Conditional Treatment Probability as the Propensity Score

What the Propensity Score Measures

In an observational study, people with different baseline characteristics may have different chances of receiving a treatment. The propensity score summarizes that assignment probability conditional on the measured covariates. It is a property of the treatment-assignment process in a specified population, not a label for the treatment a person actually received.

Let \(A\) be a binary treatment indicator, coded \(A=1\) for treated and \(A=0\) for untreated, and let \(X\) denote the set of measured covariates. For a person whose covariates are \(X\), the propensity score describes the conditional probability of treatment. The covariates used for this purpose should be defined at the relevant baseline, before treatment assignment or initiation. This timing makes the score a description of treatment assignment given prior information, rather than a description using consequences of treatment.

Because \(A\) takes only the values zero and one, its conditional mean is its conditional probability of being one: \(E[A\mid X]=P(A=1\mid X)\). Thus the propensity score can also be written \(e(X)=E[A\mid X]\). It is a probability, so it lies between zero and one, up to the usual almost-sure qualification for conditional expectations.

Three Quantities That Should Not Be Confused

The definition separates three different things. A person’s treatment status is the observed value \(A\), either zero or one. Their propensity score is the conditional probability \(e(X)\), which can be any value from zero to one. Their outcome risk is a probability involving an outcome, such as \(P(Y=1\mid X)\) for a binary outcome \(Y\). The score concerns treatment assignment; outcome risk concerns the outcome.

For example, two people with the same measured covariates have the same value of \(e(X)\) under a given population’s treatment-assignment distribution, even though one received treatment and the other did not. Their observed treatment statuses differ, but the conditional probability used to describe their assignment is the same. This does not mean their outcomes are the same, nor does it imply that the score is a probability of benefiting from treatment.

A propensity score is also not automatically a causal effect. Defining \(e(X)\) requires a binary treatment and a conditional distribution of treatment given covariates; it does not, by itself, require that treatment assignment be unconfounded or that a causal effect be identified. Those questions require additional assumptions and an appropriate analysis.

Why the Table Calculation Works

For a discrete covariate pattern \(x\) with positive probability, the definition of conditional probability gives

In a finite data set, the corresponding empirical probabilities are joint and marginal proportions. Dividing the proportion with both \(A=1\) and \(X=x\) by the proportion with \(X=x\) cancels the common total sample size, leaving the number treated in that pattern divided by the number in that pattern. This is why a covariate table gives a direct empirical calculation for discrete cells.

That calculation describes the sample’s conditional treatment fraction. Treating it as an estimate of a population propensity score requires the data to provide relevant information about the target population and treatment process. For instance, a simple random sample from that population supports a straightforward frequency interpretation; a selected or otherwise nonrepresentative sample may not. A table calculation is not the same as fitting a model, and no model-fitting procedure is needed to define the propensity score.

Continuous Covariates and Conditional Probabilities

The table calculation is especially transparent when \(X\) is discrete and a covariate pattern occurs repeatedly. With continuous covariates such as age or a laboratory measurement, an exact value \(x\) will often occur with probability zero: \(P(X=x)=0\). The ratio \(P(A=1,X=x)/P(X=x)\) is then not a usable pointwise formula. This does not mean that the propensity score is undefined for continuous covariates. The conditional probability \(P(A=1\mid X)\) is defined as a conditional expectation, and a regular conditional distribution can represent it as a function \(e(x)\), with versions agreeing almost everywhere under the distribution of \(X\).

At values \(x\) outside the support of the covariate distribution, or in other probability-zero sets, the chosen version’s value is not determined by the observed population distribution. Consequently, the simple “count treated at exactly \(x\), divided by the count at \(x\)” calculation is a discrete-table calculation, not a general definition for continuous \(X\). Estimating a score from continuous covariates requires additional methods, which are separate from defining the score.

Checks, Comparisons, and Limitations

Before interpreting a calculated score, check that treatment coding is consistent, the denominator includes all eligible people in the covariate pattern, and the covariates are measured at the intended baseline. Verify that each table row sums to its stated total and that treated counts do not exceed totals. If an empirical cell has no observations, its within-cell treatment fraction cannot be calculated from that table. A zero cell count is not evidence that the population probability is exactly zero.

Scores close to zero or one deserve attention: they indicate covariate patterns for which the observed or specified treatment probability is very low or very high. In a finite table, an estimate of zero or one may simply reflect a small or sparse cell. Such values can flag limited information about treatment alternatives in that pattern, but a score calculation alone cannot establish why the pattern occurred or resolve the consequences for a causal analysis.

The main alternatives answer different questions. The overall treatment proportion describes treatment frequency without conditioning on covariates. The observed \(A\) describes which treatment a person received. An outcome-risk quantity such as \(P(Y=1\mid X)\) concerns the outcome, not treatment assignment. Directly describing covariate patterns preserves the individual covariate information rather than summarizing it as one probability. None of these quantities is interchangeable with the propensity score.

The score’s usefulness in later causal methods depends on assumptions and design choices beyond this definition. For example, the score does not guarantee that all confounders have been measured, that treatment groups are comparable, or that a causal effect is identified. The introductory material on causal inference treats the broader requirements for causal interpretation; here, the point is to identify exactly what the score itself represents.

Common Mistakes

• Calling the score the treatment received. \(A\) is the observed zero-or-one status; \(e(X)\) is a conditional probability that can be between zero and one.
• Using outcome risk as the score. A probability involving \(Y\) describes an outcome; the propensity score conditions treatment assignment on \(X\).
• Using the overall treatment proportion for every person. That is a marginal proportion and need not equal the conditional treatment probability in a covariate pattern.
• Requiring \(P(X=x)>0\) for every possible covariate value. This condition applies to the elementary discrete-table interpretation. For continuous \(X\), conditional probabilities are defined almost surely and may be represented by a regular conditional distribution.
• Interpreting a score as proof of causal validity. The score is a treatment-assignment probability; it does not itself verify the assumptions needed for a causal conclusion.