Potential Outcomes Framework

Summary

The potential outcomes framework (Rubin causal model) defines causal effects as comparisons of counterfactual outcomes under different treatment conditions for the same unit. The key identifying assumptions — SUTVA, ignorability (unconfoundedness + overlap) — determine when causal effects can be estimated from observational data.

Overview

The potential outcomes framework is the mainstream statistical framework for causal inference, underpinning the Bayesian review by Li et al. (2022). Following the dictum “no causation without manipulation”, a cause is a pre-specified treatment or intervention that is at least hypothetically manipulable.

Setup

Consider $N$ units indexed by $i\in \{1,\dots ,N\}$. Each unit $i$ has:

• $Z_i\in \{0,1\}$ — binary treatment indicator ($Z_i=1$ active, $Z_i=0$ control)
• $X_i$ — vector of $p$ pre-treatment covariates observed before treatment
• $Y_i(1),Y_i(0)$ — potential outcomes under treatment and control respectively
• $Y_i^{\text{obs}}=Y_i(Z_i)$ — the observed outcome (only one potential outcome is observed)

The fundamental problem of causal inference: for each unit, only $Y_i(Z_i)$ is observed; $Y_i(1-Z_i)$ is missing (counterfactual).

Let $Z=(Z_1,\dots ,Z_N{)}^T$, $X=(X_1,\dots ,X_N{)}^T$.

Key Assumption: SUTVA

Assumption: Stable Unit Treatment Value Assumption (SUTVA)

There is: (i) no different version of a treatment, and (ii) no interference — unit $i$‘s potential outcomes are not affected by other units’ treatment assignments.

Under SUTVA, unit $i$ has exactly two potential outcomes: $Y_i(1)$ and $Y_i(0)$.

SUTVA rules out spillover effects and treatment heterogeneity due to dose or version. Violations occur in settings like infectious disease (interference) or drug dosage (multiple versions).

Key Assumption: Ignorability

Assumption 2.1 — Ignorability (Li et al. §2)

The assignment mechanism is ignorable if both:

(a) Unconfoundedness: $Z_i\perp \perp \{Y_i(0),Y_i(1)\}\mid X_i$, or equivalently $Z_i\perp \perp \{Y_i(0),Y_i(1)\}\mid \{X_i=x_i\}$ for all $x$.

(b) Overlap: $0<e(x)<1$ for all $x$, where $e(x)=\mathrm{Pr}(Z_i=1\mid X_i=x)$ is the propensity score.

• Unconfoundedness (also called selection on observables or no unmeasured confounding): treatment assignment is as-good-as-random conditional on observed covariates. This is fundamentally untestable from observed data.
• Overlap (also called positivity): every unit has a non-zero probability of receiving either treatment. Ensures the conditional distribution of potential outcomes is identifiable from observed data.

Together, these ensure that:

$${\mu }_z(x)\equiv \mathbb{E}[Y_i(z)\mid X_i=x]=\mathbb{E}[Y_i\mid Z_i=z,X_i=x]$$

for all $z,x$ — i.e., the potential outcome mean equals the observed conditional mean.

The Role of Covariate Overlap and Balance

Overlap and balance refers to similarity in the distribution of covariates between the two treatment groups. This is a key concept for design:

• In randomized experiments: all covariates are balanced in expectation; simple difference-in-means $\hat{\tau }$ is unbiased for ${\tau }^S$ and ${\tau }^P$.
• In observational studies: groups are often imbalanced (e.g., sicker patients get more treatment). Direct comparison gives biased causal estimates.
• Poor overlap: outcome model estimates rely on extrapolation; sensitive to model misspecification.

Design vs. Analysis

A main effort in causal inference with observational data is the design stage: ensuring overlap and balance to mimic a randomized experiment as closely as possible. The design stage does not involve the outcome — this contrasts with the analysis stage, which uses the outcome to estimate effects.

Randomized Experiments vs. Observational Studies

Feature	Randomized Experiment	Observational Study
Assignment known?	Yes, controlled	No, must be modeled
Ignorability	Holds by design	Holds approximately (best case)
Overlap	Usually good	May be poor
Confounding	Eliminated by randomization	Present; must adjust

A/B testing and randomized controlled trials are the gold standard for causal inference precisely because they eliminate confounding via randomization.

Connections

• Causal Estimands — formal treatment effect quantities defined under this framework
• Frequentist Causal Estimation — IPW, outcome modeling, doubly-robust estimators exploit ignorability
• General Structure of Bayesian CI — Bayesian inference treats missing potential outcomes as parameters to be imputed
• Sensitivity Analysis in Observational Studies — methods to assess robustness when unconfoundedness may fail

See Also

• Causal Estimands — formal definitions of ITE, SATE, CATE, PATE, MATE
• Propensity Score in Bayesian CI — propensity score $e(x)$ plays central role despite dropping from likelihood