Balancing Scores and Support

Why Balancing and Support Are Different Questions

Propensity scores are useful because they summarize treatment assignment using the measured covariates. The key result is stronger than saying that treated and control groups have similar average scores: at a given value of the true propensity score, their covariate distributions are equal in the population. But this result has a boundary. It does not supply observations from a treatment group that never occurs at a particular covariate profile, and it does not make a causal comparison possible at a score value represented in only one group.

Let \(A\) be a binary treatment, with \(A=1\) denoting treatment and \(A=0\) control, and let \(X\) be the measured pre-treatment covariates. The propensity score is the conditional probability of treatment given those covariates. It is a property of the treatment assignment process in the target population, not merely the output of a fitted model.

The balancing-score result concerns covariate distributions in the population. It does not state that each treated person has a control person with identical covariates, nor that a finite sample will display perfect balance after matching or weighting. The phrase “at a given score” also matters: it refers to score values for which both treatment groups can occur. The positivity material established that a treatment contrast requires both treatment options to have positive probability in the covariate region relevant to that contrast.

Why the True Score Balances Covariates

Write \(S=e(X)\). Because \(S\) is determined by \(X\), once \(X=x\) is known, knowing \(S\) adds no information about treatment assignment. At any score value \(s\), all covariate profiles with \(e(x)=s\) have the same treatment probability. Consequently, learning whether a unit with score \(s\) was treated does not change the relative representation of those profiles.

The argument can be written directly. For covariate values \(x\) satisfying \(e(x)=s\),

Also, averaging the treatment probability over covariate values at that score gives

Thus the probability of treatment conditional on both \(X\) and \(S\) equals the probability conditional on \(S\) alone. For binary \(A\), equality of these conditional treatment probabilities implies conditional independence of \(A\) and \(X\) given \(S\). Equivalently, wherever \(0<s<1\), the covariate distributions agree:

for any covariate set \(B\) with well-defined conditional probabilities. In continuous settings, the same statement is expressed using conditional densities or regular conditional distributions. The result is about the whole distribution of \(X\), not just its mean.

This balancing property is a treatment-assignment result; by itself it is not a causal identification result. For causal interpretation, one also needs consistency and conditional exchangeability, \(Y(a)\perp A\mid X\), for the potential outcomes \(Y(a)\), together with positivity for the target population. Under these assumptions, the covariate balance induced by the true score allows a treatment comparison conditional on score to represent a comparison not confounded by the measured \(X\). Without exchangeability, balancing measured covariates does not eliminate unmeasured confounding.

Why Balance Cannot Create Missing Support

At a covariate profile where \(e(X)=0\), treatment never occurs; where \(e(X)=1\), control never occurs. The corresponding potential outcome for the absent treatment group is not observed at that profile. Conditioning, matching, or reweighting can reorganize observed units, but none can manufacture observations from a treatment condition that has probability zero.

The theorem itself signals this limitation. At \(S=0\), the conditional distribution among treated units is undefined because there are no treated units in that score stratum. At \(S=1\), the conditional distribution among controls is undefined. The equality of treated and control covariate distributions is therefore an informative comparison only for score values with \(0<S<1\) and representation in both groups.

Score support and covariate support are related but are not identical concepts. Shared score values are necessary for comparisons made by score, but a score alone is a one-dimensional summary of \(X\). In a finite sample, a shared range of scores does not guarantee that every important covariate profile has adequate observations in both groups. Conversely, the population theorem says that at an exact true score value, the distributions of \(X\) are equal across groups; it does not guarantee that a finite sample contains enough observations to estimate that distribution reliably.

When the Score Is Estimated or the Sample Is Finite

In applications, \(e(X)\) is usually unknown and estimated. The balancing theorem is exact for the true propensity score, not automatically for an estimated score. A fitted score can fail to balance covariates if its model is misspecified, if relevant covariates are omitted, or if estimation is unstable. Even with a well-specified model, random variation means that matching or weighting in a finite sample will not generally yield exact balance.

Diagnostics, Alternatives, and Limits

A useful diagnostic question is not only “Do the score distributions overlap?” but also “Do the covariates that matter for treatment assignment and outcomes balance in the analysis sample?” Check balance after the proposed score-based procedure, using covariate summaries suited to the variables and the design. Also check that both treatment groups contribute observations in the covariate regions needed for the stated target. A good balance summary cannot compensate for an absent treatment group, and a shared score range cannot by itself establish adequate support for every covariate pattern.

Score conditioning is one design strategy among several. Direct adjustment for the full covariate set avoids compressing \(X\) to a single score but can become difficult when covariates are numerous or their relationships are complex. Score matching, subclassification, and weighting use the score to construct comparisons or change relative contributions; each still relies on appropriate covariate measurement, a suitable score model when estimated, and support for the target contrast. Outcome modeling can produce predictions in unsupported regions, but those predictions rely on extrapolation rather than observed treatment comparisons there. No choice among these methods removes a structural positivity violation.

The central limitation is therefore about what the data can compare. Balancing scores provide a principled way to balance measured covariates where treatment and control observations coexist. They do not guarantee balance in a realized sample, solve unmeasured confounding, or identify effects for covariate regions in which one treatment is absent. Interpret any score-based effect in light of both the balance actually achieved and the population for which both treatment options are represented.