Conditional Independence After Conditioning on the Score

Why Conditioning on a Score Can Be Enough

The propensity score reduces a potentially long list of baseline covariates to one number: the probability of receiving treatment given those covariates. That number is useful for causal comparisons not because it reconstructs a person's covariates, but because treatment groups have the same covariate distribution at a given true score. This tutorial develops the further implication: under exchangeability given the covariates, potential outcomes are also conditionally independent of treatment given the score.

The distinction matters. Two people can have the same propensity score but quite different covariate values. The score need not identify either person, and it need not tell us which covariate profile a person has. It can nevertheless provide a valid conditioning variable for a population comparison, provided the assumptions below hold.

The causal argument uses three assumptions. Consistency means the observed outcome equals the potential outcome under the treatment actually received: \(Y=T Y(1)+(1-T)Y(0)\). The treatment and outcome definitions must be sufficiently well specified for this relation to make sense. Conditional exchangeability given \(X\) means \((Y(0),Y(1))\perp T\mid X\): after conditioning on the measured baseline covariates, treatment assignment carries no further information about either potential outcome. Finally, positivity requires \(0<e(X)<1\) for covariate profiles in the target population. Without both treatment possibilities at a profile, its treated and untreated outcomes cannot both be learned from observed data there.

These assumptions are substantive, not consequences of calculating a propensity score. In particular, a score based on measured covariates does not address unmeasured confounding. The result below explains what follows from exchangeability given \(X\); it does not establish that exchangeability is credible in a particular application.

From Covariate Exchangeability to Score-Level Exchangeability

The balancing property says that \(T\perp X\mid e(X)\). At a score value \(s\), every covariate profile with that score has the same treatment probability \(s\). Consequently, treatment status does not further sort those profiles within the score level. The next derivation shows why this also carries potential outcomes along, even though their distributions may differ across the profiles.

To see the argument, fix a score value \(s\), a treatment value \(t\in\{0,1\}\), and an event \(A\) concerning the potential outcomes, such as \(Y(1)\leq 10\). Given \(X\), exchangeability lets us factor the probability of the potential-outcome event and treatment assignment. Also, \(P(T=t\mid X)=s^t(1-s)^{1-t}\) whenever \(e(X)=s\). Averaging over covariate profiles with that score gives:

The last line is exactly the factorization that defines conditional independence. The key step is that the treatment probability is constant across the covariate profiles being averaged at score \(s\). The potential-outcome distribution itself need not be constant across those profiles; it is averaged over the same profile mixture in both treatment groups.

Consistency then connects this population result to observed outcomes. For \(a\in\{0,1\}\), among units with \(T=a\) and score \(s\), the observed outcome is \(Y(a)\). Since \(Y(a)\perp T\mid e(X)\), its conditional distribution is the same as among all units at that score. In particular, when the relevant means exist:

Thus the difference in observed mean outcomes between treated and untreated units at score \(s\) identifies the conditional average treatment effect at that score, \(E\{Y(1)-Y(0)\mid e(X)=s\}\), under the stated assumptions. This is a population statement about conditioning on the true score. It does not say that every score value will have many observations, or that a finite-sample estimate will be precise.

Worked Causal Comparison Within a Score Level

The implication for potential outcomes can be made concrete. Suppose a hypothetical population at score \(0.40\) has the same profile proportions as above. Assume the following potential-outcome means, measured in symptom-free days during a specified follow-up period. These means may differ across profiles; exchangeability is assumed within each profile, not that outcomes are identical for everyone.

For a target population spanning multiple score values, the average treatment effect can be represented as the average of these score-specific effects over that population, when positivity and the relevant expectations hold. That does not mean a single score-level contrast automatically estimates the overall effect. The distribution of scores in the chosen target population determines how score-specific effects are combined.

When scores are continuous, an exact score value may have probability zero. Conditional expectations are still defined as functions of the score, but a finite data set cannot usually supply a sizeable group at every exact value. Practical methods therefore have to estimate comparisons across nearby scores or otherwise model the relationship. Such approximations can introduce bias; the population theorem is not a guarantee that any particular matching or grouping procedure succeeds.

Diagnostics, Alternatives, and Limits

A true score has the balancing property by definition, but applied analyses use an estimated score and a finite sample. Diagnostics should therefore examine whether measured baseline covariates are actually balanced in the comparisons being made. Check covariate distributions within the score regions or matched sets used for analysis, not just whether the treatment model predicts treatment well. A strong prediction score can still yield poor covariate balance for the intended comparison.

Other warning signs include few or no controls at high scores, few or no treated units at low scores, and large differences in covariate distributions within the regions being compared. These indicate weak overlap or an imperfect practical implementation. If some profiles have \(e(X)=0\) or \(e(X)=1\), positivity fails there: the missing treatment group cannot provide the counterfactual comparison from observed data. Restricting the target population may be considered, but it changes which population the resulting effect describes.

Within-score comparisons differ from exact matching on all covariates: the score can group distinct profiles, so it avoids requiring identical covariate vectors. Direct adjustment for covariates instead models the outcome or treatment comparison as a function of \(X\); that can be useful but relies on its own modeling choices. Weighting and matching methods also use scores to construct comparisons, but finite-sample performance depends on overlap, score estimation, and implementation. None of these approaches repairs unmeasured confounding merely by using a propensity score.

A common mistake is to reason, “Everyone with the same score has the same covariates, so matching on the score identifies comparable individuals.” That is not what the balancing result says. The correct statement is that, at a true score level, the distribution of covariates is the same across treatment groups. Individuals can have different covariates, and a score alone cannot identify their profiles. A second mistake is to treat balance as proof of no confounding: balance concerns measured covariates, while exchangeability may fail because of unmeasured variables or other design problems.