Propensity scores and overlap

Let e(L) equal P(A=1|L). Condition on a score value t. Within that score level, every baseline configuration has the same treatment probability t. Multiplying its conditional probability by t and dividing by the score-level treatment probability leaves its distribution unchanged. The same argument applies to control using one minus t. Thus the distribution of L is identical across treatment arms conditional on the true score, wherever both arms are possible. The theorem is about distributions, not exact pairwise equality of observed covariate vectors.

If treatment is exchangeable given the full baseline covariates, iterated conditioning transfers the required potential-outcome independence to the true score. Consistency and positivity are still needed to identify an effect. A score fitted with a misspecified model can fail to balance even the measured covariates, so balance diagnostics are useful after matching, weighting, or stratification. Those diagnostics assess observed features of L; they cannot establish balance of an unrecorded common cause or prove a causal graph correct.

For an overall ATE, each covariate region with target probability needs both treatment alternatives. A structural zero or one score means the observed law does not provide one needed outcome regression there. Near-zero or near-one scores preserve formal positivity but can produce unstable comparisons and extreme inverse weights. Strong overlap, bounding scores away from both endpoints, is a useful sufficient condition for many regular inference results. More observations can help with sparse regions, but cannot change a structural impossibility built into treatment assignment.

A treatment prediction model with near-perfect classification may therefore be inconvenient for ATE estimation: it can indicate that comparable alternatives rarely coexist. Prediction accuracy is not the causal design objective. Investigate whether a score reflects genuine assignment restrictions, overly flexible fitting, or a variable set that unnecessarily worsens overlap. Selection of covariates must still respect the identification argument; deleting an important confounder just to improve apparent overlap introduces another problem. Diagnostics illuminate an assumption rather than choosing it automatically.

Discarding units with extreme scores restricts the population to an overlap region. The resulting effect generally differs from the full-population ATE when effects vary. One should define the retained target and report how it differs from the original population rather than describing trimming as a purely numerical correction. Overlap weights, matching weights, and ATT weights also change the emphasis placed on covariate regions, and need their own estimand interpretation.

For ATT, the treated distribution supplies the target. Its important comparison requirement is that untreated units be available at covariate values occurring among the treated. A region with no treated units does not contribute directly to that target, although it obstructs an overall ATE if it has positive population mass. These support distinctions explain why an effect may be identified for one population and not another. A balancing score is a tool for constructing comparable groups once the target and assumptions have been chosen.