CIS Primer Question 3.3.3

Here are my solutions to question 3.3.3 of Causal Inference in Statistics: a Primer (CISP). \(\DeclareMathOperator{\do}{do}\)

The drug you have been assigned determines which ward you go to. Whether you get a lollipop is determined by which ward you go to and whether you show signs of depression. Depression is a symptom of certain risk factors. These risk factors, together with the drug you have been assigned, determine your capacity for recovery.

A causal model for the lollipop situation.
A causal model for the lollipop situation.

Since lollipop is a collider in this diagram, there are no backdoor paths from drug to recovery. In other words, it is not necessary to condition on any variables to estimate the causal effect of drug on recovery. In this case, \(\mathbb P (Y \mid \do(X)) = \mathbb P(Y \mid X)\).

If the nurse were to give out the lollipops in the day after the study, there would be no difference in the causal diagram.