Inverse-Probability Weighting for the Average Treatment Effect

Why Weight by Treatment Probability?

A treated person with a low probability of receiving treatment represents many similar people who were unlikely to be treated; a control person with a low probability of remaining untreated represents many similar people who were unlikely to be controls. Inverse-probability weighting (IPW) uses these ideas to construct weighted outcome averages for the treated and untreated states. Their difference targets the average treatment effect (ATE) for the population represented by the data.

The method requires more than a propensity score calculation. It relies on assumptions that connect observed outcomes to the potential outcomes of interest. We state those assumptions first, then derive the estimator and work through its arithmetic.

Potential Outcomes, Treatment Probabilities, and Assumptions

Definition: For unit \(i\), let \(D_i\) equal 1 if the unit receives treatment and 0 otherwise. Let \(Y_i(1)\) and \(Y_i(0)\) denote that unit’s potential outcomes under treatment and control. The observed outcome is \(Y_i=D_iY_i(1)+(1-D_i)Y_i(0)\). For pre-treatment covariates \(X_i\), the propensity score is \(e(X_i)=P(D_i=1\mid X_i)\). The average treatment effect is \(\tau=E[Y(1)-Y(0)]\), an average in the target population.

Only one potential outcome is observed for each unit. To identify the ATE from observed data, assume consistency (the observed outcome equals the potential outcome under the treatment actually received), no interference between units, conditional exchangeability \((Y(1),Y(0))\perp D\mid X\), and positivity \(0<e(X)<1\) for covariate values in the target population. The covariates used for adjustment must be measured before treatment. Conditional exchangeability rules out remaining confounding after conditioning on \(X\); positivity requires that both treatment states are possible at each relevant \(X\).

The propensity score may be known from the assignment design or estimated from observed covariates, as in propensity-score estimation. The derivation below first uses the true conditional treatment probability. Replacing it with an estimate gives the practical plug-in estimator, but does not make the assumptions unnecessary.

Deriving the IPW Identification Formula

For a treated unit, \(D=1\) and \(Y=Y(1)\). Weighting its observed outcome by \(1/e(X)\) corrects for the probability of entering the treated group. For a control unit, the corresponding weight is \(1/\{1-e(X)\}\). Conditional exchangeability lets us use observed outcomes within each treatment state to represent the relevant potential outcomes.

\begin{aligned} E\left[\left.\frac{DY}{e(X)}\right|X\right] &=\frac{1}{e(X)}E[DY(1)\mid X]\\ &=\frac{P(D=1\mid X)}{e(X)}E[Y(1)\mid D=1,X]\\ &=E[Y(1)\mid X], \end{aligned}

The last equality uses \(P(D=1\mid X)=e(X)\) and conditional exchangeability. For controls, the parallel calculation is:

\begin{aligned} E\left[\left.\frac{(1-D)Y}{1-e(X)}\right|X\right] &=\frac{1}{1-e(X)}E[(1-D)Y(0)\mid X]\\ &=E[Y(0)\mid X]. \end{aligned}

Subtracting these conditional expectations and averaging over the target population’s distribution of \(X\) gives the ATE:

\tau = E\left[ \frac{DY}{e(X)} - \frac{(1-D)Y}{1-e(X)} \right].

This is an identification result: under the stated assumptions, the expected weighted contrast equals the average contrast in potential outcomes. It motivates the sample estimator, but does not imply that every finite sample estimate will equal the population ATE.

The Sample Estimator

With \(n\) observations and propensity scores \(e_i=e(X_i)\), the Horvitz–Thompson IPW estimator of the ATE is:

\widehat{\tau}_{\mathrm{IPW}} = \frac{1}{n}\sum_{i=1}^{n} \left[ \frac{D_iY_i}{e_i} - \frac{(1-D_i)Y_i}{1-e_i} \right].

Each observation contributes either a positively weighted treated outcome or a negatively weighted control outcome—not both. The denominator is \(n\), the number of sampled units. The estimate uses the same target-population weighting over covariates for both treatment states. It is not simply the difference between the unweighted treated and control means.

If the true probabilities are known, the estimator’s expectation over treatment assignment, conditional on covariates and potential outcomes, recovers the average treatment effect for those units. With estimated propensity scores, the formula is a plug-in estimator; its finite-sample behavior also depends on how the scores were estimated. This tutorial concerns the point estimate, not its variance or confidence interval.

Worked Example: Recovering an ATE Across Two Covariate Groups

Suppose a target population has two covariate groups, \(A\) and \(B\). Group \(A\) makes up 75% of the population and has treatment probability 0.75; group \(B\) makes up 25% and has treatment probability 0.25. Assume conditional exchangeability. The mean potential outcomes are 10 and 4 in group \(A\) under treatment and control, respectively, and 6 and 2 in group \(B\).

Within group \(A\), the expected treated weighted outcome is \(E[DY/e(X)\mid A]=10\), and the expected control weighted outcome is \(E[(1-D)Y/\{1-e(X)\}\mid A]=4\). The weighted contrast is \(10-4=6\). For group \(B\), those two expected weighted outcomes are 6 and 2, giving a contrast of \(6-2=4\).

