I may be over thinking this, but in the lecture the explanation about probabilities for counterfactual outcomes makes sense but in the HW you provide a continuous outcome. any suggestions for how to think about adapting these variables to the formulas you provided? perhaps a change threshold?
thank you
In the formula for counterfactuals, you simply have “a” which is the level of the exposure, i.e A=a, and Y which is the value of the outcome on its scale. So if the outcome is binary, as in the slides, Y either equal 1 or 0. If it is continuous, Y=c, with c within the range of all the possible (continuous) values the outcome can take.In the formula for average causal effect, you would take the average (the expectation) of the potential outcomes, i.e E[Y] instead of P[Y]. Note that for a binary outcome that can only value 1 or 0, P[Y] = E[Y].A change threshold, as you are suggesting, sounds like a dichotomization of the outcome, which is definitely a possibility but changes your outcome and therefore your search question.
I am not following the explanation. I have the R class during your office hours this week. are there any examples you can point me to in text books or past problem sets to give me help with this?
Hi Laura,
Sorry for the delay in response. The TA's and I discussed. We are not sure we are understanding your questions correctly and encourage you to come to office hours. Francois has also made an example which I've asked him to post.
In the interim, here's another way to explain, in case it helps.
In the real world, we only ever can observe an outcome (binary or continuous, doesn't matter) in a single person under ONE treatment (or exposure) condition. The counterfactual framework helps us imagine that in another world, you can observe different outcomes under different treatment or exposure conditions. If one could do that, then you could compare outcomes, in the same person, under different treatment conditions.
(BTW - in case this helps - to me, this makes me think about that scene in Marvel Infinity Wars 1, where Dr. Strange says he's viewed 14 million+ different scenarios on how the war with Thanos could turn out, and in only one does the human race survive... in case that helps... I think the "multiverse concept" is a good layman's way to think about the counterfactual framework)
If your outcome is SBP (continuous), and you are interested in giving a drug A or not giving drug A, then in the counterfactual framework, you would be able to observe each individual's SBP if they had A, or if they didn't. So, maybe for person 1, if A=0 her SBP is 105; and if A=1, then her SBP is 100. In the counterfactual world, you would be able to observe these 2 blood pressures for EACH person, under those two conditions, treated vs. not, then could compare the average SBP, under the two treatment groups (in the counterfactual world).
Does that help? if not, please try to come to office hours. I may also try to be in class today closer to 1, in case we have time to chat then.
best,
JMC
Hi all,
Please find an example attached. Even with a continuous outcome (SBP), you can calculate the individual causal effect (difference of counterfactual under exposure (drug A=1) and not (no drug A=0). The average gives you then the average causal effect.
Happy to talk more about it in class. I should also be there around 1pm so feel free to come early if clarification is needed,
Cheers,
Francois
