illustration (attribution, if any possible, is at the end of the article)
When you prescribe some medication to a patient, (s)he doesn't automatically comply.
For example, due to adverse side-effects, a patient might reduce the prescribed dosage. For example, due to impatience, a patient seeing no effect might simply drop the treatment altogether. Others might sometimes forget to take the medication: among them, some might simply continue the next day as if nothing happened (maybe they didn't even realise that they forgot), others might try to 'compensate' by doubling the dose the next day. Et caetera.
So to assess if the medication is effective, you might be in trouble. You can hardly estimate whether taking the medication helps with healing if you don't know for sure whether the medication was taken or not as prescribed!
A possibility is to move back up the causal chain: estimate whether prescribing the medication seems to support healing. Rely on the "intention to treat", rather than the absorption of the treatment.
You might need to assume that a reasonable proportion of patients do comply with the prescription, but that's not necessarily too difficult an assumption to make if there are good reasons to suspect the patients would be happy to heal.
There's noise in the data, sure, but at least you can now be reliable in measuring the end points (the prescription and the healing) a lot more easily than checking on all patients, every day, to see whether they did comply on that day or not (assuming they don't lie to you…).
Would the increase in reliability more than compensate the 'noise' taken in?
One difficulty is that, now that you don't measure compliance, all the patients that heal after prescription will count as positive evidence that the prescription works.
So what if the prescription is actually causing horrible side-effects and those who survive are those "smart enough" to stop taking it in time? You're now counting as "positive evidence that the prescription works" the very people who didn't comply, the very people who prove that the medication doesn't work! You're now counting "spontaneous healing in spite of the medication" as evidence in favour of the medication!
The only way to go is to add assumptions and/or measures.
For example, even if "reported compliance" might include lies, it might be still reasonably correlated to actual compliance… giving you some useful information, if not 100% reliable. Can you estimate this correlation?
For example, you might need to assume that the effects on one patient don't affect the effects on another patient. Placebo and nocebo effects are real! If a patient 'sees' another patient improve, (s)he might get convinced that the medication is effective and strengthen the placebo effect [even if the improvement of the other had nothing to do with the medication!]. Vice-versa, seeing others not improve might convince a patient that the medication is ineffective, leading to e.g. lower compliance… even though the observer is the one patient who would benefit the most from following the prescription to the letter.
Assessing causality is trickier than it seems. Correlation is not causation, but sometimes it nonetheless indicates causation if some other assumptions stand! "Correlation is not causation" is not always true, because it's context-independent and some contexts are in fact favourable to causal identification.