Estimating Weighted Potential-Outcome Means

From Assignment Weights to Outcome Means

Inverse-probability weighting turns observed outcomes into estimates of potential-outcome means by using treatment-assignment probabilities. The weights depend on the target population: the weights for an ATE are not interchangeable with those for an ATT or ATU. This tutorial focuses on calculating the weighted means and taking their contrast; a contrast estimates an average treatment effect for the population defining those means.

Let \(A_i\) be the binary treatment indicator for unit \(i\), with \(A_i=1\) for treated and \(A_i=0\) for untreated. Let \(Y_i(1)\) and \(Y_i(0)\) be that unit’s potential outcomes under treatment and no treatment. The observed outcome is \(Y_i=A_iY_i(1)+(1-A_i)Y_i(0)\). Let \(X_i\) denote pretreatment covariates and \(e_i=P(A_i=1\mid X_i)\) the propensity score. In practice, \(e_i\) is often estimated from data; the formulas below use \(e_i\) to represent the score used to construct the weights.

For a sample of \(n\) units, write \(n_1=\sum_{i=1}^n A_i\) and \(n_0=\sum_{i=1}^n(1-A_i)\). The inverse-probability estimates below use these counts as denominators for the relevant target groups. In particular, ATE means use \(n\); ATT means use \(n_1\); and ATU means use \(n_0\). These are Horvitz–Thompson-style estimates, not versions divided by the sum of the weights.

Assumptions and Target-Specific Formulas

The causal interpretation requires assumptions about how treatment is assigned and how potential outcomes relate to the observed data. Consistency means a unit’s observed outcome equals its potential outcome under the treatment actually received. Conditional exchangeability means \((Y(1),Y(0))\) is independent of \(A\) given \(X\): after conditioning on the measured pretreatment covariates, treatment assignment carries no additional information about the potential outcomes. Positivity requires \(0<e(X)<1\) for covariate values in the target population. The propensity scores must also be correctly specified or consistently estimated for the weighting identities to apply.

Under these conditions, the ATE mean under treatment is estimated using treated outcomes weighted by \(1/e_i\); the ATE mean under no treatment uses untreated outcomes weighted by \(1/(1-e_i)\). For the ATT, the observed treated mean already describes the treated target group. To estimate its mean under no treatment, untreated outcomes receive \(e_i/(1-e_i)\), not \(1/(1-e_i)\). For the ATU, the observed untreated mean already describes the untreated target group. Its treated counterfactual mean uses \((1-e_i)/e_i\), not \(1/e_i\).

The denominators reflect the target group, while the weights reconstruct the outcome mean missing for that group. The treatment indicator in each sum ensures that only outcomes observed under the relevant treatment condition contribute. Keep the order of subtraction fixed: the treated potential-outcome mean minus the untreated potential-outcome mean.

Why the Weights Identify the Means

The central identities follow by conditioning on \(X\). For the ATE, exchangeability and consistency imply that the treated outcome distribution conditional on \(X\) represents \(Y(1)\) conditional on \(X\). Weighting by \(A/e(X)\) cancels the probability of being treated. In population notation,

Likewise, \(E[(1-A)Y/(1-e(X))\mid X]=E[Y(0)\mid X]\). Averaging these conditional means over the covariate distribution gives the two ATE means.

For the ATT counterfactual mean, the target is the covariate distribution among treated units. Conditional exchangeability gives \(E[Y\mid A=0,X]=E[Y(0)\mid X]=E[Y(0)\mid A=1,X]\). The identity

shows why the weight is \(e/(1-e)\): it makes the weighted untreated outcomes represent the treated group’s covariate distribution. Dividing the population numerator by \(P(A=1)\) gives \(E[Y(0)\mid A=1]\). Symmetrically, \(E[A(1-e(X))Y/e(X)\mid X]=(1-e(X))E[Y(1)\mid X]=E[(1-A)Y(1)\mid X]\), so division by \(P(A=0)\) gives the ATU treated mean. These identities justify the target-specific weights; with estimated scores, the sample formulas are plug-in estimates and their accuracy depends on score estimation and data support.

Worked Calculations

Diagnostics, Comparisons, and Limitations

Before interpreting a contrast, check the treatment coding, outcome units, score range, target denominator, and every weight formula. An ATE analysis should use treated weights \(1/e_i\) and untreated weights \(1/(1-e_i)\). An ATT analysis uses the observed treated mean and untreated counterfactual weights \(e_i/(1-e_i)\); an ATU analysis uses treated counterfactual weights \((1-e_i)/e_i\) and the observed untreated mean. Also check whether the weighted covariate distributions are plausible representations of the target group. Large weights, poor overlap, or weighted observations concentrated in a few units indicate that estimates may be sensitive to individual records.

Weighting is not the only approach to estimating potential-outcome means. Regression standardization models outcomes conditional on covariates and averages predicted outcomes over a target population. Matching constructs comparisons among units with similar covariates or propensity scores. Each approach relies on causal assumptions and can be vulnerable to poor overlap or model misspecification; their calculations and diagnostics differ. Weighting’s advantage is that its target-specific reweighting is explicit. Its limitation is that small assignment probabilities can produce extreme weights and imprecise, influential estimates.

A frequent mistake is to describe every inverse-probability weight as the inverse probability of receiving the observed treatment. That misses the adjustment for the target population in the ATT and ATU counterfactual means. Another mistake is to regard a calculated contrast as causal without considering exchangeability, consistency, positivity, and score estimation. The weighted arithmetic alone does not establish these assumptions, and observational data cannot rule out unmeasured confounding.