Definition of Causal Effects

Before we can estimate causal effects, we must first define precisely what we mean by a "causal effect." This may seem pedantic, but vague definitions lead to vague conclusions. The potential outcomes framework provides the mathematical precision we need.

From Intuition to Formalism

Intuitively, we understand that a treatment has a causal effect when it changes an outcome. If taking aspirin reduces your headache, we say aspirin caused the headache to go away. But how do we formalize this intuition in a way that allows for rigorous analysis?

The key insight is to think in terms of what would happen under different scenarios. For any individual, we can conceptualize:

1. The outcome they would experience if treated
2. The outcome they would experience if not treated

The difference between these two outcomes is the causal effect of treatment for that individual.

"The causal effect is defined as a contrast of the outcomes under treatment and under no treatment." — What If, Chapter 1

Why Definitions Matter

Consider this seemingly simple question: "Does the new drug reduce mortality?" Without a precise definition, this question is ambiguous:

• Reduce mortality compared to what? (Placebo? Standard care? No treatment?)
• Reduce mortality for whom? (All patients? A subgroup? The average patient?)
• Reduce mortality by how much? (Absolute reduction? Relative reduction?)

The potential outcomes framework forces us to be explicit about each of these elements, leading to clearer research questions and more interpretable results.

The Foundation: Potential Outcomes

We denote an individual's potential outcomes using superscript notation:

• $Y^{a=1}$ or simply $Y^1$: the outcome that would occur if the individual received treatment
• $Y^{a=0}$ or simply $Y^0$: the outcome that would occur if the individual did not receive treatment

These potential outcomes exist conceptually for every individual, regardless of which treatment they actually receive. This is the foundation upon which all causal effect definitions rest.