Packages and Functions

library(pacman); p_load(ggplot2)

#Jensen's (1973) between-group heritability

BGHJ <- function(PM1, PM2, MO) {
  ME <- PM1 - ((PM1 - PM2)/2)
  MOD <- PM1 - MO
  MED <- PM1 - ME
  BGHAD <- MOD/MED
  if(BGHAD > 1) BGHAD <- abs(1 + (1 - (MOD/MED)))
  return(BGHAD)}

#Jensen's (1973) adjusted between-group heritability

BGHJA <- function(PM1, PM2, MO, PA, PB, PM) {
 ME <- abs(PM1 - PM2)
 AD <- (PB + (1 - PA)) / PM
 MEA <- ME * AD
 BGHAD <- MO/MEA
 if(BGHAD > 1) BGHAD <- abs(1 + (1 - (MOD/MED)))
 return(abs(BGHAD))}

#Scarr's (1977) between-group heritability

BGHS <- function(M1, M2, SD2, OE, d) {
  MB <- M1 - M2
  SDD <- MB/SD2
  EE <- d/SDD
  BGHA <- OE/EE
  return(abs(BGHA))}

Rationale

A new global admixture study for cognitive ability has been posted on BiorXiv (Connor & Pesta, 2021). I have looked at some other work on this topic and generated estimates of between-group heritability based on it (https://rpubs.com/JLLJ/BGH). Here are the estimates for this new study; they are produced as described in that linked page. All values come from the first table in version 2 of their preprint except for a correlation I had to ask for. Effect sizes are in Hedge’s g and the White group’s mean is set to 0 for comparison. This is only done because of potential differences in the variances that I don’t wish to mess with the analysis. These are only done for the Black and White groups since the rest are a bit more dubious and sometimes represented by only a handful of people.

Numbers

Jensen’s Methods

Jensen <- BGHJ(0, -1.121, -0.412); Jensen #Naive estimate

## [1] 0.735058

(0.97+0.18)/2 #Expected ancestry proportion in "mixed-race" Black plus White individuals based on the rest of the sample

## [1] 0.575

JensenA <- BGHJA(0, -1.121, -0.412, 0.97, 0.18, 0.58) #Estimate adjusted for empirical ancestry proportions

We’re not given many significant digits here, so perhaps the slight increase over 100% is due to that rounding.

Scarr’s Method

(1.121)/((0.97-0.18)/0.11) #Expected correlation in the Black group assuming 100% between-group heritability

## [1] 0.1560886

Scarr <- BGHS(0.97, 0.18, 0.11, 0.14, 1.121); Scarr #Estimate derived by assessing the deviation of the observed correlation from prediction

## [1] 0.8969264

I would also like to utilize the numbers from the much smaller PING study (Kirkegaard et al., 2019). Unfortunately, they do not provide and the samples are not large enough to give data for mixed-race individuals. The sample sizes used here are n = 567 for the White group, 138 for the Black group, and the values for Cohen’s d and Hedge’s g are 1.096 and 1.092, respectively. The sample sizes are much smaller than in the other studies, so the results are considerably more subject to sampling error.

(1.092)/((0.97-0.17)/0.11)

## [1] 0.15015

ScarrPING <- BGHS(0.97, 0.17, 0.11, 0.17, 1.092); ScarrPING

## [1] 1.132201

Plotting the Results

df <- data.frame("BGS" = c(Jensen, JensenA, Scarr, ScarrPING), "Name" = c("Jensen", "Adjusted Jensen", "Scarr", "Scarr - PING"))

ggplot(df, aes(x = Name, y = BGS, fill = Name)) + geom_bar(stat = "identity", show.legend = F) + coord_cartesian(ylim = c(0, 1)) + xlab("Method") + ylab("Between-group Heritability") + theme_bw() + scale_fill_manual(values = c("#E69F00", "#999999", "#3C94CF", "#E32528"))

Conclusion

It seems the between-group heritability is very high in these studies, if imprecisely estimated in the earlier PING study. Rounding may have led to somewhat strange results and environments may lead to larger than expected deficits, inflating between-group heritabilities in the PING case, though I suspect the small sample size in that study has more explanatory power.

References

Connor, G., & Pesta, B. J. (2021). Linear and partially linear models of behavioural trait variation using admixture regression. BioRxiv, 2021.05.14.444173. https://doi.org/10.1101/2021.05.14.444173

Kirkegaard, E. O. W., Woodley of Menie, M. A., Williams, R. L., Fuerst, J., & Meisenberg, G. (2019). Biogeographic Ancestry, Cognitive Ability and Socioeconomic Outcomes. Psych, 1(1), 1–25. https://doi.org/10.3390/psych1010001

Between-Group Heritability from a New Global Admixture Study