[ex3xu]

You're right that it's a low sample size, and the authors themselves do warn that diagnostic rates of ASD in children may be inflated:

> With the increasing reported frequency of ASD in the pedi-atric population one cannot rule out a possible coincidence of the diagnosis of ASD in patients with propionic acidemia.

Regarding the calculation, Wikipedia says 1 in 35000 for PA the US, rather than 3500, and the researchers use an incidence rate of .5% for autism rather than 1%. So the number of people is 1000, not 20000. The researchers say they're calculating off a 5/8 ASD/PA incidence rate, so the calculation is something like (1000/7 billion)^5*(some negligible amount)^3.

[gus_massa]

* 1000 cases in the word instead of 20000

OK. I'm fine with that. My idea was to show that if we assume independence, it is not difficult that someone can find 12 cases with both illness.

* About the calculation:

Is that in the paper? [I assume that "some negligible amount" means "close to 1".]

This calculation is the probability that if you pick 12 persons at random from the whole word population you find a group with 5 person with PA+ASD and 3 PA+dubiousASD and 4 with PA+noASD. There is a missing (1-epsilon)^4 that doesn't matter. And you need to add some combinatorial number, like (12! / 5! 3! 4!) ~= 30000.

Even assuming that "1-epsilon" and "some negligible amount" are equal to 1, the result of your calculation is 6E-35 that is much much much smaller than the reported result 4E-13 (after using the combinatorial number you get or 2E-30 that is still smaller). So they clearly used another calculation.

Anyway, your calculation is wrong because you can't use the probabilities of the general population in a calculation of a group that you cherrypicked.

[ex3xu]

They mention the 5 ASD out of 8 PA sample size, and they explicitly cite the .005 rate of ASD, but I think their PA incidence number is different than the one available on Wikipedia, and I couldn't find it in the paper so that may be where the discrepancy is from. I may be wrong but I don't think we need the combinatorics for this particular calculation since their calculation is trying to disprove that these events are independent rather than calculate the joint conditional probability. Obviously I can't comment on whether any cherry picking was done, or if these just happen to be the available sample of available children with this rare metabolic disorder. I'd agree and I think the authors of the study would agree that the sample size is too small for any definitive conclusions, but I think the body of evidence when considering the rest of the available PPA studies make a reasonable case for it to be a finding of interest.