Wednesday, June 10, 2009

And People Forget Bayes Once Again

There's been a lot of psychological research that boils down to one simple statement: people without statistical training don't get Bayes's Theorem... and even people with statistical training tend to ignore its implications in day-to-day life. Put another way, people ignore base-rate data when evaluating probability.

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.


  1. You're absolutely right that the majority of the screen-positive children would not go on to be diagnosed with autism.

    I estimate it would be about 12%. But, at this level of ascertainment, a screen-positive result would give you an odds ratio of about 12.0, which is very big.

    That's a comparable or bigger odds ratio than, say, being a sibling of an autistic person. There's probably no other recognized factor of any kind with a bigger odds ratio. So they would consider this type of screening "useful."

    The ideal gap, which would be 100% matches and 0% false positives, is probably an impossibility.

    Daedalus2u argues that the false negative rate is what matters, because of liability.

  2. Actually, the odds ratio would be more like 28.5.

  3. That calculation, as I said, was based on the percentages you gave and the commonly cited 1/150 prevalence rate. I just plugged them into Bayes's theorem.

  4. It won't be long before 1% is considered the consensus prevalence. But using Bayes Theorem I'm sure is unintuitive to follow for most people.

    Easier method. Say there are 100,000 screened children. Of these 1,000 go on to be diagnosed with autism. 600 of the 1,000 are screen-positive. But 5% (4950) of the remaining 99,000 are also screen-positive. In total, there are 5,550 screen-positive children, of whom 600 (10.8%) go on to be diagnosed with autism.

    There are 94,450 screen-negative children, though. Of these, only 400 (0.42%) will go on to be diagnosed with autism.

    With this hypothetical test, they could say a child has a 10% chance to be diagnosed with autism later in life vs. 0.42% chance (OR=23.8). There's nothing in the world of autism that even remotely approaches this. There's no question researchers would be falling over themselves if they had a test this accurate.