I'm starting to read Causal Inference in Statistics, A Primer by Judea Pearl et. al. I have a masters in math, but I have never taken a statistic course. I'm a bit confused by one of the early study questions, and there's no one I can ask about it, so I'm hoping that someone on this site will critique my answers for me. (This is not a homework problem. I'm a retiree, just keeping my mind active.) Note that there are no specific data given in the problems.
a)There are two treatments for kidney stones, Treatment A and Treatment B. Doctors are more likely to prescribe Treatment A on large (and therefore more severe) stones and more likely to prescribe Treatment B on small stones. Should a patient who doesn't know the size of his or her stone examine the general population data, or the size-specific data when determining which treatment will be more effective?
b)There are two doctors in a small town. Each has performed 100 surgeries in his career, which are of two types: one very easy surgery and one very difficult surgery. The first doctor performs the easy surgery much more often than the difficult surgery and the second performs the difficult surgery more often than the easy surgery. You need surgery, but you don't know if your case is easy or difficult. Should you consult the success rate of each doctor over all cases, or should you consult the success rates for the easy and difficult cases separately, to maximize the chance of a successful surgery?
As to part a) it's reasonable to suppose that there are drawbacks to Treatment A as compared to Treatment B, or why isn't it prescribed all the time? So, it seems to me that I can't make an intelligent decision without knowing the size of my kidney stone. I would expect the data to show Treatment A to be more effective on large stones, and at least as effective on small stones, but I wouldn't want to assume the presumed risks of Treatment A if my stone is small. Assuming that small stones can almost always be treated successfully, I would expect Treatment B to show a higher success rate in the general population, but I wouldn't want to adopt Treatment B if I have a large stone.
It seems to me that the data are useless unless I know the size of my stone. Is this the answer to the question, perhaps? The whole thing seems rather pointless, because I can't go into the pharmacy and buy either treatment over the counter. My doctor will prescribe it, and if he can't (or won't) tell me the size of the stone, I will change doctors.
As for part b) it's clear that you want to look at the rates for the procedures separately, but the rates alone aren't enough. Suppose the first doctor has performed the difficult surgery just once, with a successful outcome, and the second doctor has performed it 37 times, with 35 successes. I would be awfully inclined to go with the second doctor, but I'd want to how 35 out of 37 compares to national norms, and also if the 2 failures occurred early in his career (while he was still learning) or more recently (after he started drinking heavily).
Is this sort of discussion what is called for by the problems, or is a more cut-and-dried answer expected? If I'm lucky enough to have an instructor read this, how would you grade my answer?