0.7,0.3,0.3)\). The Brier score is the mean squared difference between the observed treatment and its predicted probability:

\operatorname{Brier} = \frac{1}{n}\sum_{i=1}^{n}(A_i-\hat e_i)^2.

For Model C, the squared errors are \(0.01, 0.16, 0.16, 0.01\), giving

\operatorname{Brier}_C = \frac{0.01+0.16+0.16+0.01}{4} = 0.085.

For Model D, each squared error is \(0.09\), so \(\operatorname{Brier}_D=0.09\). On this small example, Model C has the lower Brier score. The mean log loss is another proper probability score, defined here as

-\frac{1}{n}\sum_{i=1}^{n} \{A_i\log(\hat e_i)+(1-A_i)\log(1-\hat e_i)\}.

For Model C it is \(-\{\log(0.9)+\log(0.6)+\log(0.6)+\log(0.9)\}/4\approx0.3081\). For Model D it is \(-\log(0.7)\approx0.3567\). Both scores favor Model C here. These calculations assess predictions of observed treatment assignment. They do not test exchangeability, and four observations are far too few to support a reliable model comparison.

Assessing a Propensity-Score Fit for Causal Use

Prediction diagnostics and causal diagnostics answer different questions. Calibration asks whether cases assigned a given predicted probability receive treatment at approximately that frequency. Brier score and log loss assess the accuracy of probability predictions, with log loss penalizing confident errors particularly strongly. Discrimination measures whether treated individuals tend to receive higher scores than controls; for example, the area under the receiver operating characteristic curve summarizes ranking. A model can rank well but have probabilities that are poorly calibrated, so discrimination alone is not a sufficient assessment of probability estimates.

For AIPW, also examine the implications of the score for the covariate distribution. Balance diagnostics compare measured covariates across treatment groups after the relevant weighting or adjustment. Inspect covariates that informed the causal design, including important nonlinear terms or interactions where appropriate. Check the score distributions by treatment group for evidence that the target population has covariate profiles poorly represented in one group. The earlier material on balance diagnostics develops these comparisons in detail.

A low prediction error does not guarantee balance, and an excellent balance summary does not establish that all confounders were measured or modeled correctly. No diagnostic based only on observed treatment and measured covariates can prove conditional exchangeability. Positivity is also an assumption about the target population and treatment assignment, not a property certified by a single summary statistic. These checks are evidence for evaluating a design, not substitutes for substantive knowledge.

Worked Example: Accurate Treatment Prediction With Confounding

Suppose, hypothetically, that all individuals have the same measured \(X\), and 90% receive treatment. A model predicting \(\hat e(X)=0.9\) is calibrated in this population and classifies treatment correctly for 90% of individuals if probabilities above 0.5 are classified as treated. However, suppose an unmeasured binary severity variable \(U\) is present. Among treated individuals, 90% have \(U=1\); among controls, 10% have \(U=1\). Let the potential outcomes, in outcome units, be

Y(0)=10U,\qquad Y(1)=2+10U.

The true treatment effect for every individual is \(Y(1)-Y(0)=2\). Yet the observed treated mean is \(2+10(0.9)=11\), while the observed control mean is \(10(0.1)=1\). The observed difference is \(11-1=10\) outcome units, not the true causal effect of 2 units. The score predicts treatment frequency well given the measured \(X\), but \(U\) creates confounding that conditional exchangeability given \(X\) does not remove. This example shows why prediction quality cannot establish causal identification.

Common Pitfalls and Practical Choices

Choosing covariates only to maximize classification. A variable can predict treatment without helping control confounding, and a treatment-assignment model selected only for prediction may not give satisfactory balance. Choose covariates with the causal question in view, then examine balance.
Treating a high AUC as proof of a valid score. AUC measures ranking, not calibration, exchangeability, or positivity. It cannot show that the causal assumptions hold.
Adding post-treatment variables. Conditioning on variables affected by treatment can distort the causal comparison. The propensity model should be based on appropriately measured pre-treatment information.
Assuming diagnostics prove the model is correct. Calibration, prediction scores, and balance reveal aspects of observed data fit. None can rule out unmeasured confounding or establish every modeling assumption.
Interpreting one diagnostic in isolation. Review the covariate specification, probability predictions, balance, overlap, and substantive causal assumptions together. A favorable result on one dimension does not override a serious concern on another.

The main alternatives are not competing tests of the same thing. Outcome-regression diagnostics assess the fitted outcome predictions; propensity-model diagnostics assess treatment probabilities and their consequences for adjustment. AIPW combines these nuisance fits, but the propensity-model checks remain useful even when the outcome model is also fitted. A prediction-focused model-selection rule is not a replacement for either causal reasoning or balance assessment.

The practical objective is not to find a propensity model with the best possible treatment classification score. It is to specify a defensible model for measured treatment assignment, assess its probabilities and their impact on the treated and control covariate distributions, and state the causal assumptions separately. Difficulties in overlap or finite-sample behavior require additional attention; they should not be hidden by reporting a favorable prediction metric.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A logistic model gives \(\operatorname{logit}\{\hat e(X)\}=0.2-0.5X\). What is the fitted propensity score when \(X=1\)?
In one sample, a model has lower Brier score but lower AUC than another model. Which model is necessarily better for causal identification?
State the conditional expectation identity that explains why inverse propensity weighting can balance a function of measured covariates.
A score is well calibrated for measured \(X\), but an unmeasured cause of treatment and outcome remains. Does calibration establish exchangeability?

Solutions. 1. The linear predictor is \(0.2-0.5=-0.3\), so \(\hat e(X)=\exp(-0.3)/\{1+\exp(-0.3)\}\approx0.4256\). 2. Neither is necessarily better for causal identification. Brier score and AUC assess different aspects of treatment prediction; neither establishes exchangeability or positivity, and balance and causal assumptions must also be assessed. 3. For \(e(X)>0\), \(E[A h(X)/e(X)\mid X]=h(X)E(A\mid X)/e(X)=h(X)\). 4. No. Calibration concerns predicted treatment frequencies given measured information. It cannot establish that treatment is independent of potential outcomes conditional on \(X\), especially when an unmeasured common cause remains.