# Method of determining base values of traits in isolated populations

Prelude: I came across a discussion about the correct formula for calculating the average IQ of offspring, which goes something like the following

$$100 + \frac35 \left( \left(\frac{\text{father's IQ + mother's IQ}}{2} \right) - 100 \right)$$

but it does not matter for this question, and I do not know or particularly care if this is the correct formula.

As you all know, there is regression to the mean, i.e., if both parents have a freakishly high IQ of, say, 160, but come both from a "base population" with average IQ 100, the formula calculates the average IQ of the offspring to be lower than the arithmetic mean, because the parents are both outliers of their own "base population". Now, the above formula supposedly takes regression to the mean into account (and the mean it takes into account is IQ 100). As some people then pointed out, there is assortative mating, and very smart people descend often from very smart people, so the formula does not work for couples of "good pedigree".

I take it that this regression to the mean happens because these outliers with high IQ still carry some of the genes of the base population with IQ 100 and those are likely to be inherited by their children, thus lowering the IQ of offspring in comparison to their high-IQ parents.

My questions are:

1. How many generations of ancestors of average IQ, say, 110, do you have to have, in order for the regression of the mean to go towards IQ 110?

2. More generally and more interestingly, how do you determine what is the average value of a trait of a reasonably isolated subpopulation, that stems from and still lives inside a larger population? I am specifically interested in subpopulations as small as families, families with mean values of traits that deviate significantly from that of the overall population. Obviously, the first family-generation with a mean IQ of 110 does not make it likely that the mean of the offspring will be 110 as well if the direct ancestors, i.e., the zeroth generations and those before average around 100. At what stage or how many generations in can we reasonably assume that the IQ 110 is the value that the mean regresses to, under the assumption that there is constant assortative mating regarding intelligence and no freakish outliers occur in the family tree?

I guess it boils down to the following. How big do the two family trees of a couple have to be in order to reliably infer the average value of a trait — like intelligence, highly heritable, polygenous — in future offspring?

(No eugenics, no one gets selected other than by their mates, no evil scientists lurking.)

• Consider a couple where the male has an IQ of 125 and the female has an IQ of 100. What average IQ does the formula give? Commented Nov 18, 2021 at 6:28

# Update

I put in an answer to this question, with some reservations noted below, as there is a somewhat straightforward genetics answer to at least part of it. I'll leave the answer up for a little while as a sort of placeholder, up to the point that the question hopefully gets taken down.

I'll first note the strong "yikes!" aspect of this question, which is closely related to e.g. eugenics.

## Question 1

However, I believe that you will be most interested in the "breeder's equation", which is I believe the source of some of your values:

$$\Delta Z = h^2S$$

$$\Delta Z$$ = change in mean trait value in a population per unit time (generation usually)

$$h^2$$ = narrow-sense heritability

$$S$$ = strength of selection.

I suggest going to that resource for more information.

I will quickly note that this model is strongly dependent on a number of assumptions:

1. You have accurately estimated the narrow-sense heritability.
2. You have a large, panmictic population (you have raised this with assortative mating).
3. a lot of other stuff

(1) in particular is a big deal for IQ or really nearly any human, where there are reported heritability estimates in the 50% range from twin studies, but we honestly don't have a good estimate of this stuff, and there is really strong confounding from environmental variation (contra evo psych). I would not expect ("normal") IQ variation to respond very strongly to selection if environments were held constant. (There are obviously very clear examples of systematic environmental effects on IQ).

When I say "normal" variation, I mean among people who do not have cognitive disabilities related to specific variants of large effect (such as abnormal karyotypes). If you include those, you will get a large initial response to selection due to selecting on that unambiguous variation.

## Question 2

Why not just measure the trait of interest?