For an email campaign, do customer tenure, Black-Friday timing and clicking the email belong in the adjustment set for purchases?

## The setup Treatment T is "a campaign email was sent to this customer". Outcome Y is "the customer purchased within seven days". You did not randomise: an in-house targeting rule decided who got the email. You want the causal effect of sending on purchasing, and you have three candidate variables to control for. The right method is not intuition about "what predicts the outcome" — it is to draw the graph and check each node against the two conditions of the backdoor criterion: 1. No node in the adjustment set is a descendant of T. 2. The set blocks every backdoor path from T to Y — every path whose first edge points into T. ## Node 1: customer tenure How long someone has been a customer plausibly affects **whether they were emailed**: targeting rules favour established customers, or conversely exclude them from win-back campaigns. It also plausibly affects **whether they purchase**, independently of any email: established customers buy more often. That is the fork `Send <- Tenure -> Purchase`. It is a path from T to Y whose first edge points into T, so it is a backdoor path, and it is open because a fork transmits association unless its middle node is conditioned on. Leaving tenure out means part of the apparent "email effect" is really just the tendency of the sort of customer who gets emailed to buy anyway. Tenure is measured before the send and is not caused by it, so condition (1) is satisfied. **Tenure goes in.** ## Node 2: sent during Black Friday This is the same structure at a different level of the system. A shopping peak raises the volume of campaign sends *and* raises purchase propensity for reasons that have nothing to do with any particular email — the whole market is buying. So `Send <- BlackFriday -> Purchase` is a second open backdoor path with a shared cause in the middle. This one is easy to miss because it is a property of *when* rather than *who*, and it often lives in a different table from the customer attributes. It is nonetheless the single most dangerous confounder in campaign analysis: a campaign that fired during a peak will look spectacular until the timing is accounted for. It precedes the send in the causal order, so condition (1) is fine. **Black-Friday timing goes in.** ## Node 3: clicked the email Here the criterion answers before you have to think about mechanisms at all. A customer cannot click an email that was never sent, so `Send -> Clicked`. Clicking is a **descendant of the treatment**, and condition (1) bans descendants of T from the adjustment set. **Clicks stay out.** It is worth being able to say *why* the rule exists here, in the vocabulary of this criterion. Clicks lie on the route by which the treatment reaches the outcome — the email works partly by getting opened and clicked. The backdoor criterion is built to leave that route intact while closing the routes that carry no effect; conditioning on a node on the causal route closes what you are trying to measure. Practically, "clicked" is also undefined or structurally zero for every untreated unit, so comparing clickers to non-recipients is not a like-for-like contrast at all. The temptation is strong because clicks correlate beautifully with purchases. Predictive strength is not the criterion. A variable's admissibility is decided by where it sits in the graph relative to T, not by how much of Y it explains. ## The resulting analysis Adjustment set: `{tenure, Black-Friday timing}`. The identified quantity is `P(Purchase | do(Send)) = sum over z of P(Purchase | Send, Z=z) * P(Z=z)` with Z ranging over the joint levels of the two variables, averaged over the population distribution of those levels. ## What to say about the parts you cannot see A strong answer does not stop at the three variables handed to you. It states the remaining threat: if the targeting rule used a score built from browsing behaviour that also predicts purchasing, and that score is not in your data, there is an unmeasured node with arrows into both T and Y, and no set of measured variables closes that path. In campaign analysis this is the usual situation, because targeting is deliberately built to select likely buyers. The correct move is to recover the actual targeting inputs — they exist, they are logged somewhere, and they are exactly the variables the criterion asks for — rather than to add whatever features are convenient. If they cannot be recovered, say the effect is not identified and argue for a holdout group on the next campaign instead of an elaborate observational model. ## The interview signal This question separates people who apply a rule from people who reason about the mechanism. Weak answers control for clicks because it improves fit; medium answers get tenure but miss timing because it is not a customer-level column; strong answers walk each node against both conditions, name the unmeasured targeting score as the residual threat, and propose the holdout that would settle it.