Potential outcomes framework

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

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 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.

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

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 $\mathbb{E}[Y(1)-Y(0)\mid A=1]$.