Outcome standardization ($R(a)=\int P(Y=1\mid A=a,X=x)d{F}_{\mathrm{target}}(x)$)

Outcome standardization averages fitted conditional outcome risks over a specified target population's pretreatment covariate distribution. The same target distribution is used for each treatment strategy. Under conditional exchangeability, the positivity assumption and consistency, this average identifies the target mean of the corresponding potential outcomes. For a target consisting of treated subjects, average over their covariate distribution. A conditional odds ratio cannot generally be applied directly to an aggregate mean risk: the nonlinear odds-to-probability conversion must be performed before averaging. Outcome standardization supplies that individual-risk calculation.

Let $p_{gs}$ be a group-specific conditional probability in stratum $s$, and let $w_s\ge 0$ with ${\sum }_s{w}_s=1$. Average each group's conditional probabilities using the same weights. ThenThis is a descriptive comparison under a common covariate composition. Its difference from a crude contrast need not be a causal effect. Causal outcome standardization additionally requires its identification assumptions.

Suppose group $g$ has stratum weights $w_{gs}$ summing to one and conditional means $p_{gs}$. Write ${\bar{w}}_s=(w_{1s}+w_{0s})/2$ and ${\bar{p}}_s=(p_{1s}+p_{0s})/2$. The crude difference $D={\sum }_s{w}_{1s}{p}_{1s}-{\sum }_s{w}_{0s}{p}_{0s}$ has the displayed decomposition. Expanding the two products in each stratum cancels the cross terms and leaves $w_{1s}{p}_{1s}-w_{0s}{p}_{0s}$, proving the identity. The first term uses common average-group weights; the second records a composition difference. Neither term alone identifies a causal pathway.