Total causal effect of each cause on each outcome, summing over all directed paths. Compare with the direct weight matrix to see how much indirect routing amplifies or dampens each relationship.
Direct weights measure one-step relationships. But causes rarely act through a single edge — they propagate through chains of mediators, each multiplying or attenuating the signal. The path effect matrix computes the total causal effect of each node on every other, summing contributions across all directed paths.
This is what a do() query delivers at the population level: the expected change in Y per unit change in do(X), accounting for all intermediate steps. Comparing this matrix to the weight matrix shows where indirect mechanisms dominate the direct connection — the places where an intervention's consequences arrive through routes the decision-maker may not have anticipated.