The Overlap Population as an Estimand

Why Define an Overlap Population?

An average treatment effect is always an average over some population. In observational data, however, treatment assignment may be nearly deterministic for some covariate profiles: people with those profiles are almost always treated or almost always untreated. Comparing treatment outcomes for such profiles requires information that may be scarce or absent in the other group. The overlap-population estimand makes the target explicit: it emphasizes profiles for which both treatment assignments are plausible.

The introductory material on the average treatment effect in the overlap population defines the ATO. Here the focus is its interpretation through treatment-assignment probabilities and the way those probabilities determine the target population. The overlap population is not simply the set of people who happened to receive either treatment, nor is it automatically a subgroup defined by a fixed propensity-score cutoff.

Definition: Let $A\in\{0,1\}$ denote treatment, $X$ the pretreatment covariates, and $e(X)=P(A=1\mid X)$ the propensity score. The overlap population has covariate distribution proportional to $e(X)\{1-e(X)\}$ times the original covariate distribution. Its average treatment effect, often called the average treatment effect in the overlap population (ATO), is $$ \tau_{\mathrm{ATO}} = \frac{E\left[e(X)\{1-e(X)\}\{Y(1)-Y(0)\}\right]} {E\left[e(X)\{1-e(X)\}\right]}. $$ Here $Y(1)$ and $Y(0)$ are the potential outcomes under treatment and control.

The factor $e(X)\{1-e(X)\}$ is largest when the treatment probability is one-half and approaches zero as that probability approaches either zero or one. It therefore gives more target-population weight to covariate profiles where treated and untreated people can both be found. It does not say that every person with a propensity score near one-half has the same importance: the original prevalence of that profile also matters.

From Assignment Probabilities to the Target

For precision, write $f_X(x)$ for the original population distribution of $X$, and define $\kappa=E[e(X)\{1-e(X)\}]$. If $\kappa>0$, the overlap-population covariate density is

f_{\mathrm{ATO}}(x) = \frac{e(x)\{1-e(x)\}f_X(x)}{\kappa}.

This is a reweighted population distribution, not a claim that the original population has literally been replaced. The factor $f_X(x)$ retains how common a covariate profile is in the population; the additional factor $e(x)\{1-e(x)\}$ tilts the target toward profiles with better treatment-group overlap. The ATO is the mean of the conditional treatment effect $E[Y(1)-Y(0)\mid X=x]$ under this target distribution.

A useful way to see why the factor arises is to consider the weighted covariate distributions among treated and control units. In the population, treated units with covariates near $x$ occur in proportion to $e(x)f_X(x)$. Weighting those units by $1-e(x)$ makes their covariate mass proportional to $e(x)\{1-e(x)\}f_X(x)$. Control units occur in proportion to $\{1-e(x)\}f_X(x)$; weighting them by $e(x)$ produces that same mass. Thus these two choices of weights target the same covariate distribution.

Key identity: Overlap weights are $1-e(X)$ for treated units and $e(X)$ for control units. After weighting, each group has covariate mass proportional to $e(X)\{1-e(X)\}f_X(X)$. The equality concerns the target covariate distribution; it does not by itself guarantee that the observed groups are causally comparable.

Assumptions for a Causal Interpretation

An estimand can be defined without assumptions about how treatment was assigned. To identify the ATO from observed outcomes, however, the observed treatment groups must provide information about the relevant potential outcomes. Let $Y=AY(1)+\{1-A\}Y(0)$. The standard conditions are:

Consistency: A person's observed outcome equals the potential outcome under the treatment actually received, as expressed by $Y=AY(1)+\{1-A\}Y(0)$.
No interference: One person's treatment does not change another person's potential outcome, and the treatment versions are sufficiently well defined for the potential outcomes to be meaningful.
Conditional exchangeability: $\{Y(1),Y(0)\}$ is independent of $A$ given $X$. In words, after conditioning on the measured pretreatment covariates, treatment assignment carries no further information about the potential outcomes.
Relevant positivity: There is positive probability of treatment and control for covariate profiles that receive positive weight in the overlap target. In particular, $0<e(X)<1$ wherever $e(X)\{1-e(X)\}f_X(X)$ contributes positive mass.

Under these conditions, the conditional mean outcome among treated units with covariates $X=x$ identifies $E[Y(1)\mid X=x]$, and the corresponding control mean identifies $E[Y(0)\mid X=x]$. The ATO then averages their difference under $f_{\mathrm{ATO}}(x)$. Profiles with $e(x)=0$ or $e(x)=1$ receive zero overlap weight; the ATO does not require estimating their conditional treatment effects. If $\kappa=0$, there is no population mass with treatment-assignment uncertainty and this normalized estimand is undefined.

