Conditional Effects as Weighted Population Effects

Why Population Weights Matter

A conditional treatment effect describes the average causal effect for units with a particular value of the pretreatment covariates. A population effect averages those conditional effects across some target population. The target is important: averaging over everyone, over those treated, over those untreated, or over an overlap-weighted population can produce different answers when treatment effects vary with covariates.

This tutorial derives those differences through population weights. The formulas are useful even when conditional effects are not directly observed: they state exactly which people contribute to each estimand and how heavily. They do not, by themselves, establish that an effect can be identified from observed data.

Multiplying \(h\) by any positive finite constant leaves \(\psi_h\) unchanged. Thus, weights specify relative emphasis across covariate values, not an absolute scale. The weighted target population has covariate distribution proportional to \(h(x)\) times the original population distribution of \(X\).

Deriving the Four Population Effects

The average treatment effect (ATE) averages the conditional effect over the full population, so its weight is \(h(x)=1\). The ATT averages over the covariate distribution among treated units; the ATC does the same among controls. To express those distributions in the full population, use \(P(A=1\mid X)=e(X)\) and \(P(A=0\mid X)=1-e(X)\).

For example, the ATT identity follows by averaging \(\tau(X)\) over the covariate distribution of treated units. For any integrable function \(g\), \(E[g(X)\mid A=1]=E[g(X)I(A=1)]/P(A=1)\). Taking the expectation conditional on \(X\) in the numerator gives \(E[g(X)e(X)]\); also \(P(A=1)=E[e(X)]\). Set \(g(X)=\tau(X)\) to obtain the ATT formula. Replacing treated with control gives the ATC formula.

The ATO uses the overlap weight \(e(X)(1-e(X))\). It gives greatest relative weight to covariate profiles with substantial probability of receiving either treatment, and zero weight where the propensity is exactly zero or one. Its target covariate distribution is the original distribution reweighted by \(e(X)(1-e(X))\).

If causal effects are to be identified from observed data, additional assumptions are needed. A common set is consistency (the observed outcome equals the potential outcome under the treatment actually received), conditional exchangeability \(\{Y(1),Y(0)\}\perp A\mid X\), and positivity in the target population. For the ATE, positivity ordinarily requires \(0<e(X)<1\) throughout the population of interest. For ATT identification, treated covariate profiles must have a positive chance of control; for ATC identification, control profiles must have a positive chance of treatment. The ATO targets covariate profiles with \(0<e(X)<1\), so it does not require positivity at profiles it assigns zero weight. These identification conditions do not change the definitions of the estimands above.

A General Weighted Effect and Its Calculation

The same reasoning applies to any nonnegative weight function \(h(X)\) that meets the integrability conditions. Such a function defines a target population whose covariate profiles are more or less represented according to \(h\). This perspective makes explicit that a weighted effect is not an “adjusted effect” with no target: it is an average for a particular reweighted population.

Why the ATE and ATT Can Differ

The ATT and ATE coincide when the propensity score is unrelated to the conditional treatment effect in the population, but treatment-effect homogeneity is not required for coincidence. A useful identity makes the role of their association precise. Provided the expectations exist and \(P(A=1)>0\),

To verify it, expand the covariance: \(\operatorname{Cov}(e(X),\tau(X))=E[e(X)\tau(X)]-E[e(X)]E[\tau(X)]\). Divide by \(E[e(X)]\), then add \(E[\tau(X)]\); the result is \(E[e(X)\tau(X)]/E[e(X)]\), the ATT. Positive association between propensity and effect therefore makes ATT exceed ATE; negative association makes it smaller. This is a statement about how effects and treatment prevalence vary across \(X\), not a claim that the propensity score causes the effect.

Choosing and Diagnosing the Target

The four effects differ in their target covariate distributions, not merely in the algebra used to estimate them. The table summarizes their population weights.

This is a population-level comparison. In an application, investigators may estimate effects through outcome models, propensity-based weighting, matching, or other procedures. Those approaches can target different populations, and their validity depends on design and modeling assumptions. The formulas here specify the estimands; choosing an estimation method does not automatically make its target equal to the ATE, ATT, ATC, or ATO.

Several checks help reveal a mismatch between the intended target and the effect being reported:

• State the target population before interpreting the estimate. A result for treated units does not automatically describe everyone.
• Check treatment prevalence and the denominator for the chosen estimand. A zero denominator means that target population is not represented.
• For identification, examine whether both treatment groups occur in the covariate profiles relevant to the target. Sparse representation can make estimates unstable even when a formal positivity condition holds.
• Check whether the conditional effect varies across important covariates. If it does, different population weights can yield substantively different averages.
• Check that the effect has a finite expectation under the target weights. A finite normalizing denominator cannot repair an undefined or infinite numerator.

A common mistake is to say, “The ATT is the ATE among treated people,” and then calculate the unweighted average of conditional effects across the overall population. That uses weight 1 and gives the ATE. The correction is to average using the treated covariate distribution, equivalently weighting the full population by \(e(X)\) and dividing by \(E[e(X)]\). Another mistake is to assume that a well-defined weighted estimand is automatically identified from observed outcomes. The definition concerns potential outcomes and the target distribution; identification additionally requires assumptions such as consistency, conditional exchangeability, and suitable positivity.