Symmetric decomposition of a stratified mean difference ($D={\sum }_s{\bar{w}}_s(p_{1s}-p_{0s})+{\sum }_s(w_{1s}-w_{0s}){\bar{p}}_s$)

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.