Average the group contrasts using their shares in the target population:

\tau=0.75(6)+0.25(4)=4.5+1=5.5.

Equivalently, the population treated weighted mean is \(0.75(10)+0.25(6)=9\), and the control weighted mean is \(0.75(4)+0.25(2)=3.5\); their difference is \(9-3.5=5.5\). Thus, the ATE is 5.5 outcome units. This calculation shows why both weighted groups must represent the same covariate distribution.

Worked Example: Calculating the IPW Estimate in a Small Sample

Consider four hypothetical observations. The outcome is measured in points, and the propensity scores are treated as given for this calculation.

Unit	\(D_i\)	\(Y_i\)	\(e_i\)
1	1	12	0.60
2	0	4	0.20
3	1	9	0.30
4	0	6	0.50

For unit 1, the treated contribution is \(12/0.60=20\). Unit 2 is a control, so its contribution is \(-4/(1-0.20)=-5\). Unit 3 contributes \(9/0.30=30\), and unit 4 contributes \(-6/(1-0.50)=-12\). Therefore:

\widehat{\tau}_{\mathrm{IPW}} = \frac{20-5+30-12}{4} = \frac{33}{4} = 8.25\text{ points}.

The estimate says that the IPW-weighted treated outcome average exceeds the IPW-weighted control outcome average by 8.25 points for the target represented by this estimator. It is not a claim that every individual gains 8.25 points. For comparison, the unweighted treated mean is \((12+9)/2=10.5\), the unweighted control mean is \((4+6)/2=5\), and their difference is 5.5 points. The two contrasts differ because IPW changes how much each observation contributes.

What the Weights Do—and What They Cannot Fix

The treated observation’s inverse-probability weight is \(1/e_i\); the control observation’s is \(1/(1-e_i)\). A treated unit with \(e_i=0.2\), for example, receives weight 5, while a control unit with \(e_i=0.8\) receives weight 5. In each case, the weight is large because that observed treatment state was relatively unlikely given the covariates.

Weighting can correct imbalances in observed pre-treatment covariates when the propensity model and identification assumptions are adequate. It cannot eliminate confounding from unmeasured variables, repair a misspecified propensity model automatically, or create evidence about treatment states absent from the data. Positivity is especially important: if a covariate profile almost determines treatment, the inverse probability for the uncommon state becomes very large.

Worked Example: Seeing the Effect of a Large Weight

Suppose four hypothetical observations have the following treatment status, outcome, and propensity score: \((D,Y,e)=(1,8,0.10),(0,3,0.90),(1,5,0.50),(0,4,0.50)\). The first treated observation has weight \(1/0.10=10\); the second, a control with \(e=0.90\), has weight \(1/(1-0.90)=10\).

Their signed contributions are \(8/0.10=80\), \(-3/0.10=-30\), \(5/0.50=10\), and \(-4/0.50=-8\). Hence:

\widehat{\tau}_{\mathrm{IPW}} = \frac{80-30+10-8}{4} = \frac{52}{4} = 13\text{ outcome units}.

The two observations with propensity scores near the extremes account for most of the estimate’s magnitude. The arithmetic is valid, but this small-sample estimate may be highly sensitive to those observations. A large weight is not proof that the estimate is wrong; it is a warning that the data contain limited information for some covariate profiles and that the assumptions and overlap deserve scrutiny.

Common Mistakes and Interpretation

Using the same inverse probability for both groups. The treated weight is \(1/e_i\); the control weight is \(1/(1-e_i)\). The probabilities correspond to the treatment state actually observed.
Assuming weighting guarantees causal identification. The correction relies on consistency, conditional exchangeability, positivity, and suitable propensity probabilities. Weighting observed covariates does not adjust for unmeasured confounding by itself.
Ignoring the target population. The ATE averages \(Y(1)-Y(0)\) over the target population’s covariate distribution. Changing the target population can change the estimand.
Reading a point estimate as an individual effect or a certainty statement. An estimate of 8.25 points is a weighted average contrast, not a gain for every person and not a statement about the probability that the true ATE equals 8.25.

IPW is useful because it expresses causal adjustment as a transparent weighted contrast. Its interpretation remains tied to the target population and assumptions, while its numerical stability depends in part on how close the estimated treatment probabilities are to zero or one. A point estimate should therefore be accompanied in applied work by checks of the propensity model, overlap, and covariate balance.

Check Your Understanding

Use the definitions and estimator above to answer these questions.

Write the IPW contribution for a control observation with outcome \(Y_i\) and propensity score \(e_i\).
Which assumption allows observed treated outcomes, conditional on \(X\), to represent \(Y(1)\) for the target population?
A treated observation has \(Y_i=15\) and \(e_i=0.5\). What is its signed contribution before division by \(n\)?
Why can a propensity score close to zero produce an unstable ATE estimate?
In one sentence, interpret an estimated ATE of 3.2 points without implying that every individual’s treatment effect is 3.2 points.