frontdoor criterion

Sometimes the confounder between treatment and outcome is invisible, so no backdoor set exists and adjustment seems impossible. The frontdoor criterion finds a way through anyway, provided the effect of the treatment travels entirely through some measured intermediate step. You estimate two clean legs separately, how the treatment moves the mediator and how the mediator moves the outcome, and multiply them, routing the causal effect through the open front door rather than around the blocked back ones.

A mediator M satisfies the criterion if M intercepts all directed paths from T to Y, there is no unblocked backdoor path from T to M, and every backdoor path from M to Y is blocked by T. The first leg, the effect of T on M, is unconfounded by assumption; the second, the effect of M on Y, is confounded only by T, which you can adjust for. Composing the two by summation gives a do-free formula even though the T-to-Y confounder is never observed.

The classic illustration is smoking, tar deposits, and cancer with an unobserved genetic confounder: if smoking causes cancer only through tar, the effect is recoverable. In practice fully mediating, confounder-free mediators are rare, which is why the frontdoor is celebrated as a proof of identifiability's reach more often than it is deployed as a workhorse.