Use Propensity Weights to Create Comparable Groups

Why Weight the Exposure Groups?

In an observational study, people who receive an exposure or treatment may differ from those who do not. If those differences include causes of the outcome, comparing the groups’ observed outcome means can mix the exposure effect with pre-existing differences. Propensity weighting uses the estimated probability of exposure, given measured covariates, to reweight the groups so their covariate distributions represent a specified target population.

The target matters. For an average treatment effect, the comparison aims to represent the full eligible population. For an average treatment effect among the treated, or ATT, it aims to represent the people who actually received the exposure. These targets use different weights, particularly for the unexposed group.

Definition: Let \(A\) be a binary exposure, with \(A=1\) for exposed and \(A=0\) for unexposed. Let \(X\) denote measured covariates recorded before exposure, and let \(Y\) be the outcome. The propensity score is \(e(X)=P(A=1\mid X)\), the probability of exposure conditional on those covariates. An inverse-probability weight gives more representation to observations whose exposure status is less common for their covariate profile.

The propensity score is typically estimated from an exposure-assignment model, as in the earlier material on estimating propensity scores. In the calculations below, take the estimated scores as given. The weights do not change anyone’s recorded exposure or outcome; they change each observation’s contribution to a group summary.

What the Weights Target

Write \(Y(1)\) and \(Y(0)\) for the outcomes a person would have under exposure and no exposure, respectively. The population average treatment effect is \(E[Y(1)-Y(0)]\). The ATT is \(E[Y(1)-Y(0)\mid A=1]\). These are causal estimands, not simply differences between observed group means.

A causal interpretation of propensity weighting requires appropriate identification conditions. Consistency means that a person’s observed outcome equals the potential outcome under the exposure actually received; the exposure must be sufficiently well-defined for that statement to make sense. No interference means one person’s exposure does not change another person’s outcome. Conditional exchangeability requires that, given \(X\), exposure assignment is independent of the potential outcomes. Positivity requires \(0<e(X)<1\) for covariate profiles in the target population. In practice, the propensity model must also estimate the relevant exposure probabilities adequately. Weighting measured covariates cannot, by itself, remove confounding by unmeasured variables.

Definition: For an ATE target, the unstabilized inverse-probability weights are \(1/e(X)\) for exposed observations and \(1/[1-e(X)]\) for unexposed observations. For an ATT target, the weights are \(1\) for exposed observations and \(e(X)/[1-e(X)]\) for unexposed observations. These formulas apply only to the observed exposure group: each person receives the weight corresponding to their actual \(A\).

Under the true propensity score, the ATE weights make each exposure group represent the covariate distribution of the full target population; with estimated propensity scores, they aim to approximate that balance in a finite sample. Exposed people with a low chance of exposure receive large weights; unexposed people with a high chance of exposure receive large weights. The ATT weights use the treated group itself as the target: exposed people retain weight 1, while unexposed people are reweighted to represent the treated group’s covariate distribution.

Weighted Outcome Means and Causal Contrasts

A common finite-sample calculation is a normalized, or ratio, weighted mean. The normalization divides the weighted outcome total by the sum of the weights in that group. For the ATE, calculate one weighted mean among exposed people and another among unexposed people, using their respective ATE weights. Their difference estimates the ATE. For the ATT, the exposed mean is the ordinary mean among exposed people, and the unexposed mean uses the ATT weights; their difference estimates the ATT.

\widehat{\mu}_{1,\mathrm{ATE}} = \frac{\sum_{i=1}^{n} A_iY_i/e(X_i)} {\sum_{i=1}^{n} A_i/e(X_i)}, \qquad \widehat{\mu}_{0,\mathrm{ATE}} = \frac{\sum_{i=1}^{n} (1-A_i)Y_i/[1-e(X_i)]} {\sum_{i=1}^{n} (1-A_i)/[1-e(X_i)]}

\widehat{\mathrm{ATE}}=\widehat{\mu}_{1,\mathrm{ATE}}-\widehat{\mu}_{0,\mathrm{ATE}}, \qquad \widehat{\mathrm{ATT}} = \frac{\sum_{i=1}^{n} A_iY_i}{\sum_{i=1}^{n} A_i} - \frac{\sum_{i=1}^{n}(1-A_i)\{e(X_i)/[1-e(X_i)]\}Y_i} {\sum_{i=1}^{n}(1-A_i)e(X_i)/[1-e(X_i)]}

The first two formulas use the ATE weights; the final formula uses the ATT weights. These normalized estimators are sometimes called Hájek estimators. Other inverse-probability estimators use weighted totals without dividing by the observed sum of weights. Do not mix a weighted-total formula with a normalized-mean calculation: state which estimator is being used and calculate its numerator and denominator consistently.

