Additional thoughts on the Ted Hill paper

First, I’d like to thank the large number of commenters on my previous post for keeping the discussion surprisingly calm and respectful given the topic discussed. In that spirit, and to try to practise the scientific integrity that I claimed to care about, I want to acknowledge that my views about the paper have changed somewhat as a result of the discussion. My understanding of the story of what happened to the paper has changed even more now that some of those attacked in Ted Hill’s Quillette article have responded, but about that I only want to repeat what I said in one or two comments on the previous post: that my personal view is that one should not “unaccept” or “unpublish” a paper unless something was improper about the way it was accepted or published, and that that is also the view of the people who were alleged to have tried to suppress Ted Hill’s paper on political grounds. I would also remark that whatever happened at NYJM would not have happened if all decisions had to be taken collectively by the whole editorial board, which is the policy on several journals I have been on the board of. According to Igor Rivin, the policy at NYJM is very different: “No approval for the full board is required, or ever obtained. The approval of the Editor in Chief is not required.” I find this quite extraordinary: it would seem to be a basic safeguard that decisions should be taken by more than one person — ideally many more.

To return to the paper, I now see that the selectivity hypothesis, which I said I found implausible, was actually quite reasonable. If you look carefully at my previous post, you will see that I actually started to realize that even when writing it, and it would have been more sensible to omit that criticism entirely, but by the time it occurred to me that ancient human females could well have been selective in a way that could (in a toy model) be reasonably approximated by Hill’s hypothesis, I had become too wedded to what I had already written — a basic writer’s mistake, made in this case partly because I had only a short window of time in which to write the post. I’m actually quite glad I left the criticism in, since I learnt quite a lot from the numerous comments that defended the hypothesis.

I had a similar experience with a second criticism: the idea of dividing the population up into two subpopulations. That still bothers me somewhat, since in reality we all have large numbers of genes that interact in complicated ways and it is not clear that a one-dimensional model will be appropriate for a high-dimensional feature space. But perhaps for a toy model intended to start a discussion that is all right.

While I’m at it, some commenters on the previous post came away with the impression that I was against toy models. I agree with the following words, which appeared in a book that was published in 2002.

There are many ways of modelling a given physical situation and we must use a mixture of experience and further theoretical considerations to decide what a given model is likely to teach us about the world itself. When choosing a model, one priority is to make its behaviour correspond closely to the actual, observed behaviour of the world. However, other factors, such as simplicity and mathematical elegance, can often be more important. Indeed, there are very useful models with almost no resemblance to the world at all …

But that’s not surprising, since I was the author of the book.

But there is a third feature of Hill’s model that I still find puzzling. Some people have tried to justify it to me, but I found that either I understood the justifications and found them unconvincing or I didn’t understand them. I don’t rule out the possibility that some of the ones I didn’t understand were reasonable defences of this aspect of the model, but let me lay out once again the difficulty I have.

To do this I’ll briefly recall Hill’s model. You have two subpopulations and of, let us say, the males of a species. (It is not important for the model that they are male, but that is how Hill hopes the model will be applied.) The distribution of desirability of subpopulation is more spread out than that of subpopulation , so if the females of the species choose to reproduce only with males above a rather high percentile of desirability, they will pick a greater proportion of subpopulation than of subpopulation .

A quick aside is that what I have just written is more or less the entire actual content (as opposed to surrounding discussion) of Hill’s paper. Of course, he has to give a precise definition of “more spread out”, but it is very easy to come up with a definition that will give the desired conclusion after a one-line argument, and that is what he does. He also gives a continuous-time version of the process. But I’m not sure what adding a bit of mathematical window dressing really adds, since the argument in the previous paragraph is easy to understand and obviously correct. But of course without that window dressing the essay couldn’t hope to sell itself as a mathematics paper.

The curious feature of the model, and the one that I still find hard to accept, is that Hill assumes, and absolutely needs to assume, that the only thing that can change is the sizes of the subpopulations and not the distributions of desirability within those populations. So if, for example, what makes a male desirable is height, and if the average heights in the two populations are the same, then even though females refuse to reproduce with anybody who isn’t unusually tall, the average height of males remains the same.

The only way this strange consequence can work, as far as I can see, is if instead of there being a gene (or combination of genes) that makes men tall, there is a gene that has some complicated effect of which a side-effect is that the height of men is more variable, and moreover there aren’t other genes that simply cause tallness.

It is hard to imagine what the complicated effect might be in the case of height, but it is not impossible to come up with speculations about mathematical ability. For example, maybe men have, as has been suggested, a tendency to be a bit further along the autism spectrum than women, which causes some of them to become very good at mathematics and others to lack the social skills to attract a mate. But even by the standards of evolutionary just-so stories, that is not a very good one. Our prehistoric ancestors were not doing higher mathematics, so we would need to think of some way that being on the spectrum could have caused a man at that time to become highly attractive to women. One has to go through such contortions to make the story work, when all along there is the much more straightforward possibility that there is some complex mix of genes that go towards making somebody intelligent, and that if prehistoric women went for intelligent men, then those genes would be selected for. But if that is what happened, then the proportion of less intelligent men would go down, and therefore the variability would go down.

While writing this, I have realized that there is a crucial assumption of Hill’s, the importance of which I had not appreciated. It’s that the medians of his two subpopulations are the same. Suppose instead that the individuals in male population are on average more desirable than the individuals in male population . Then even if population is less variable than population , if females are selective, it may very well be that a far higher proportion of population is chosen than of population , and therefore a tendency for the variability of the combined population to decrease. In fact, we don’t even need to assume that is less variable than : if the population as a whole becomes dominated by , it may well be less variable than the original combination of populations and .

So for Hill’s model to work, it needs a fairly strange and unintuitive combination of hypotheses. Therefore, if he proposes it as a potential explanation for greater variability amongst males, he needs to argue that this combination of hypotheses might actually have occurred for many important features. For example, if it is to explain greater variability for males in mathematics test scores, then he appears to need to argue (i) that there was a gene that made our prehistoric male ancestors more variable with respect to some property that at one end of the scale made them more desirable to females, (ii) that this gene had no effect on average levels of desirability, (iii) that today this curious property has as a side-effect greater variability in mathematics test scores, and (iv) this tendency to increase variability is not outweighed by reduction of variability due to selection of other genes that do affect average levels. (Although he explicitly says that he is not trying to explain any particular instance of greater variability amongst males, most of the references he gives concerning such variability are to do with intellectual ability, and if he can’t give a convincing story about that, then why have all those references?)

Thus, what I object to is not the very idea of a toy model, but more that with this particular toy model I have to make a number of what seem to me to be highly implausible assumptions to get it to work. And I don’t mean the usual kind of entirely legitimate simplifying assumptions. Rather, I’m talking about artificial assumptions that seem to be there only to get the model to do what Hill wants it to do. If some of the hypotheses above that seem implausible to me have in fact been observed by biologists, it seems to me that Hill should have included references to the relevant literature in his copious bibliography.

As with my previous post, I am not assuming that everything I’ve just written is right, and will be happy to be challenged on the points above.