A data science team is designing an observational study to measure the total causal effect of sending a promotional email (E) on repeat purchases in the following month (Y). Their directed acyclic graph is:
B → E, B → Y, E → S → Y
where
B = prior browsing intensity
S = post-email customer satisfaction score
Using Pearl's back-door criterion, which set of variables is sufficient-and no larger than necessary-to adjust for in order to obtain an unbiased estimate of the total effect of E on Y?
Condition on prior browsing intensity (B) only
Condition on both B and S
Make no covariate adjustment
Condition on post-email satisfaction score (S) only
The back-door criterion requires that the adjustment set 1) blocks every path from the treatment (E) to the outcome (Y) that enters E through a back door, and 2) contains no descendants of the treatment. In the graph, the only back-door path is E ← B → Y. Conditioning on B alone blocks this path. The variable S must not be conditioned on because it is a descendant (mediator) of E; including it would block part of the causal pathway and bias the total-effect estimate. Omitting B leaves the back-door path open, and adjusting for both B and S introduces the same mediator bias. Therefore, adjusting for B only is both sufficient and necessary.
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