# Is individual causal effect identifiable when there is no unmeasured confounder?

From the first section of the causal inference book by Hernan and Robins (link:https://cdn1.sph.harvard.edu/wp-content/uploads/sites/1268/2018/12/hernanrobins_v1.10.37.pdf), I read that the individual causal effect is generally not identifiable. However, the authors do mention that the crossover randomized experiment allows the identification of individual causal effects.

My question is, in observational setting or randomized controlled trial (both without the crossover design), if there is no unmeasured confounder, can we identify the individual causal effects?

• The linked article says " individual causal effects cannot be identified" just before the section 1.2. Jan 4, 2019 at 1:29

Individual effects are identifiable depending on the causal assumptions you can sustain and the type of data you can collect. In fact, we make inferences of individual causal effects all the time: if you give a glass of an unknown chemical to a perfectly healthy person, and the person suddenly dies, most people would agree that it is a reasonable inference that the chemical was the cause of death of that person (which could be later confirmed by autopsy). Can you tell what is the assumption that allows you to make this inference?

The bottom line is that identification of causal effects, be it average effects, conditional effects or individual effects depends on causal assumptions (see here and here). The difference between identifying average effects and individual effects is the strength of assumptions that you need to make. While you can make inference of average effects with just mild qualitative assumptions --- such as a DAG, with no need for parametric information about the form of the functions or distribution of error terms --- inference about individual causal effects require more details. See for instance, this answer.

Here let me give you two examples: in a linear causal model, all individual treatment effects are identified --- linearity implies constant effects, so the individual treatment effect equals the average treatment effect. Another example, related to the fine point in Hernan's book: if you treat the same individual more than once with a reversible treatment, and if you assume that the individual does not change during time periods, you can also estimate the treatment effect for that individual. So, as you can see, it all depends on the causal assumptions you can sustain and the type of data you can collect.

• in the answer you cited link, the unmeasured factor u makes the individual causal effect unidentifiable. In my understanding, u is an unmeasured confounder. It seems to me that if we measure all confounders, the individual causal effect is alway identifiable. Is this right? Jan 9, 2019 at 22:07
• @Hammer.Wang no, $u$ is not an unmeasured confounder, it is an unobserved factor that interacts with $x$ but does not confound $x$ (in the example $x$ was randomized, so it is not confounded). Jan 10, 2019 at 1:43

The short answer here is yes. If we are in a hypothetical world where we have perfect information over all confounding factors, then we can recover the true causal relationship between two variables.

However, this is almost never true. And this is why many social sciences -- economics being probably the best example -- employ quasi-experimental designs to recover causal relationships.

edit -- sorry, I overlooked that you asked specifically about individual causal effects. What I wrote is true if one is happy with recovering the ATE.

• (-1) This is incorrect, you need more information than knowing all confounding factors, see a counterexample here (stats.stackexchange.com/questions/379799/…) where you have a randomized trial (no confounders) yet the individual causal effect (counterfactual) is not identifiable. Jan 4, 2019 at 5:45
• ....the entire point of a randomized control trial is that you randomly assign treatment and control. And so you have a counterfactual. Should i specify that one needs variance in data? Thst seems stupid. He asked about recovering a causal relationship. The answer is clearly yes.
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Jan 4, 2019 at 6:22
• 123 you cannot identify individual effects just with a randomized trial, what you get is an average treatment effect. The individual effects can be very different from the average, see the example in the previous link, where the average effect is zero but each individual effect is not zero. Jan 4, 2019 at 6:27
• @CarlosCinelli -- overlooked that the OP was specifically interested in individual causal effects. What I wrote is true of the ATE. edited my answer. Thanks.
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Jan 4, 2019 at 6:44