Why the Reweighting Works

The key idea is that weighting changes the covariate distribution represented by each group. For the ATE, the covariate distribution among exposed people is proportional to their original distribution times \(1/e(X)\). By Bayes’ rule, the original distribution among exposed people is proportional to \(e(X)\) times the population distribution of \(X\). Multiplying by \(1/e(X)\) therefore leaves a distribution proportional to the population distribution. The same argument applies to unexposed people using \(1/[1-e(X)]\).

f(x\mid A=1)\frac{1}{e(x)} \ \propto\ f(x)e(x)\frac{1}{e(x)} \ =\ f(x), \qquad f(x\mid A=0)\frac{1}{1-e(x)} \ \propto\ f(x)[1-e(x)]\frac{1}{1-e(x)} \ =\ f(x)

Here \(f(x)\) denotes the population distribution of the covariates. The proportionality constants disappear when each group’s weighted distribution is normalized. Thus, both weighted groups target the full population’s covariate distribution. For the ATT, Bayes’ rule instead shows that the unexposed covariate distribution multiplied by the odds \(e(X)/[1-e(X)]\) is proportional to the exposed covariate distribution. The exposed group already represents that target, so its ATT weight can remain 1.

Worked Calculations

Worked Example: ATE Weights Across Two Covariate Profiles

Suppose a hypothetical sample has two covariate profiles, low and high risk, with estimated propensity scores \(0.25\) and \(0.75\), respectively. The sample has 2 exposed and 6 unexposed people with the low-risk profile, and 6 exposed and 2 unexposed people with the high-risk profile. Their hypothetical outcome means are as follows.

Profile	Estimated propensity score	Exposed count	Exposed outcome mean	Unexposed count	Unexposed outcome mean
Low risk	0.25	2	11	6	8
High risk	0.75	6	15	2	12

For an exposed person with the low-risk profile, the ATE weight is \(1/0.25=4\); for an exposed person with the high-risk profile, it is \(1/0.75=1.333\). The total exposed weight is \(2(4)+6(4/3)=8+8=16\). The weighted exposed outcome mean is

\widehat{\mu}_{1,\mathrm{ATE}} = \frac{2(4)(11)+6(4/3)(15)}{16} = \frac{88+120}{16} = 13.0

For an unexposed person with the low-risk profile, the weight is \(1/(1-0.25)=1.333\); for an unexposed person with the high-risk profile, it is \(1/(1-0.75)=4\). The total unexposed weight is \(6(4/3)+2(4)=8+8=16\), and the weighted mean is

\widehat{\mu}_{0,\mathrm{ATE}} = \frac{6(4/3)(8)+2(4)(12)}{16} = \frac{64+96}{16} = 10.0

The estimated ATE is \(13.0-10.0=3.0\) outcome units. In each weighted group, the low-risk profile contributes weight 8 and the high-risk profile contributes weight 8. Thus each group represents a weighted population that is 50% low risk and 50% high risk, matching the full sample’s profile distribution. A causal reading still depends on the stated identification assumptions.

Worked Example: ATT Weights and the Treated Target

In a hypothetical sample, 8 exposed people have outcomes summing to 116 points. Four unexposed people have a propensity score of \(2/3\) each, and their outcomes are 8, 10, 11, and 14 points. Calculate the normalized ATT contrast.

