In an attempt to estimate the effectiveness of a new drug, a randomized experiment is conducted. In all, 50% of the patients are assigned to receive the new drug and 50% to receive a placebo. A day before the actual experiment, a nurse hands out lollipops to some patients who show signs of depression, mostly among those who have been assigned to treatment the next day (i.e., the nurse’s round happened to take her through the treatment-bound ward). Strangely, the experimental data revealed a Simpson’s reversal: Although the drug proved beneficial to the population as a whole, drug takers were less likely to recover than nontakers, among both lollipop receivers and lollipop nonreceivers.
Assume lollipop sucking in itself has no effect on recovery. Is taking the drug beneficial?
I understand why the subgroup of drug takers that received lollipops could have a lower rate of recovery than placebo takers who didn't receive lollipops; it's because they are more likely to suffer from depression in the first place, which affects recovery rates.
But how would you explain why the subgroup of drug takers who didn't receive lollipops could have a lower rate of recovery than placebo takers who didn't receive lollipops?