The Mathematics of Robison's Males

I'm not sure if anyone has remarked on this yet (someone should have by now—it's pretty obvious), but John Elder Robison's observation that most families with an autistic child have more boys than girls is almost certainly true. It's not a mystery, it's simple mathematics.

Start with the assumption that on average every newborn has a 50% chance of being male and a 50% chance of being female. Then add the assumption that on average every autistic newborn has an 80% chance of being male and a 20% chance of being female. In the absence of other information, the following two scenarios will emerge as mathematical results:

• Given that a family has at least one autistic child, the family will likely have more males than females.
• Given that a family has no autistic child, the family will likely have more females than males.

The effect is more pronounced (and thus more readily observable) in the first scenario. But this gets balanced by the fact that the second scenario is far more common.

Note that when we ask questions such as what is the male/female ratio in families given that at least one child has autism, we are in the land of conditional probabilities. A 50/50 male/female ratio is fine if there are no other conditions, but by saying that the family has (or does not have) a child with autism, you are creating conditional probabilities, and in this case those conditional probabilities do not come out to be 50/50.

It's easiest to see the phenomenon in families that have at least one child with autism. Some of those families will have only one child. In that case, we know that the one child has autism and so 80% of those families will have more males than females and 20% of those families will have more females than males.

With two children, we have two cases to consider. In one case, one of the children has autism (80% male/20% female) and one does not (roughly 49.7% male/50.3% female). Put those odds together and you will find that about 39.8% of such families have more males than females, 50.1% have equal sex distribution, and 10.1% have more females than males. The other case is that both children have autism, and in this case it comes out as 64% have more males than females, 32% have equal sex distribution, and 4% have more females than males.

For families with more than two children with at least one of them having autism, the calculations will be similar with similar results: the odds will always be greater that the family has more male children than female children. Again, there is nothing mysterious about this outcome, it's just simple mathematics.

In contrast to all these scenarios, we will have the situations where the family does not have a child with autism. In all those instances, the results will skew slightly towards having more females in the family than males. The numbers aren't as dramatic in these scenarios, but of course we have to keep in mind that these families are far more common than the families who have at least one autistic child.

This example is a good reminder that science is based on a foundation of logic and mathematics. I see an awful lot of scientific work that appears to be fine as far as the scientific technique goes but is utterly abysmal when it comes to the underlying logic and mathematics. In other words, even if you do RCTs out the ass, if the work is still based on lousy logic and lousy mathematics, then the science is going to be lousy too.