Potential outcomes framework

The potential outcomes framework represents causal effects by comparing outcomes assigned to the same unit under different hypothetical interventions.

Principal stratification groups units by the joint values of their potential values of an intermediate variable under every treatment assignment. With a binary instrument and binary treatment, the principal strata are always-takers, never-takers, compliers, and defiers.

A principal stratum is one group in a principal stratification.

A potential outcome $Y(a)$ is the outcome that would be observed under intervention $a$.

A causal effect compares potential outcomes under different interventions.

A treatment effect is homogeneous when the relevant contrast of potential outcomes is constant across units. For an additive treatment, this takes the form $Y(a)-Y(0)=\beta a$.

A causal null hypothesis states that an intervention has no effect on a specified potential-outcome contrast. A sharp causal null sets every unit's treated and untreated potential outcomes equal; an average causal null sets their population mean difference to zero.

A structural zero is an outcome value fixed at zero by the data-generating mechanism under a specified intervention or condition, rather than a zero arising by chance.

The average treatment effect is $\mathbb{E}[Y(1)-Y(0)]$ in a target population.

The conditional average treatment effect at covariates $X=x$ is

The overlap-weighted average treatment effect averages the conditional average treatment effect with weight proportional to $e(X)\{1-e(X)\}$, where $e(X)$ is the propensity score. It emphasizes covariate strata in which both treatments are plausible.

A heterogeneous treatment effect varies across units or covariate values rather than being constant throughout the population. The conditional average treatment effect describes one common form of this heterogeneity.

The average treatment effect on the treated is $\mathbb{E}[Y(1)-Y(0)\mid A=1]$.

For binary potential outcomes, the causal risk ratio compares intervention risks: