Why Derive the Target Population?
A weighted contrast is easiest to interpret once we know whose treatment effect it averages. Overlap weights are designed to emphasize covariate profiles for which either treatment assignment was plausible. Deriving their target makes this interpretation precise: the target is not generally the full study population, nor simply the treated or untreated group.
Building on the introductory material on Overlap Weights for Binary Treatment, this tutorial derives the target covariate distribution and shows why the corresponding weighted contrast equals an average treatment effect in that population. The target is shaped by both the original covariate distribution and the propensity score.
The factor \(e(x)\{1-e(x)\}\) is largest at \(e(x)=0.5\) and approaches zero as the propensity score approaches either 0 or 1. Thus, the target distribution gives greater relative representation to covariate profiles with more treatment equipoise. It still includes the source density \(f_X(x)\): a profile cannot receive substantial target mass if it was rare in the source population, even if its propensity score is near one half.
Assumptions and the Target-Density Derivation
The target density is defined mathematically from the source population and its true propensity score. Its causal interpretation requires assumptions. We use consistency, so a person's observed outcome equals the potential outcome for the treatment actually received. We also require conditional exchangeability, \(Y(a)\perp A\mid X\) for \(a=0,1\), on the covariate profiles included in the overlap target. This says that, conditional on the measured baseline covariates, treatment assignment carries no further information about either potential outcome.
Finally, \(C>0\) is required so the target density can be normalized. Wherever \(f_{\mathrm{OW}}(x)>0\), both treatment probabilities must be positive: \(0<e(x)<1\). Profiles with \(e(x)=0\) or \(e(x)=1\) receive zero target density. Exchangeability and consistency are not guaranteed by weighting; they are substantive assumptions that must be defended using study design and subject-matter knowledge.
The treated overlap weight is \(1-e(X)\), and the control overlap weight is \(e(X)\). To see why these weights produce the same covariate target, consider the weighted covariate density within each treatment group. Before normalization, the treated density is proportional to \(\{1-e(x)\}P(A=1\mid X=x)f_X(x)\). Since \(P(A=1\mid X=x)=e(x)\), this becomes \(e(x)\{1-e(x)\}f_X(x)\). For controls it is proportional to \(e(x)P(A=0\mid X=x)f_X(x)\), which is the same expression because \(P(A=0\mid X=x)=1-e(x)\). Both densities normalize by \(C\), giving \(f_{\mathrm{OW}}(x)\).
This is the key justification: after weighting, the treated and control groups have the same covariate distribution. The weighted outcome contrast therefore compares their outcome means over that shared target, rather than comparing means over different covariate distributions.
From the Weighted Contrast to the Causal Effect
In the population, the normalized weighted treated outcome mean is \(E[(1-e(X))AY]/E[(1-e(X))A]\). Conditional on \(X=x\), the denominator contribution is \(\{1-e(x)\}e(x)\), while the numerator contribution is \(\{1-e(x)\}e(x)E[Y\mid A=1,X=x]\). Under consistency and conditional exchangeability, \(E[Y\mid A=1,X=x]=\mu_1(x)\). The corresponding control mean uses weight \(e(X)\) and equals the target average of \(\mu_0(x)\). Subtracting the means gives the following result.
This is a population identity using the true propensity score. In a sample, the usual normalized estimator replaces expectations with weighted sample averages and uses estimated propensity scores. Its success then also depends on estimating the propensity scores adequately and having enough information in both treatment groups across the target covariate profiles.
Worked Example: Finding the Target in Three Covariate Groups
Suppose a hypothetical population has three covariate groups. Their source-population proportions are \(0.40\), \(0.20\), and \(0.40\); their propensity scores are \(0.10\), \(0.50\), and \(0.90\), respectively. Suppose the conditional treatment effects, measured in outcome units, are \(1\), \(4\), and \(10\). Assume consistency and conditional exchangeability within these groups.
For each group, multiply its source proportion by \(e(1-e)\). The unnormalized target masses are \(0.40(0.10)(0.90)=0.036\), \(0.20(0.50)(0.50)=0.050\), and \(0.40(0.90)(0.10)=0.036\). Thus \(C=0.036+0.050+0.036=0.122\). Dividing by \(C\), the normalized target proportions are \(0.036/0.122=0.2951\), \(0.050/0.122=0.4098\), and \(0.036/0.122=0.2951\).
The overlap-population effect is therefore \(0.2951(1)+0.4098(4)+0.2951(10)=4.885\) outcome units (rounding the displayed proportions). The full-population average treatment effect is \(0.40(1)+0.20(4)+0.40(10)=5.200\) units, so the two targets differ. The overlap target gives relatively more weight to the group with propensity score \(0.50\); it does not discard the other groups because their scores are not exactly one half.
Worked Example: A Continuous Propensity Score
Consider a hypothetical population where \(X\) is uniformly distributed on \([0,1]\), so \(f_X(x)=1\), and the propensity score is \(e(x)=x\). For \(0<x<1\), both treatments have positive probability. Assume consistency and conditional exchangeability on this support.
