Potential outcome ($Y(a)$)

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

A causal effect compares potential outcomes under different specified interventions. For a binary treatment, a unit-level additive effect is $Y(1)-Y(0)$. Only one of those outcomes is observed for a given unit, so an observed treated-versus-untreated difference is not automatically the effect. For example, patients receiving treatment can differ in baseline severity from untreated patients. The average treatment effect averages the contrast over a target population; identifying it from data requires assumptions linking treatment assignment and the unobserved potential outcomes.

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 in a target population is $\mathbb{E}[Y(1)-Y(0)]$, the average additive contrast between two potential outcomes. It does not require every individual effect to be the same. If half the population has effect two and half effect minus one, the average treatment effect is $1/2$. Under consistency in causal inference, conditional exchangeability given pretreatment covariates $X$, and positivity, it is identified by ${\mathbb{E}}_X[\mathbb{E}(Y\mid A=1,X)-\mathbb{E}(Y\mid A=0,X)]$. The outer average uses the specified target population, not whichever group happens to be observed most often.

The conditional average treatment effect at pretreatment covariates $X=x$ is $\tau (x)=\mathbb{E}[Y(1)-Y(0)\mid X=x]$. It describes average effect variation between strata rather than identifying a particular individual effect. If $\tau (x)=2$ in one equally common stratum and $-1$ in another, the average treatment effect is $1/2$. Under conditional exchangeability, consistency in causal inference and positivity, conditional outcome means identify $\tau (x)$. A predictor measured after treatment cannot simply be substituted for a pretreatment covariate without changing the causal question.

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 $\mathbb{E}[Y(1)-Y(0)\mid A=1]$, averaging over those who actually receive treatment. Their untreated potential outcomes are counterfactual. This can differ from the average treatment effect when effects vary and treatment uptake changes the population mix. Under conditional exchangeability, consistency in causal inference and adequate untreated overlap, it equals $\mathbb{E}[Y\mid A=1]-\mathbb{E}[\mathbb{E}(Y\mid A=0,X)\mid A=1]$. The second term standardizes untreated outcomes to the treated covariate distribution rather than using the crude untreated mean.

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