📊Causal Inference Unit 2 Review

A counterfactual is the outcome that would have happened if circumstances had been different. In causal inference, this means asking: what would have happened to this person if they had received a different treatment?

This idea is the foundation of causal reasoning. You can't measure a treatment's effect without having something to compare it to, and that comparison is the counterfactual.

Counterfactuals vs observed outcomes

Observed outcomes are what actually happened and got recorded. If a patient took the drug, their observed outcome is their health after taking it.
Counterfactual outcomes are what would have happened under a different condition. For that same patient, the counterfactual is their health if they had not taken the drug.
The causal effect for that individual is the difference between these two quantities. The catch: you can only ever observe one of them.

Counterfactuals in causal inference

Counterfactuals let you define causal effects precisely. Rather than vaguely saying "the treatment helped," you can specify exactly what comparison you're making: the outcome under treatment versus the outcome under control, for the same unit.

Even though you can never directly observe both outcomes for a single person, the counterfactual framing gives you a clear target. All the estimation methods you'll encounter (randomization, matching, etc.) are strategies for getting at this fundamentally unobservable comparison.

Potential outcomes framework

The potential outcomes framework takes the intuition behind counterfactuals and makes it mathematically precise. It gives you notation, definitions, and assumptions that let you move from "what if" reasoning to actual estimation of causal effects.

This framework is used across statistics, economics, epidemiology, and political science. It's sometimes called the Rubin causal model (more on that below).

Defining potential outcomes

For each individual $i$ , you define a potential outcome for every possible treatment level. With a simple binary treatment (treated vs. control):

$Y_i(1)$ = the outcome individual $i$ would have under treatment
$Y_i(0)$ = the outcome individual $i$ would have under control

These potential outcomes exist conceptually for every person, regardless of which treatment they actually receive. The notation $Y_i(t)$ generalizes this to any treatment level $t$ .

Potential outcomes vs observed outcomes

The observed outcome for individual $i$ is just one of their potential outcomes, determined by which treatment they actually got. If individual $i$ receives treatment ( $T_i = 1$ ), you observe $Y_i(1)$ but not $Y_i(0)$ . If they're in the control group ( $T_i = 0$ ), you observe $Y_i(0)$ but not $Y_i(1)$ .

You can write this compactly as:

$Y_i^{obs} = T_i \cdot Y_i(1) + (1 - T_i) \cdot Y_i(0)$

The unobserved potential outcome is the counterfactual. This "missing data" structure is what makes causal inference hard.

Assumptions of the potential outcomes framework

The framework relies on several assumptions to connect potential outcomes to what you can actually estimate:

SUTVA (Stable Unit Treatment Value Assumption): Each person's potential outcomes depend only on their own treatment assignment, not on anyone else's. There's also only one version of each treatment level. (More detail in the SUTVA section below.)
Consistency: The observed outcome for someone who received treatment $t$ equals their potential outcome $Y_i(t)$ . In other words, the potential outcome notation actually maps onto reality.
No interference: One person's treatment doesn't affect another person's outcomes. (This is often folded into SUTVA.)

Individual-level causal effects

The most intuitive notion of a causal effect is at the individual level: how much did this specific treatment change this specific person's outcome?

Definition of individual-level causal effects

The individual causal effect for person $i$ is:

$\tau_i = Y_i(1) - Y_i(0)$

This is simply the difference between what happens to person $i$ under treatment and what happens under control. Notice that $\tau_i$ can vary from person to person. A drug might help some patients a lot and others not at all.

Fundamental problem of causal inference

Here's the core difficulty: you can never observe both $Y_i(1)$ and $Y_i(0)$ for the same person at the same time. Once someone takes the drug, you can't rewind time and see what would have happened without it.

This is called the fundamental problem of causal inference (a term coined by Paul Holland). For every individual, at least one potential outcome is always missing. Individual-level causal effects are therefore never directly observable.

Estimating individual-level causal effects

Because of the fundamental problem, you can't pin down $\tau_i$ for any specific person. But you can estimate average effects across groups of people. The key insight: if you randomly assign treatment, the treatment and control groups will be comparable on average, so the difference in group means estimates the average causal effect. This shifts the goal from individual effects to population-level summaries like the ATE.

Average treatment effect (ATE)

Since individual causal effects are unobservable, the average treatment effect (ATE) is the most common estimand in causal inference. It tells you the expected causal effect of the treatment across an entire population.

Definition of ATE

The ATE is defined as:

$ATE = E[Y_i(1) - Y_i(0)]$

For a finite population of $N$ individuals:

$ATE = \frac{1}{N} \sum_{i=1}^{N} \left( Y_i(1) - Y_i(0) \right)$

This is the average of all individual-level causal effects. It answers: if you could apply the treatment to everyone and also withhold it from everyone, what would the average difference in outcomes be?

ATE vs individual-level causal effects

The ATE is a summary. It smooths over individual variation. Two treatments could have the same ATE but very different distributions of individual effects. One treatment might help everyone a little; another might help half the population a lot and hurt the other half.

The practical advantage of the ATE is that it's estimable. You don't need to observe both potential outcomes for any single person. You just need groups that are comparable on average.

Estimating ATE from potential outcomes

Randomized experiments are the most straightforward approach. Random assignment makes treatment independent of potential outcomes, so the difference in group means is an unbiased estimator of the ATE.
Observational studies require additional assumptions and adjustments. Methods like propensity score weighting, matching, and regression can be used to control for confounders and approximate what randomization would have achieved.

Stable unit treatment value assumption (SUTVA)

SUTVA is one of the most important (and most easily violated) assumptions in the potential outcomes framework. Without it, potential outcomes aren't even well-defined.

