A Speculative Explanation of the Missing Heritability Problem II

Taking quincunx seriously

Francis Galton invented the Galton Board, the familiar toy in which ball bearings bounce down through a matrix of pins, winding up in bins that approximate a normal distribution. Galton never actually called the device a quincunx, an old word that refers to the pattern of five dots on a die, which is reproduced in the staggered rows of pins.

If you are interested in a deep dive into the history of Galton and the quincunx, I highly recommend Gregory Radick’s Disputed Inheritance. Until recently, I had always thought of the quincunx as just a clever way to demonstrate the binomial theorem, according to which a large number of random binary events (bounce left or right from a pin) can add up to a quasi-continuous normally distributed quantitative measure. Radick argues that Galton had a deeper meaning in mind, seeing the quincunx as a real model of biological development, and that his interest in the quincunx suggests that he was less of a determinist than he is usually thought of. I’m not so sure on the latter point— I would say that Galton was ambivalent. At his best, he could certainly be sophisticated about the complexity of genetics and evolution in the development of organisms, but when he had to put his money down he always did so on the strictly deterministic side. Radick knows more about all this than I do.

In any event, I agree that the quincunx is a real model of development. I see it this way: the quincunx challenges us to think about the relationship between the realities of biological and behavioral development on the one hand, and the statistical models we use to study them on the other. The binomial theorem demonstrated by the quincunx was the basis for RA Fisher’s infinitesimal model that unified the Mendelian and biometric models of genetics. Many tiny Mendelian events, added together, can produce normally distributed continuous traits that do not display classical Mendelian segregation.

But that is a statistical model; is it actually how development works? I remember my one and only extended conversation with the great Paul Meehl. Because Meehl thought of me (accurately) as an Irving Gottesman protege, he wanted to talk about the genetics of schizophrenia, about which he endorsed a sophisticated, albeit manifestly false, single gene theory. Irv Gottesman endorsed the diathesis-stress theory, ultimately based on Fisher’s infinitesimal model. Many tiny genetic effects added up to normally distributed latent liability, and then a combination of genetic and environmental events locates individuals somewhere on that distribution. If their location exceeded a certain threshold, they develop the disorder.

Meehl’s analysis of the situation, as I recall it, was, “Name me one fucking medical disorder that is known to work that way!” Diathesis-stress is a statistical model, and a good one, but it is indeed hard to come up with a biological process in which tiny additive effects accumulate into a risk with a threshold. Meehl found the idea so ridiculous that he was willing to commit to an alternative theory that was obviously wrong (though, as I say, quite impressive on its own terms) on the day he formulated it. Meehl wanted real disorders to have real causes, like mumps, not a bunch of invisible statistical effects. He was suspicious of explanations that lacked real causes, and devoted the second half of his career to taxometrics, which was an attempt to discover the big causes that underlie complex phenotypes. (Spoiler alert: in the domain of human behavior, there aren’t any.)

The quincunx is another instantiation of the infinitesimal model. In contrast to diathesis-stress, it emphasizes the sequential nature of development (top pins have to be traversed before getting to bottom pins) and de-emphasizes the threshold part in favor of the normally distributed outcome. My two papers with the philosopher Jonathan Kaplan explore the implications of taking the quincunx seriously as a model of development, not just as a toy for demonstrating the binomial theorem. I posted links to them on Bluesky.

The first, more philosophical, paper explores the implications of quincunx development for notions of causation. I would sum it up this way: what are the causes that explain why some balls wind up in the fifth bin from the left? What do those balls have in common? I would say there are no causes, and the balls in Bin #5 have essentially nothing in common. It is just that, in a system that works according to certain rules, different outcomes can occur. It is an example of something I said once about causation of human behavior

…individual differences in complex human characteristics do not, in general, have causes, neither genetic nor environmental. Complex human behaviour emerges out of a hyper-complex developmental network into which individual genes and individual environmental events are inputs. The systematic causal effects of any of those inputs are lost in the developmental complexity of the network. Causal explanations of complex differences among humans are therefore not going to be found in individual genes or environments any more than explanations of plate tectonics can be found in the chemical composition of individual rocks.

This is what, in Chapter 10 of my book, I call an outcome, as opposed to an essence. Mumps has a deep Meehlian cause, an essence; underlying whatever variation one might find in mumps symptoms, everyone carries a particular virus. In a medical sense it is a good thing when a disease has an essence, because there is a causal image of the phenotype down in the genes or bloodstream or wherever, something we can target and fix. But not all phenotypes have essences. Divorce, to trot out my standard example, has no essence. It is the equivalent of Bin #5 in the quincunx. You fall down the quincunx of life, and you either wind up in the divorce bin or you don’t. Different pins may have an effect on the likelihood that you are going to wind up in the divorce bin, but none of them bear any direct causal relationship to the outcome. This is why science, in principle, is never going to prevent unhappy marriages: there is nothing down there to fix.

Next time: heritability of quincunx behavior. I will finally get to the missing heritability problem.