>
Since they clearly could alter the schedule, offering a limited number of later slots and comparing results would seem like the prudent response.There's a difference between a doctor entertaining a medically-irrelevant suggestion from a patient (or patient's family), vs. assuming that the subsequent improvement was related to it, and then making that decision for some other patients (or suggesting it to them). The former is being accommodating, the latter is making treatment changes without good reason.
Improvement or no change aren't the only two possible outcomes for a patient. They could also get worse. What's worse, often neither improvement nor decline are obviously related to the treatment, or treatment changes.
Maybe it's the circadian rhythm thing. Maybe it's some delayed effects of something unrelated about the patient, that just coincided with your intervention. Maybe it's just a response to a change - any change. Or maybe it's just completely random. The point is, you don't know. You might feel like you do, or maybe it really looks obvious - but from N=1 you don't actually know, not enough to potentially bet other people's health on it.
Because maybe you do go ahead, and make a schedule change to another few patients - and few days later, suddenly and for no apparent reason, one of them goes into critical condition and dies soon after. Good luck convincing the grieving family, your colleagues, the board - and your own conscience - that the schedule change could not have possibly caused this. You won't, because you don't actually know.