The normalizing constant is \(C=\int_0^1x(1-x)\,dx=[x^2/2-x^3/3]_0^1=1/6\). Consequently, \(f_{\mathrm{OW}}(x)=6x(1-x)\) for \(0<x<1\). For example, the target probability that \(X\) falls between \(1/4\) and \(3/4\) is \(6\int_{1/4}^{3/4}x(1-x)\,dx=0.6875\), or \(68.75\%\). The source population assigns probability \(0.50\) to that interval, so the overlap target concentrates more mass there.
Suppose the conditional effect is \(\tau(x)=2+4x^2\) outcome units. Its overlap average is \(6\int_0^1(2+4x^2)x(1-x)\,dx\). The constant part contributes \(2\), because \(f_{\mathrm{OW}}\) integrates to one. For the second part, \(E_{\mathrm{OW}}[X^2]=6\int_0^1(x^3-x^4)\,dx=6(1/4-1/5)=0.3000\). Thus \(\tau_{\mathrm{OW}}=2+4(0.3000)=3.200\) outcome units. By comparison, the source-population average is \(\int_0^1(2+4x^2)\,dx=3.333\) units.
Worked Example: Checking the Weighted Outcome Contrast
Consider a hypothetical population split equally between two covariate groups. Their propensity scores are \(0.20\) and \(0.80\). In the first group, the conditional mean outcomes under treatment and control are \(12\) and \(10\); in the second, they are \(18\) and \(14\). Assume consistency and conditional exchangeability.
Each group has unnormalized overlap mass \(0.50(0.20)(0.80)=0.080\), so \(C=0.160\). The target gives each group proportion \(0.080/0.160=0.50\). The weighted treated outcome mean is \(\{0.080(12)+0.080(18)\}/0.160=2.400/0.160=15.00\). The weighted control outcome mean is \(\{0.080(10)+0.080(14)\}/0.160=1.920/0.160=12.00\). Their contrast is \(15.00-12.00=3.000\) outcome units. Directly averaging the two group effects gives \(\{(12-10)+(18-14)\}/2=3.000\) units, confirming the identity.
How This Target Differs From Other Targets
Inverse-probability weighting can target different populations by changing the weights. The distinction is in the target distribution, not simply in the arithmetic used to calculate a weighted mean.
| Target | Covariate density | Interpretation |
|---|---|---|
| ATE | \(f_X(x)\) | Average effect in the source population |
| ATT | \(e(x)f_X(x)/E[e(X)]\) | Average effect among those treated |
| ATU | \(\{1-e(x)\}f_X(x)/E[1-e(X)]\) | Average effect among those untreated |
| Overlap effect | \(e(x)\{1-e(x)\}f_X(x)/C\) | Average effect in the overlap population |
The ATE, ATT, and ATU targets answer different questions and can differ when treatment effects vary with \(X\). Overlap weights do not automatically estimate the full-population ATE merely because both treatment groups contribute. They estimate the overlap effect unless the relevant target distributions or treatment effects make the parameters coincide.
Diagnostics, Failure Modes, and Limits
The target-density derivation relies on the true propensity score, but in practice it is estimated. Inspect propensity-score distributions by treatment group and assess whether the estimated scores distinguish areas with little empirical overlap. Also check whether weighted covariate distributions are similar between groups. Good balance is useful evidence that the weighting has worked on measured covariates; it does not establish exchangeability or rule out unmeasured confounding.
Overlap weights are bounded between zero and one, which avoids the very large individual weights that inverse-probability ATE weights can produce when estimated scores are near zero or one. Bounded weights do not guarantee useful precision: if very few observations lie in the region emphasized by the target, the effective information can still be limited. Report the target clearly, describe the propensity-score and balance diagnostics, and consider uncertainty due to estimating the propensity score.
- Wrong target claim: Calling the overlap effect the ATE without justification changes the population to which the conclusion applies.
- Overstated equipoise: The target weights covariate profiles by \(e(1-e)f_X\); it does not put equal weight on all profiles with moderate scores.
- Unmeasured confounding: Balanced measured covariates do not guarantee that treatment groups are exchangeable with respect to unmeasured causes of the outcome.
- Model problems: A poorly estimated propensity score can yield misleading weights, balance, and target interpretation. Overlap weighting is not a substitute for checking model performance.
- Limited generalizability: The resulting effect describes the overlap target, which may not represent patients or settings with very different treatment propensities.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- What does \(C=E[e(X)\{1-e(X)\}]\) normalize, and why must it be positive?
- In a group with propensity score \(0.20\), what are the overlap weights for treated and control units?
- Does overlap weighting generally estimate the ATE? State the reason.
- Name two assumptions needed to interpret the overlap-weighted contrast causally.
Solutions. (1) \(C\) normalizes the unnormalized covariate density \(e(x)\{1-e(x)\}f_X(x)\); if \(C=0\), no positive-mass overlap target can be formed. (2) A treated unit receives weight \(1-0.20=0.80\), and a control unit receives weight \(0.20\). (3) Generally no: it averages effects under the overlap density, not the source density \(f_X\), unless the targets coincide or treatment effects make the averages equal. (4) Consistency and conditional exchangeability are required; the target must also have positive mass and treatment support wherever its density is positive.