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Definition of SUTVA

SUTVA has two components:

No interference: Person $i$ 's potential outcomes depend only on their own treatment, not on whether person $j$ was treated. Your outcome under treatment is the same regardless of how many other people around you are also treated.
No hidden variations of treatment: There's only one version of "treatment" and one version of "control." If different people receive slightly different forms of the treatment, the potential outcomes become ambiguous.

Together, these ensure that $Y_i(t)$ is a single, well-defined quantity for each person and treatment level.

Importance of SUTVA in causal inference

When SUTVA holds, you can cleanly link observed outcomes to potential outcomes and estimate causal effects. If it fails, the potential outcomes themselves become ill-defined or depend on the full vector of everyone's treatment assignments, which makes estimation far more complex.

Violations of SUTVA

SUTVA violations are common in practice. Recognizing them matters for study design and interpretation.

Interference (spillover effects): A vaccinated person reduces disease transmission to their unvaccinated neighbor, so the neighbor's outcome depends on the vaccinated person's treatment. Similarly, in an education study, a tutored student might share what they learned with classmates.
Treatment variation: If "the treatment" is a job training program but different sites run it differently, then there isn't a single $Y_i(1)$ . The potential outcome depends on which version of the program the person receives.

When you suspect SUTVA violations, you may need to redefine the unit of analysis (e.g., treat a whole classroom as one unit) or explicitly model the interference structure.

Rubin causal model

The Rubin causal model (RCM), developed by Donald Rubin, is the formal framework that organizes potential outcomes, treatment assignment mechanisms, and assumptions into a coherent system for causal inference.

Overview of Rubin causal model

The RCM defines causal effects as comparisons of potential outcomes and places the treatment assignment mechanism at the center of the analysis. The core question is always: how was treatment assigned, and does that assignment process allow you to identify causal effects?

Potential outcomes in Rubin causal model

In the RCM, potential outcomes are treated as fixed (non-random) attributes of each individual. What's random is the treatment assignment. The observed outcome is then determined by which treatment the person happened to receive. This perspective makes it clear that causal inference is fundamentally a missing data problem.

Assumptions of Rubin causal model

The RCM requires three assumptions for identification of causal effects:

SUTVA: No interference between units and a single version of each treatment.
Ignorability (unconfoundedness): Conditional on observed covariates $X$ , treatment assignment is independent of potential outcomes: $T_i \perp (Y_i(1), Y_i(0)) \mid X_i$ . In a randomized experiment, this holds by design. In observational studies, it's an assumption you need to argue for.
Positivity (overlap): Every individual has a nonzero probability of receiving each treatment level: $0 < P(T_i = 1 \mid X_i) < 1$ . If some subgroup always or never receives treatment, you can't estimate causal effects for that subgroup.

Neyman-Rubin causal model

The Neyman-Rubin causal model integrates Jerzy Neyman's randomization-based inference with Rubin's potential outcomes framework. It's especially relevant for analyzing randomized experiments.

Comparison to Rubin causal model

The Neyman-Rubin model shares the same potential outcomes setup but emphasizes randomization as the basis for inference rather than relying on modeling assumptions. Where the broader Rubin framework accommodates observational studies through ignorability assumptions, the Neyman-Rubin approach draws its inferential power directly from the known randomization procedure.

It also focuses on finite population inference, meaning it targets the ATE for the specific group of individuals in the study, not a hypothetical superpopulation.

Randomization in Neyman-Rubin causal model

Randomization does the heavy lifting here. When you randomly assign treatment:

Treatment is guaranteed to be independent of potential outcomes (ignorability holds by design)
You don't need to model the outcome or control for confounders
The randomization distribution itself provides the basis for hypothesis tests and confidence intervals

This is why randomized experiments are considered the gold standard for causal inference.

Estimating causal effects in Neyman-Rubin model

The standard estimator is the difference in sample means:

$\hat{ATE} = \bar{Y}_1 - \bar{Y}_0$

where $\bar{Y}_1$ is the average outcome in the treatment group and $\bar{Y}_0$ is the average outcome in the control group. Under random assignment, this is an unbiased estimator of the ATE.

To quantify uncertainty, you can use:

Randomization inference (permutation tests): Assess what the difference in means would look like under the null hypothesis by considering all possible random assignments
Neyman's conservative variance estimator: Provides standard errors that account for sampling variability, though they tend to be slightly conservative because individual treatment effects are unobservable

Applications of counterfactuals

The potential outcomes framework isn't just theoretical. It structures how researchers across many fields design studies and evaluate interventions.

Counterfactuals in policy evaluation

Policy evaluation asks: did this policy cause the observed change in outcomes? The counterfactual is what would have happened without the policy. For example, to evaluate a minimum wage increase on employment, you'd compare employment levels after the increase to what employment would have been without it. Methods like difference-in-differences and regression discontinuity are designed to construct credible counterfactuals in these settings.

Counterfactuals in medical research

Randomized controlled trials (RCTs) are the standard in medicine precisely because they create clean counterfactual comparisons. Patients are randomly assigned to treatment or placebo, so the placebo group's outcomes serve as the counterfactual for the treatment group. When RCTs aren't feasible (due to ethical or practical constraints), observational methods grounded in the potential outcomes framework are used instead.

Social scientists use counterfactual reasoning to evaluate programs like job training, early childhood education, or crime prevention strategies. The challenge is that random assignment is often difficult in these contexts, so researchers rely on quasi-experimental designs (instrumental variables, natural experiments, matching) to approximate the counterfactual comparison that a randomized experiment would provide.

📊Causal Inference Unit 2 Review