Deriving the Overlap-Weighted Outcome Contrast

The connection between the target definition and weighted observed outcomes follows directly from conditioning on $X$. For the treated group, the expected weighted outcome has the form

E[A\{1-e(X)\}Y] = E\left[e(X)\{1-e(X)\}E\{Y(1)\mid X\}\right].

To justify this equality, condition on $X$. Consistency makes $Y=Y(1)$ when $A=1$, and conditional exchangeability gives $E[Y(1)\mid A=1,X]=E[Y(1)\mid X]$. Also $E[A\mid X]=e(X)$. Therefore the conditional expectation of the left-hand integrand is $e(X)\{1-e(X)\}E[Y(1)\mid X]$. For controls, the same reasoning gives

E[\{1-A\}e(X)Y] = E\left[e(X)\{1-e(X)\}E\{Y(0)\mid X\}\right].

Both weighted groups have total expected weight $\kappa$, since $E[A\{1-e(X)\}]=E[\{1-A\}e(X)]=\kappa$. Subtracting their normalized outcome means therefore gives

\frac{E[A\{1-e(X)\}Y]}{\kappa} - \frac{E[\{1-A\}e(X)Y]}{\kappa} = \frac{E[e(X)\{1-e(X)\}\{Y(1)-Y(0)\}]}{\kappa} = \tau_{\mathrm{ATO}}.

This identity motivates overlap weighting as an estimator, but it does not establish the assumptions from the data. In particular, weighting cannot remove confounding by an unmeasured variable, and estimated propensity scores must adequately represent treatment assignment conditional on the measured covariates.

Worked Example: Computing the Target Across Two Profiles

Consider a hypothetical population with two covariate profiles. Profile L occurs in 60% of the population, has treatment probability $e(L)=0.10$, and has conditional average treatment effect 2 outcome units. Profile H occurs in 40%, has $e(H)=0.80$, and has conditional average treatment effect 10 outcome units. Assume the conditions needed for causal identification hold.

The unnormalized overlap mass for each profile is its population proportion times $e(1-e)$. For L this is $0.60(0.10)(0.90)=0.0540$. For H it is $0.40(0.80)(0.20)=0.0640$. The total is $\kappa=0.0540+0.0640=0.1180$, so the normalized target proportions are

P_{\mathrm{ATO}}(L)=\frac{0.0540}{0.1180}=0.4576, \qquad P_{\mathrm{ATO}}(H)=\frac{0.0640}{0.1180}=0.5424.

The ATO is consequently $0.4576(2)+0.5424(10)=6.339$ outcome units, equivalently $\{0.0540(2)+0.0640(10)\}/0.1180=6.339$. For comparison, the population ATE is $0.60(2)+0.40(10)=5.200$ outcome units. The difference arises because the overlap target gives profile H somewhat more weight than its 40% share of the original population. This target is defined by both profile prevalence and treatment-assignment probabilities, not by effect size.

Worked Example: Showing Why Both Weighted Groups Target the Same Profiles

In a second hypothetical population, two profiles each make up half the population. Their treatment probabilities are 0.20 and 0.80. Suppose the potential-outcome means are $E[Y(0)\mid X]=3$ and $E[Y(1)\mid X]=5$ for the first profile, and 8 and 14 for the second. Assume consistency, no interference, conditional exchangeability, and the stated treatment probabilities.

Each profile has overlap factor $e(1-e)=0.20(0.80)=0.16$. Multiplying by its 0.50 population share gives unnormalized mass 0.080 for each, so the overlap target assigns probability 0.50 to each profile. The ATO is therefore $0.50(5-3)+0.50(14-8)=4.000$ outcome units.

The weighted treated outcome contributions are proportional, profile by profile, to $0.50(0.20)(0.80)(5)=0.400$ and $0.50(0.80)(0.20)(14)=1.120$. Dividing their sum by the treated total weight $0.080+0.080=0.160$ gives a weighted treated mean of $1.520/0.160=9.500$. For controls, the corresponding contributions are $0.080(3)=0.240$ and $0.080(8)=0.640$, yielding $0.880/0.160=5.500$. Their difference is $9.500-5.500=4.000$ outcome units, matching the ATO. The calculation uses conditional outcome means; an actual finite-sample estimate will also reflect sampling variation.

