The goal here is to distinguish between these three causal graphs:
(A) Pure placebo
X --> belief
/
|
v
Effect
(B) Direct effect
X --> belief
\
|
v
Effect
(C) Both direct and placebo effects
X --> belief
\ /
| |
v v
Effect
Unfortunately, you cannot possibly distinguish these if you only measure X and the effect, without either observing or controlling peoples’ belief (such as by giving them placebos so they all believe they’re getting X, or asking them survey questions about whether they think X will work). There are some theorems about which causal graphs can and can’t be distinguished from which types of observations in Pearl’s book Causality.
The goal here is to distinguish between these three causal graphs:
Unfortunately, you cannot possibly distinguish these if you only measure X and the effect, without either observing or controlling peoples’ belief (such as by giving them placebos so they all believe they’re getting X, or asking them survey questions about whether they think X will work). There are some theorems about which causal graphs can and can’t be distinguished from which types of observations in Pearl’s book Causality.