= Symmetric decomposition of a stratified mean difference
{title2=$D=\sum_s\bar w_s(p_{1s}-p_{0s})+\sum_s(w_{1s}-w_{0s})\bar p_s$}
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.
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