Worked Example: A Large Effect Can Receive Little Target Weight

Suppose a hypothetical population has a profile that occurs in 20% of the population, has $e=0.02$, and has a conditional treatment effect of 20 outcome units. The remaining 80% has $e=0.50$ and an effect of 4 units. Assume the causal identification conditions hold.

The first profile's unnormalized overlap mass is $0.20(0.02)(0.98)=0.00392$. The second profile's mass is $0.80(0.50)(0.50)=0.2000$, giving $\kappa=0.20392$. The target shares are $0.00392/0.20392=0.01922$ and $0.2000/0.20392=0.9808$, respectively. Thus

\tau_{\mathrm{ATO}} = \frac{0.00392(20)+0.2000(4)}{0.20392} = 4.308 \text{ outcome units}.

The rare, high-effect profile contributes little because treatment is very uncommon there, not because its effect is small. This is a feature of the target definition: the ATO describes effects where the data contain comparatively good treatment-group overlap, rather than maximizing or preserving the influence of every high-effect profile.

How the ATO Differs From Other Average Effects

The choice among these estimands changes whose effects are averaged. In the table, each target's covariate density is proportional to the original density multiplied by the displayed factor; each factor is normalized by its own expectation.

Estimand	Target-weight factor	Profiles emphasized
ATE	1	The original population
ATT	e(X)	Profiles represented among treated units
ATC	1-e(X)	Profiles represented among control units
ATO	e(X){1-e(X)}	Profiles with treatment and control both plausible

The ATO is not a compromise that must lie between the ATT and ATC, nor does it equal the ATE unless the corresponding weighted averages of conditional effects happen to agree. As with any change of target population, treatment-effect heterogeneity can make the numerical estimands differ. The ATO answers a specific question: what is the average effect in a population tilted toward covariate profiles with treatment-assignment overlap?

Diagnostics, Limitations, and a Common Mistake

In an analysis, inspect the estimated propensity-score distributions by treatment group and assess whether covariates are balanced after applying the overlap weights. Also examine how much of the original population contributes to the target and whether the fitted assignment model produces credible probabilities. Overlap weights are bounded between zero and one for each unit, so they avoid the very large inverse-probability weights associated with propensity scores near zero or one. Bounded weights do not guarantee adequate information: an overlap target can still be concentrated in a small set of profiles, and the analysis can still be sensitive to sparse data, model misspecification, or unmeasured confounding.

A common mistake is to say, “The ATO is the effect for people whose propensity score is close to 0.5.” That is not its exact definition. It is an average under a continuously reweighted covariate distribution, with weight $e(X)\{1-e(X)\}$ multiplied by the original prevalence of each profile. A cutoff-defined subgroup would be a different target. Another mistake is to interpret strong overlap weighting as fixing confounding: causal interpretation still depends on the identification assumptions, especially conditional exchangeability given the measured covariates.

Key takeaway: The overlap population is defined by the original covariate distribution tilted by $e(X)\{1-e(X)\}$. Its ATO emphasizes profiles where both treatment assignments are plausible; it does not automatically describe the original population or any group selected by an arbitrary propensity-score cutoff.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

If $e(x)=0.30$, what is the unnormalized overlap factor at that profile? How does it compare with the factor at $e(x)=0.50$?
A profile has population prevalence 0.25, treatment probability 0.40, and conditional treatment effect 3 units. What is its unnormalized contribution to the ATO denominator and numerator?
Does a high conditional treatment effect guarantee that a profile receives high weight in the ATO? Explain.
State the conditional exchangeability condition and explain why overlap weighting does not make it unnecessary.
Why is the overlap population not generally the same thing as the set of people with estimated propensity scores inside a chosen interval?

Solutions. (1) The factor is $0.30(0.70)=0.2100$; at 0.50 it is $0.50(0.50)=0.2500$, its maximum. (2) The overlap mass is $0.25(0.40)(0.60)=0.0600$. The numerator contribution is $0.0600(3)=0.1800$ outcome units. (3) No. Weight depends on both profile prevalence and $e(x)\{1-e(x)\}$, not on the treatment effect. (4) Conditional exchangeability is $\{Y(1),Y(0)\}$ independent of $A$ given $X$. Weighting changes the target distribution; it does not control unmeasured differences in potential outcomes within covariate profiles. (5) The ATO uses continuous weights proportional to $e(x)\{1-e(x)\}f_X(x)$; an interval rule instead defines a cutoff-selected subgroup and generally induces a different target.