Each exposed person receives weight 1. Their weighted mean is \(116/8=14.5\) points. Each unexposed person receives the ATT weight

\frac{e(X)}{1-e(X)} = \frac{2/3}{1-2/3} = 2

The four unexposed observations therefore have total weight \(4(2)=8\), and their weighted outcome total is \(2(8+10+11+14)=2(43)=86\). Their normalized weighted mean is \(86/8=10.75\) points. The estimated ATT is therefore

14.5-10.75=3.75\text{ points}

The calculation targets the treated population: the treated mean represents the treated people directly, while the reweighted unexposed mean uses the ATT weights to represent their covariate distribution. The 3.75-point contrast is an estimate, not proof that exposure caused the difference; its causal interpretation requires the identification assumptions.

Worked Example: Calculate Both Types of Weight for Individual Records

Consider four hypothetical individuals. Two are exposed and two are unexposed. Their estimated propensity scores and outcomes are shown below. For this small illustration, the scores need not reproduce the sample exposure proportion within each profile; estimated probabilities describe exposure chances, not a rule that every finite sample must follow them exactly.

Individual	Exposure \(A\)	Propensity score \(e(X)\)	Outcome \(Y\)
1	1	0.25	10
2	1	0.75	16
3	0	0.25	8
4	0	0.75	12

For the ATE, the exposed weights are \(1/0.25=4\) and \(1/0.75=4/3\). Their weighted mean is \([4(10)+(4/3)(16)]/[4+4/3]=(184/3)/(16/3)=11.5\). The unexposed weights are \(1/(1-0.25)=4/3\) and \(1/(1-0.75)=4\). Their weighted mean is \([(4/3)(8)+4(12)]/[4/3+4]=(176/3)/(16/3)=11.0\). The ATE contrast is \(11.5-11.0=0.5\) outcome units.

For the ATT, the exposed people each have weight 1, so their mean is \((10+16)/2=13\). The unexposed weights are \(0.25/0.75=1/3\) and \(0.75/0.25=3\). Their weighted mean is \([(1/3)(8)+3(12)]/(1/3+3)=(116/3)/(10/3)=11.6\). The ATT contrast is \(13-11.6=1.4\) outcome units. The two answers differ because the estimands target different populations and use different weights.

Practical Interpretation and a Common Pitfall

Weights can be very large when a propensity score is near 0 for an exposed person or near 1 for an unexposed person. Such observations represent covariate profiles that are uncommon in their observed exposure group. A few large weights can dominate a weighted mean, making it unstable and causing the effective information in the weighted sample to be much smaller than the raw sample size suggests. Positivity is therefore not merely a theoretical condition: poor overlap can make the intended comparison unreliable.

Stabilized weights are sometimes used to change the overall scale of the weights while retaining the same within-group weighted means. For example, multiplying all exposed ATE weights by the same positive constant multiplies both the weighted outcome total and the sum of weights by that constant, so their ratio is unchanged. This rescaling does not solve poor overlap or correct a misspecified propensity model.

Common mistake: using the ATE control weight for the ATT. The reasoning “all unexposed people get \(1/[1-e(X)]\)” applies to the ATE weighting scheme, not the ATT scheme. For the ATT, exposed people have weight 1 and unexposed people have weight \(e(X)/[1-e(X)]\), because the target is the treated population.
Common mistake: treating a weighted contrast as automatically causal. Weighting adjusts only for the covariates represented in the propensity model and only under suitable identification conditions. A weighted difference alone does not establish exchangeability, positivity, or correct model specification.
Common mistake: interpreting a weight as a person’s probability of treatment. The propensity score is a probability. The weight is a function of that probability, assigned according to the person’s observed exposure status.

Key takeaway: Choose the target population first. Use \(1/e(X)\) for exposed and \(1/[1-e(X)]\) for unexposed observations to target the ATE. Use weight 1 for exposed and \(e(X)/[1-e(X)]\) for unexposed observations to target the ATT. In both cases, interpret the weighted contrast in light of identification assumptions and the quality of the propensity scores.

Check Your Understanding

Answer each question by identifying the target and showing the relevant weight or calculation.

For an unexposed person with \(e(X)=0.60\), calculate the ATE weight and the ATT weight.
In one sentence, describe the covariate distribution that each group represents after ATE weighting.
Why does the ATT scheme leave exposed individuals with weight 1?
A weighted contrast is large, but an important confounder was not measured. Explain why the contrast need not be a valid causal estimate.
What practical concern arises when an exposed individual has an estimated propensity score close to zero?