He had a sudden cardiac arrest. How does this change the probability that he has a particular genetic condition?

Megan McArdle writes:

I have a friend with a probability problem I don’t know how to solve. He’s 37 and just keeled over with sudden cardiac arrest, and is trying to figure out how to assess the probability that he has a given condition as his doctors work through his case. He knows I’ve been sharply critical of doctors’ failures to properly assess the Type I/Type II tradeoff, so he reached out to me, but we quickly got into math questions above my pay grade, so I volunteered to ask if you would sketch out the correct statistical approach.

To be clear, he’s an engineer, so he’s not asking you to do the work for him! Just to sketch out in a few words how he might approach information gathering and setting up a problem like “given that you’ve had sudden cardiac arrest, what’s the likelihood that a result on a particular genetic test is a false positive?”

My reply:

I agree that the conditional probability should change, given the knowledge that he had the cardiac arrest. Unfortunately it’s hard for me to be helpful here because there are too many moving parts: of course the probability of the heart attack conditional on having the condition or not, but also the relevance of the genetic test to his health condition. This is the kind of problem that is addressed in the medical decision making literature, but I don’t think I have anything useful to add here, beyond emphasizing that the calculation of any such probability is an intermediate step in this person’s goal of figuring out what he should do next regarding his heart condition.

I’m posting the question here, in case any of you can point to useful materials on this. In addition to the patient’s immediate steps in staying alive and healthy, this is a general statistical issue that has to be coming up in medical testing all the time, in that tests are often done in the context of something that happened to you, so maybe there is some general resource on this topic?

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