How does the front-door criterion identify a causal effect when the confounder is unmeasured?

## Why another criterion is needed The backdoor criterion requires a measured set that blocks every path from treatment T to outcome Y beginning with an arrow into T. If an unmeasured variable U has arrows into both T and Y, that path can only be closed by U itself, and adjustment on measured variables is helpless. Most people stop there and declare the effect unidentified. Sometimes it is not: if the mechanism by which T reaches Y passes entirely through a variable you *can* measure, the effect can be recovered through the front door instead. ## The canonical graph Pearl's textbook example uses smoking, tar deposits in the lungs, and lung cancer: - `Smoking -> Tar -> Cancer` (the mechanism, fully mediated) - `Genotype -> Smoking` and `Genotype -> Cancer` (an unmeasured common cause) - No `Smoking -> Cancer` edge that bypasses tar, and no `Genotype -> Tar` edge The genotype is a hypothetical unobserved trait that both predisposes people to smoke and independently raises cancer risk. It makes the smoking-to-cancer association confounded and unadjustable, since it is never measured. Tar deposits, by contrast, are measurable. ## The three conditions A set M satisfies the **front-door criterion** relative to the ordered pair (T, Y) when: 1. **M intercepts every directed path from T to Y.** The mediator carries the entire effect; nothing sneaks past it. 2. **There is no unblocked backdoor path from T to M.** The first stage, T's effect on M, is itself unconfounded. 3. **Every backdoor path from M to Y is blocked by T.** Conditioning on the treatment suffices to make the second stage, M's effect on Y, unconfounded. Check them on the smoking graph. Condition 1: cancer is reached from smoking only via tar, by assumption. Condition 2: the only route from smoking to tar other than the direct arrow would go back through the genotype, but the genotype has no arrow into tar, so nothing is open. Condition 3: the backdoor path `Tar <- Smoking <- Genotype -> Cancer` runs through smoking, so conditioning on smoking closes it. ## The estimand When the conditions hold, the interventional distribution is `P(y | do(t)) = sum over m of P(m | t) * [ sum over t' of P(y | m, t') * P(t') ]` Read it as two chained stages: - **Stage one**, `P(m | t)`: how much the treatment shifts the mediator. Unconfounded by condition 2, so the raw conditional distribution is already causal. - **Stage two**, the bracketed term: the effect of the mediator on the outcome, obtained by backdoor adjustment for T — note the inner sum uses the *marginal* distribution `P(t')` over treatment values, not the value t fixed in stage one. That reweighting is essential and the most common place to go wrong. Multiplying and summing over m composes the mechanism: the treatment moves the mediator, the mediator moves the outcome, and the product is the total effect. The unmeasured confounder is never conditioned on and never needs to be measured. ## Why it is rare in practice The conditions are strong and largely untestable: - **Complete mediation.** Any direct edge from T to Y that bypasses M breaks condition 1, and total mediation is a heroic assumption in most social and business settings. Smoking almost certainly harms lungs by routes other than tar, which is why the example is presented as an illustration rather than as epidemiology. - **An unconfounded first stage.** The unmeasured confounder must not touch M. If whatever drives treatment also drives the mediator, condition 2 fails. - **A well-measured mediator.** Measurement error in M attenuates both stages and there is no natural correction. When you cannot defend those, the honest response is to report the effect as unidentified and quantify how much unmeasured confounding it would take to change your conclusion. ## What a good answer sounds like Name the three conditions, check them against a concrete graph, write the two-stage estimand and note that the second stage averages over the marginal distribution of the treatment. Then volunteer the weakness: complete mediation is the assumption that usually fails, and it is not testable from the data. The criterion is genuinely useful when the mechanism is engineered and instrumented — for instance when the treatment can only reach the outcome through a logged, mandatory step — and misused everywhere else.