# Clarification on Counterfactual Outcomes in Causal Inference

I’m studying the textbook Causal Inference: What If by Miguel A. Hernán, James M. Robins. On page 4, I came across a passage that seems nonsensical. The authors claim that, for each individual, the counterfactual outcome corresponding to the treatment they actually received is equal to the observed outcome:

To make our causal intuition amenable to mathematical and statistical analysis we will introduce some notation. Consider a dichotomous treatment variable $$A$$ (1: treated, 0: untreated) and a dichotomous outcome variable $$Y$$ (1: death, 0: survival). In this book we refer to variables such as $$A$$ and $$Y$$ that may have different values for different individuals as random variables. Let $$Y^{a=1}$$ (read $$Y$$ under treatment $$a = 1$$) be the outcome variable that would have been observed under the treatment value $$a = 1$$, and $$Y^{a=0}$$ (read $$Y$$ under treatment $$a = 0$$) the outcome variable that would have been observed under the treatment value $$a = 0$$. $$Y^{a=1}$$ and $$Y^{a=0}$$ are also random variables. Zeus has $$Y^{a=1} = 1$$ and $$Y^{a=0} = 0$$ because he died when treated but would have survived if untreated, while Hera has $$Y^{a=1} = 0$$ and $$Y^{a=0} = 0$$ because she survived when treated and would also have survived if untreated.

We can now provide a formal definition of a causal effect for an individual: The treatment $$A$$ has a causal effect on an individual’s outcome $$Y$$ if $$Y^{a=1} \neq Y^{a=0}$$ for the individual. Thus, the treatment has a causal effect on Zeus’s outcome because $$Y^{a=1} = 1 \neq 0 = Y^{a=0}$$, but not on Hera’s outcome because $$Y^{a=1} = 0 = Y^{a=0}$$. The variables $$Y^{a=1}$$ and $$Y^{a=0}$$ are referred to as potential outcomes or as counterfactual outcomes. Some authors prefer the term “potential outcomes” to emphasize that, depending on the treatment that is received, either of these two outcomes can be potentially observed. Other authors prefer the term “counterfactual outcomes” to emphasize that these outcomes represent situations that may not actually occur (that is, counter-to-the-fact situations).

For each individual, one of the counterfactual outcomes—the one that corresponds to the treatment value that the individual did receive—is actually factual. For example, because Zeus was actually treated $$(A = 1)$$, his counterfactual outcome under treatment $$Y^{a=1} = 1$$ is equal to his observed (actual) outcome $$Y = 1$$. That is, an individual with observed treatment $$A$$ equal to $$a$$, has observed outcome $$Y$$ equal to his counterfactual outcome $$Y^a$$. This equality can be succinctly expressed as $$Y = Y^a$$ where $$Y^a$$ denotes the counterfactual $$Y^a$$ evaluated at the value $$a$$ corresponding to the individual’s observed treatment $$A$$. The equality $$Y = Y^a$$ is referred to as consistency.

This seems nonsensical to me because, by definition, a counterfactual outcome is what would have happened if the opposite treatment had been received. Are the authors correct here (am I misunderstanding)? Or should I be looking for another textbook?

## 1 Answer

Different fields sometimes adopt different terminology for the same concepts. This can be very annoying when you read papers from other fields, so I might be biased in favour of “potential outcomes” simply because that’s the term used within my own field (economics).

That being said, the third paragraph

For each individual, one of the counterfactual outcomes—the one that corresponds to the treatment value that the individual did receive—is actually factual...

does demonstrate that “counterfactual outcomes” is a rather awkward choice which can lead to misunderstandings.

If you can look past this, I’ve heard that it is a good book.

Edit: A brief explanation of “potential outcomes”. For a binary treatment, each individual has two potential outcomes. One without treatment ($$Y_{0i}$$) and one with treatment ($$Y_{1i}$$). We will ever only observe one outcome ($$Y_{i}$$). If the individual is treated, $$Y_i = Y_{1i}$$; if not treated $$Y_i = Y_{0i}$$.

In their example, Jupiter (only epidemiologists say Zeus) would die if treated ($$Y_{1i}=1)$$ but survive if untreated ($$Y_{0i}=0$$), making the treatment effect on Jupiter $$Y_{1i}-Y_{0i} = 1$$. Again, this individual treatment effect we will never be able to observe, but using this notation, it is possible to show that when, e.g., treatment is randomly assigned the difference in means between treatment and control group will yield $$E[Y_{1i}-Y_{0i}]$$

• Wait, but I still don’t understand what the authors are trying to say here. Are they defining the “counterfactual” outcome as actually being the “factual” outcome? I’m very confused. Commented Aug 15 at 18:18
• They are using “counterfactual outcome” synonymously with “potential outcome”. The factual outcome will always be one of the two potential outcomes. Commented Aug 15 at 20:14
• I also found the "counterfactual outcome" terminology extremely confusing until I realized that only one of the 2 outcomes was counterfactual. Nice summary (even though I prefer Zeus). (+1)
– EdM
Commented Aug 15 at 20:39
• Even "potential outcomes" isn't ideal for the actual outcome. I've seen "conditional outcomes" in philosophy writing about these sorts of things, which would be better except that stats causal people really want to distinguish probabilities of conditional outcomes from observable conditional probabilities. Commented Aug 15 at 22:58