Standardization with common weights ($p_g^{(w)}={\sum }_s{w}_s{p}_{gs}$)

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.