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.

With binary assignment or encouragement $Z$, potential treatment receipt $D(z)$ is the treatment a unit would actually receive under assignment $z$. It differs from the assignment itself when there is noncompliance. Only $D=D(Z)$ is observed for that unit, under the usual consistency in causal inference convention.

The pair $(D(0),D(1))$ defines four principal strata: never-takers $(0,0)$, compliers $(0,1)$, defiers $(1,0)$ and always-takers $(1,1)$. A patient who takes a drug only when encouraged is a complier; one who obtains it under either assignment is an always-taker. Randomization of $Z$ does not itself randomize actual treatment receipt. If $D(1)\ge D(0)$ for every unit, defiers are excluded andunder randomized assignment. Thus the effect of encouragement on uptake identifies the complier fraction, not automatically the effect of receiving treatment on outcomes.^[ref]

A principal stratum is defined by the joint potential outcomes of an intermediate variable under all compared treatments, rather than by its one observed value. For binary assignment $A$ and intermediate response $S$, a basic principal stratum is the set of units with a specified pair $(S(0),S(1))$. Each unit belongs to the same principal stratum whichever assignment it receives, although only $S(A)$ is observed.^[ref]

For example, let $S$ record whether a unit takes a treatment when offered encouragement. The pairs $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$ define never-takers, compliers, defiers and always-takers, respectively. An instrumental-variable monotonicity assumption rules out defiers. A comparison conditional on observed receipt $S=1$ generally mixes compliers with always-takers and can mix them in different proportions across assignment arms. It therefore does not automatically identify a causal effect within either principal stratum.

A never-taker is a unit whose potential treatment receipts are $D(0)=D(1)=0$: it receives no treatment under either value of a binary assignment or encouragement $Z$. This is the $(0,0)$ principal stratum, defined before observing which assignment actually occurs.

For example, a unit may decline participation whether or not it receives an invitation. Among units with observed $Z=0,D=0$, never-takers are mixed with compliers, so a single observation does not reveal the principal stratum. Under an exclusion restriction through treatment receipt, assignment has no outcome effect on a never-taker because receipt stays fixed; that is an additional causal assumption, not part of the definition.^[ref]

A complier is a unit with potential treatment receipts $D(0)=0$ and $D(1)=1$: binary encouragement $Z$ changes its receipt from no treatment to treatment. This $(0,1)$ pair defines a principal stratum, rather than a behavior inferred from one observed assignment.

For example, a unit that participates exactly when invited is a complier. Under instrumental-variable monotonicity, which excludes defiers,Thus, with randomized assignment, the difference in treatment-receipt rates estimates the complier fraction. To interpret an outcome-rate ratio as the local average treatment effect, one additionally needs the relevant independence and exclusion assumptions and a nonzero complier fraction. An observed encouraged participant may instead be an always-taker.^[ref]

A defier is a unit with potential treatment receipts $D(0)=1$ and $D(1)=0$: it takes the treatment without encouragement but does not take it when encouraged. This is the $(1,0)$ principal stratum. The word specifies potential behavior under a particular binary intervention; it does not describe a general personality trait.

A defier contributes $-1$ to $D(1)-D(0)$, whereas a complier contributes $+1$. Consequently,The usual instrumental-variable monotonicity assumption sets the second probability to zero. A positive average effect of encouragement on receipt does not prove that there are no defiers: positive and negative individual changes can coexist.^[ref]

An always-taker is a unit with potential treatment receipts $D(0)=D(1)=1$: it takes the treatment under either value of binary encouragement $Z$. This $(1,1)$ pair is a principal stratum defined by both potential receipts.

For example, a unit may participate whether or not it receives an invitation. Encouraged participants combine always-takers and compliers; unencouraged participants combine always-takers and defiers. Under instrumental-variable monotonicity, the latter group consists only of always-takers. Even then, comparing observed participants across assignment arms mixes different strata and is not automatically a within-stratum causal effect. The exclusion restriction, when imposed, says assignment cannot affect an always-taker's outcome through any channel other than the treatment whose receipt is unchanged.^[ref]

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: