Causal inference begins by defining outcomes under interventions before asking whether they can be learned from observed data. Potential outcomes separate the scientific contrast, the assumptions that identify it, and the statistical procedure used to estimate it.
Units, treatments, and counterfactual outcomes
Definition. For a binary treatment \(A\in\{0,1\}\), the potential outcome \(Y(a)\) is the outcome that would be observed for the same unit under intervention \(A=a\). The individual treatment effect is \[ \tau=Y(1)-Y(0). \] The observed data for one unit are \(O=(X,A,Y)\), where \(X\) is measured before treatment.
Potential outcomes are random variables on a superpopulation probability space. In a finite-population analysis, the \(Y_i(a)\) may instead be treated as fixed attributes of the sampled units and randomness comes solely from treatment assignment.
Consistency and the observation equation
Definition. Consistency asserts \[ A=a\quad\Longrightarrow\quad Y=Y(a). \] For binary treatment this is equivalent to \[ Y=AY(1)+(1-A)Y(0). \]
Remark. Consistency contains two substantive requirements: the intervention \(a\) must be sufficiently well defined, and one unit's treatment must not alter another unit's outcome. The latter is often separated as the no-interference part of SUTVA. Versions, spillovers, and network effects require potential outcomes indexed by a richer exposure vector.
Causal estimands
Definition. Important contrasts include \[ \begin{aligned} \operatorname{ATE}&=\mathbb E\{Y(1)-Y(0)\},\\ \operatorname{ATT}&=\mathbb E\{Y(1)-Y(0)\mid A=1\},\\ \operatorname{CATE}(x)&=\mathbb E\{Y(1)-Y(0)\mid X=x\},\\ \operatorname{QTE}_\tau&= F_{Y(1)}^{-1}(\tau)-F_{Y(0)}^{-1}(\tau). \end{aligned} \] The probability of benefit is \(P\{Y(1)>Y(0)\}\), a functional of the joint law of the two potential outcomes.
The marginal laws of \(Y(1)\) and \(Y(0)\) identify the ATE and marginal quantile effect. They do not identify the distribution of the individual effect, because that depends on the unobserved coupling between \(Y(1)\) and \(Y(0)\).
Proposition. Knowing both marginal distributions \(F_1,F_0\) does not generally identify \(P\{Y(1)>Y(0)\}\) or \(\operatorname{Var}\{Y(1)-Y(0)\}\).
Proof. Let both marginals be \(\operatorname{Bernoulli}(1/2)\). Under the comonotone coupling \(Y(1)=Y(0)\), the probability of benefit and effect variance are zero. Under the countermonotone coupling \(Y(1)=1-Y(0)\), the probability of benefit is \(1/2\) and the effect variance is one. The observed marginals agree in the two constructions.
The fundamental missing-data structure
Proposition. Consistency alone does not identify \(\mathbb EY(1)\) from the observed law of \((A,Y)\).
Proof. The observed law identifies \(Y(1)\mid A=1\), but never observes \(Y(1)\mid A=0\). One may alter the latter distribution while leaving \(Y=AY(1)+(1-A)Y(0)\) unchanged. For example, keep all observed treated outcomes fixed and assign arbitrary counterfactual treated outcomes to controls. These completions induce the same observed law but different \(\math EY(1)\).
Randomization identifies marginal means
Theorem. If \(A\perp\!\!\!\perp\{Y(0),Y(1)\}\), consistency holds, and \(0<P(A=1)<1\), then \[ \mathbb EY(a)=\mathbb E(Y\mid A=a),\qquad a=0,1, \] and therefore \[ \operatorname{ATE}= \mathbb E(Y\mid A=1)-\mathbb E(Y\mid A=0). \]
Proof. Independence gives \(\mathbb EY(a)=\mathbb E\{Y(a)\mid A=a\}\). Consistency replaces \(Y(a)\) by \(Y\) on the event \(A=a\). Subtract the two identified means.
Finite-population randomization
Theorem. For \(N\) fixed units, assign exactly \(n_1\) to treatment uniformly among all such assignments and let \(n_0=N-n_1\). The difference in sample means \[ \widehat\tau=\overline Y_1^{\mathrm{obs}}- \overline Y_0^{\mathrm{obs}} \] is unbiased for \(\tau_N=N^{-1}\sum_i\{Y_i(1)-Y_i(0)\}\), with variance \[ \operatorname{Var}(\widehat\tau) =\frac{S_1^2}{n_1}+\frac{S_0^2}{n_0} -\frac{S_\tau^2}{N}, \] where \(S_a^2\) and \(S_\tau^2\) are finite-population variances.
Proof. Each unit has treatment probability \(n_1/N\), so linearity gives unbiasedness of each arm mean. The covariance structure of sampling without replacement gives \(\operatorname{Var}(\overline Y_a)= (1/n_a-1/N)S_a^2\). Expanding the covariance between the two arm means and using \(S_\tau^2=S_1^2+S_0^2-2S_{10}\) yields the formula.
The unobservable \(S_\tau^2\) prevents exact variance estimation without assumptions on effect heterogeneity. Dropping its nonnegative term gives the conservative Neyman variance estimator.
Multiple treatments and dose-response curves
Definition. For treatment \(A\in\mathcal A\), define the dose-response function \[ \mu(a)=\mathbb EY(a). \] For continuous \(A\), point interventions can be statistically delicate because \(P(A=a)=0\); identification uses conditional densities and positivity in neighborhoods, while stable estimands may average \(\mu(a)\) over a stochastic intervention.
Interference and exposure mappings
Definition. With \(N\) interacting units, the unrestricted outcome is \(Y_i(\mathbf a)\), indexed by the full assignment vector. An exposure mapping \(g_i(\mathbf a)\) posits \[ Y_i(\mathbf a)=Y_i\{a_i,g_i(\mathbf a)\}. \] Direct and spillover effects compare potential outcomes while changing different components of this exposure.
This is not a minor notation change: ordinary consistency and propensity-score arguments fail if one suppresses interference that is present. The estimand must be indexed by the intervention actually contemplated.