Worked AIPW Calculation for a Binary Treatment

From Predictions to an AIPW Estimate

An augmented inverse-probability weighted (AIPW) calculation combines two kinds of predictions for each person: a propensity score, which is the predicted probability of receiving treatment given covariates, and two treatment-specific outcome predictions. The calculation then adds a residual correction for the outcome actually observed. Working through the contributions one person at a time makes the estimator easier to verify and helps reveal when a small number of observations dominate the result.

Let \(A_i\) indicate treatment for person \(i\), with \(A_i=1\) for treated and \(A_i=0\) for untreated. Let \(Y_i\) be that person’s observed outcome and \(X_i\) their covariates. Write \(Y_i(1)\) and \(Y_i(0)\) for the potential outcomes under treatment and control. The target is the average treatment effect (ATE), \(E[Y(1)-Y(0)]\), for the population represented by the data. The observed outcome is \(Y_i=A_iY_i(1)+(1-A_i)Y_i(0)\), by consistency.

The first part, \(\hat m_1(X_i)-\hat m_0(X_i)\), is the predicted treatment-control contrast for person \(i\). If that person was treated, the residual \(Y_i-\hat m_1(X_i)\) corrects the treated prediction; if untreated, \(Y_i-\hat m_0(X_i)\) corrects the control prediction. The indicators \(A_i\) and \(1-A_i\) ensure that only the residual from the observed treatment group is used. The propensity weights scale that residual correction. They do not create observations of the unobserved potential outcome.

For the estimate to have a causal interpretation, the standard identification conditions are required: consistency and no interference between people; conditional exchangeability, \((Y(1),Y(0))\) independent of \(A\) given \(X\); and positivity, \(0<P(A=1\mid X)<1\) for covariate values in the target population. In addition, the nuisance predictions must be estimated in a way that supports the desired statistical properties. The data table alone cannot establish exchangeability or rule out unmeasured confounding. For the arithmetic, check that every propensity prediction is strictly between zero and one and that predictions and outcomes refer to the same person and covariate values.

Why the Residual Correction Works

The structure of the estimator can be justified by considering its population counterpart. If the propensity model gives the true \(e(X)=P(A=1\mid X)\), then, conditional on \(X\), the expected treated residual correction is \(E[Y(1)-m_1(X)\mid X]\). The expected control correction before its minus sign is \(E[Y(0)-m_0(X)\mid X]\). Adding these to \(m_1(X)-m_0(X)\) gives \(E[Y(1)-Y(0)\mid X]\). Averaging over \(X\) gives the ATE.

This calculation uses consistency and exchangeability to relate observed residuals in each treatment group to the corresponding potential outcomes. Alternatively, if both outcome predictions equal their conditional means, the conditional expected residuals are zero, so the expected correction terms vanish. These population arguments explain the estimator’s augmentation; they do not guarantee that a particular small-sample estimate is stable or that the identification assumptions hold.

Binary Outcomes and Risk Differences

The same calculation applies when \(Y\) is binary. In that case, the outcome predictions are predicted probabilities, and the ATE is measured as a risk difference. Each contribution can fall outside the interval from \(-1\) to \(1\); the AIPW contribution is a corrected estimating quantity, not an individual’s predicted probability or individual treatment effect.

Checks Before Reporting the Number

A hand calculation is useful partly because it exposes input and implementation errors. Before interpreting the result, check the following:

• Prediction columns: \(\hat m_1(X_i)\) must be the prediction under treatment and \(\hat m_0(X_i)\) the prediction under control, for the same person’s covariates.
• Observed residual: use \(Y_i-\hat m_1(X_i)\) for a treated person and \(Y_i-\hat m_0(X_i)\) for a control person. The other potential outcome is not observed and does not supply a residual.
• Signs and denominators: the treated correction is added with denominator \(\hat e(X_i)\); the control correction is subtracted with denominator \(1-\hat e(X_i)\).
• Score range and influence: scores must be strictly between zero and one. Very small \(\hat e(X_i)\) for a treated person, or very small \(1-\hat e(X_i)\) for a control person, can produce a large correction. Inspect the individual contributions rather than relying only on the final average.
• Interpretation: retain the outcome’s units. A numerical estimate does not verify exchangeability, consistency, or positivity, and a small dataset may give an unstable estimate even when the arithmetic is correct.

The main alternatives answer related questions with different calculations. Outcome regression averages predicted contrasts and does not use these residual corrections. Inverse-probability weighting uses weighted observed outcomes without the outcome-model augmentation. AIPW combines the two components, but its finite-sample result can still be sensitive to extreme scores or inaccurate predictions. For inference, the estimate alone is not enough: a standard error and interval must use an appropriate variance procedure, including any clustering in the data.

The introductory material on AIPW under correct and misspecified nuisance models gives the broader interpretation of the estimator’s two nuisance components. For a numerical calculation, the essential discipline is to compute and record each person’s full contribution before averaging; that makes sign, denominator, and influential-observation errors easier to locate.