Backdoor identification and adjustment

Consistency connects the observed outcome in arm a to Y(a). Conditional exchangeability says Y(a) is independent of actual treatment given L, for each treatment level. This allows the observed arm-specific conditional mean to represent the corresponding potential-outcome mean within each covariate stratum. Positivity requires both treatment alternatives to occur with nonzero probability at almost every L value in the target population. Otherwise some needed conditional means are unavailable without additional extrapolation restrictions.

Apply iterated expectation to Y(a), use exchangeability to condition on A=a without changing its conditional mean, and then invoke consistency. This yields the adjustment formula below. A discrete sum becomes an integral for continuous covariates. It identifies an overall population mean only when the averaging distribution is that target population’s covariate distribution. Averaging the treated mean using treated covariates and the control mean using control covariates would return the original incomparable contrast rather than the same standardized target.

In a causally sufficient DAG, Pearl’s classical backdoor criterion asks for a set containing no descendants of treatment and blocking every path from treatment to outcome that begins with an arrow entering treatment. Such a set supplies a sufficient graphical route to adjustment. In a fork L→A and L→Y, conditioning on L blocks the confounding path while retaining the directed treatment-to-outcome paths whose total effect is sought. The criterion is sufficient, rather than a claim that it characterizes every valid set under all later graphical criteria.

Conditioning on a mediator along A→M→Y generally changes a total-effect question, because it removes part of the treatment’s mechanism. Conditioning on a collider or its descendant can open an otherwise blocked path. These are reasons to examine the whole graph, not a recipe to select variables solely by their predictive significance. If an unmeasured common cause leaves a backdoor path open, adding more measured descendants need not fix it. An accepted graph still rests on scientific assumptions about omitted arrows and hidden variables.

If high-baseline-risk units are more likely to receive treatment, a treated group can have a high observed outcome rate even when treatment raises or lowers each stratum’s mean by a modest amount. Standardization holds the mixture of baseline strata constant across the two treatment alternatives. The arithmetic can reveal how composition generated an observed difference. It cannot prove that the recorded L includes every common cause, that treatment versions are comparable, or that selection into observation is harmless.

Regression adjustment, stratification, and weighting are alternative ways to estimate related identifying functionals, not competing substitutes for exchangeability. With continuous or high-dimensional L, estimates depend on smoothness, model flexibility, and the amount of overlapping data. A precise result reports its target covariate distribution, identifies which variables precede treatment, and separates empirical overlap checks from the untestable assertion that residual confounding is absent. This makes the adjustment formula assessable without claiming that a fitted coefficient automatically answers an intervention question.