# There is no breeder’s equation for environmental change

This post is about why heritability coefficients of human traits can’t tell us what to do. Yes, it is pretty much an elaborate subtweet.

Let us begin in a different place, where heritability coefficients are useful, if only a little. Imagine there is selection going on. It can be natural or artificial, but it’s selection the old-fashioned way: there is some trait of an individual that makes it more or less likely to successfully reproduce. We’re looking at one generation of selection: there is one parent generation, some of which reproduce and give rise to the offspring generation.

Then, if we have a well-behaved quantitative trait, no systematic difference between the environments that the two generations experience (also, no previous selection; this is one reason I said ‘if only a little’), we can get an estimate of the response to selection, that is how the mean of the trait will change between the generations:

$R = h^2S$

R is the response. S, the selection differential, is the difference between the mean all of the parental generation and the selected parents, and thus measures the strength of the selection. h2 is the infamous heritability, which measures the accuracy of the selection.

That is, the heritability coefficient tells you how well the selection of parents reflect the offspring traits. A heritability coefficient of 1 would mean that selection is perfect; you can just look at the parental individuals, pick the ones you like, and get the whole selection differential as a response. A heritability coefficient of 0 means that looking at the parents tells you nothing about what their offspring will be like, and selection thus does nothing.

Conceptually, the power of the breeder’s equation comes from the mathematical properties of selection, and the quantitative genetic assumptions of a linear parent–offspring relationship. (If you’re a true connoisseur of theoretical genetics or a glutton for punishment, you can derive it from the Price equation; see Walsh & Lynch (2018).) It allows you to look (one generation at a time) into the future only because we understand what selection does and assume reasonable things about inheritance.

We don’t have that kind machinery for environmental change.

Now, another way to phrase the meaning of the heritability coefficient is that it is a ratio of variances, namely the additive genetic variance (which measures the trait variation that runs in families) divided by the total variance (which measures the total variation in the population, duh). This is equally valid, more confusing, and also more relevant when we’re talking about something like a population of humans, where no breeding program is going on.

Thus, the heritability coefficient is telling us, in a specific highly geeky sense, how much of trait variation is due to inheritance. Anything we can measure about a population will have a heritability coefficient associated with it. What does this tell us? Say, if drug-related crime has yay big heritability, does that tell us anything about preventing drug-related crime? If heritability is high, does that mean interventions are useless?

The answers should be evident from the way I phrased those rhetorical questions and from the above discussion: There is no theoretical genetics machinery that allows us to predict the future if the environment changes. We are not doing selection on environments, so the mathematics of selection don’t help us. Environments are not inherited according to the rules of quantitative genetics. Nothing prevents a trait from being eminently heritable and respond even stronger to changes in the environment, or vice versa.

(There is also the argument that quantitative genetic modelling of human traits matters because it helps control for genetic differences when estimating other factors. One has to be more sympathetic towards that, because who can argue against accurate measurement? But ought implies can. For quantitative genetic models to be better, they need to solve the problems of identifying variance components and overcoming population stratification.)

Much criticism of heritability in humans concern estimation problems. These criticisms may be valid (estimation is hard) or silly (of course, lots of human traits have substantial heritabilities), but I think they miss the point. Even if accurately estimated, heritabilities don’t do us much good. They don’t help us with the genetic component, because we’re not doing breeding. They don’t help us with the environmental component, because there is no breeder’s equation for environmental change.