Deriving the Propensity Score Balancing Property

Why the Balancing Property Holds

The propensity score is useful because it compresses information about measured covariates into a single probability. The central mathematical fact is that, within a value of the true propensity score, the distribution of those covariates is the same for treated and untreated units. This is a population result about how treatment is assigned given measured covariates; it is not a claim that treatment assignment was random in the full population.

Let \(A\) be a binary treatment indicator, with \(A=1\) for treated units and \(A=0\) for untreated units. Let \(X\) denote the vector of measured pretreatment covariates. The propensity score is the conditional probability of treatment given those covariates.

The equality is understood for score values where both treatment groups occur. In particular, when discussing both groups within a score stratum, require positivity: \(0<e(X)<1\) in that stratum. At a score of zero there are no treated units, and at a score of one there are no untreated units, so a two-group comparison cannot be made there.

Derivation From Conditional Probability

The proof uses the definition of \(e(X)\) and the fact that conditioning on a score value restricts the conditional treatment probability to that value. To make the argument precise without relying on the probability of an individual continuous score being positive, consider any set \(B\) of covariate values and a score stratum \(S\) on which \(e(X)=s\), where \(0<s<1\). The argument applies to discrete strata directly and, more generally, to conditional distributions defined almost surely.

First, among units in that score stratum, the treatment probability is \(s\). This follows by averaging the treatment probability given \(X\) over the stratum:

Now calculate the probability that a treated unit in the stratum has covariates in \(B\). Conditional expectation gives the numerator, and the preceding result gives the denominator:

Here \(\mathbf{1}\{X\in B\}\) is one when \(X\) lies in \(B\) and zero otherwise. Because \(e(X)\) is determined by \(X\), conditioning on both \(X\) and \(e(X)\) does not change \(P(A=1\mid X)=e(X)\). On the stratum, \(e(X)=s\), so the factor inside the expectation is \(s\).

The untreated group has the same covariate distribution. Its probability of occurring in the stratum is \(1-s\), so the corresponding calculation is:

Thus both conditional distributions equal the covariate distribution in the stratum. Since this holds for every covariate set \(B\), the treated and untreated distributions of \(X\) agree there. That is the balancing property: the true score makes treatment groups alike in measured covariates within score strata.

A Numerical Example With a Binary Covariate

A covariate need not be the only determinant of the score. The following hypothetical population has two binary covariates, \(X\) and \(Z\). The treatment probabilities are specified by covariate pattern. Two patterns share a score of \(0.60\), allowing us to compare the distribution of the binary covariate \(X\) between treatment groups within that score stratum.

Focus on the score stratum \(e(X,Z)=0.60\). Its two patterns have population masses \(0.30\) for \((X,Z)=(0,1)\) and \(0.40\) for \((1,0)\), for a total of \(0.70\). The stratum contains both treatment groups because its treatment probability is neither zero nor one.

The treated mass in the first pattern is \(0.30(0.60)=0.18\), and in the second it is \(0.40(0.60)=0.24\). The total treated mass in the stratum is \(0.18+0.24=0.42\). Thus:

For untreated units, the masses are \(0.30(0.40)=0.12\) for \(X=0\) and \(0.40(0.40)=0.16\) for \(X=1\). Their total is \(0.28\), giving:

The proportion with \(X=1\) is therefore about \(57.14\%\) in each group within this stratum. The matching proportions are not a coincidence: both covariate-pattern masses were multiplied by the same treatment probability, \(0.60\), for treated units, or by the same untreated probability, \(0.40\), for controls. Dividing by each group’s total removes that common factor. The same reasoning applies to any set of covariate patterns within a fixed true-score stratum.

What the Result Does and Does Not Guarantee

The proof is an exact population result for \(e(X)=P(A=1\mid X)\). In an application, the score is generally estimated. A fitted score need not equal the true conditional probability, and it need not balance covariates exactly in a finite sample. Balance should therefore be checked in the data after the chosen matching, weighting, or stratification procedure; a model that produces plausible treatment probabilities is not, by itself, evidence that balance has been achieved.

The proof also concerns the measured covariates included in \(X\), not unmeasured characteristics. It does not show that treatment groups have the same potential outcomes, nor does it establish that an outcome comparison is causal. Those conclusions require additional assumptions and design or analysis choices. The balancing property is a statement about covariate distributions conditional on the true score.

It is helpful to distinguish score-based balancing from direct balancing on all covariates. Comparing treatment groups at identical values of the entire vector \(X\) targets covariate equality directly, but exact comparisons may be difficult when \(X\) has many components. The score reduces that vector to a scalar for the balancing argument. This reduction is useful only insofar as the score is the true conditional treatment probability, or an estimate that works adequately for the intended analysis.

Common Pitfalls and Diagnostics

• Using the wrong probability. The balancing proof uses \(e(X)=P(A=1\mid X)\), not any arbitrary summary or treatment-prediction score. If the score does not equal the conditional treatment probability, the algebra does not establish balance.
• Ignoring positivity. At \(e(X)=0\) or \(e(X)=1\), one treatment group is absent. Conditional distributions for both groups in that stratum cannot be compared.
• Confusing a population result with a sample guarantee. The exact result concerns the true score and population distributions. Estimated scores and finite samples can leave residual differences; inspect covariate balance in the resulting groups.
• Interpreting balance as causal identification. Balanced measured covariates do not demonstrate balance on unmeasured factors or establish assumptions about potential outcomes.

A useful diagnostic is to compare covariate distributions between treatment groups within the score strata or matched sets actually used. If meaningful differences remain, the empirical procedure has not delivered the balance suggested by the ideal population result. Such differences may reflect estimation error, sparse data, an inadequate score specification, or insufficient overlap; the theorem alone cannot identify which cause is responsible.