I was reminded rather strongly of this by a comment on Natural Variation. To wit:
"There's probably no genotype that exactly matches what we call autism. But the phenotype is highly heritable, apparently, so in principle it should be possible to find a genotype->phenotype mapping that is more convincing, e.g. 60% matches and 5% false positives. When they come up with a gap of this size, they'll probably start seriously talking about genetic screening."
If we develop a prenatal test that correctly identifies autistic children 60% of the time and delivers a false positive 5% of the time, what are the odds of a foetus developing into an autistic child given that the test returned a positive?
If you ask most people, they'd scratch their heads in confusion. Many would settle on 95%, some would settle on 60%, and a few would answer with a figure slightly over 92% (60% divided by the sum of 60% and 5%, or 60/65).
The actual answer, however, is none of these... for the simple reason that there are a lot more non-autistic children than autistic children. That five percent would be effectively multiplied by all of the non-autistic children it was used to test.
That answer depends highly on prevalance estimates, but let's not go there. For the sake of simplicity, I'm going to use the popularized 1/150 figure. Suffice it to say, however, that the prevalence of autism depends on how you define the word "autism"... and a lot of other things.
This gives us:
- P(A), or the probability of a foetus developing into an autistic child without testing, as 1/150, or about .6%.
- P(A'), or the probability of a foetus developing into a non-autistic child without testing, as 149/150, or about 99.3%.
- P(BA), or the probability of an autistic foetus getting a positive test result, as 60%.
- P(BA'), or the probability of a non-autistic foetus getting a positive test result, as 5%.
- P(B), or the probability of a randomly-selected foetus, autistic or not, getting a positive test result. This can be calculated from the above, as P(BA)P(A)+P(BA')P(A'). Doing the math, this works out to 161/3000, or about 5.37%.
Given that last figure, you should see where this is going.
In the end, the probability of a positive result on the test above correctly indicating an autistic child is a spectacular 12/161... or about 7.5%. The other 92.5% of the time... the test would be indicating that a non-autistic child is autistic.