Pith: My deduction is that for a typical trait varied due to drift phenotypic variation is roughly ~ quantitative genetic variance*h^2, where quantitative genetic variance is roughly 2 SNP Fst; thus, assuming a typical narrow heritability of 0.5, one gets phenotypic variation ~ SNP Fst. And this is why the (phenotypic) craniometric continental eta-squared is ~ 0.12.
The rewrite was:
As for the soundness of the argument:
A-1. When it comes to discussions of genetic contributions to phenotypic differences between groups, what is of relevance are the differences in the specific genes associated with specific traits, not the average genetic differences between groups. Regarding racial groups, genetic variation at a typical locus will have no functional consequence since a typical locus is selectively neutral. As such, average genetic variation will tend to measure neutral mutations and so index the time of divergence and the degree of isolation between populations (Sarich and Miele, 2004). The upshot is that the average genetic variation across loci does not allow one to predict the amount of differentiation in loci that were under selection – the very ones that are typically relevant when it comes to discussions of behavioral genetic and many other socially significant differences, ones which presumably were and are subject to selective pressure. With regards to these, one must look at differentiation in specific genetic regions (for example: Wu and Zhang, 2011) and the specific genes that code for differences. To give a concrete example in which overall genetic differentiation is unindicative of differentiation with respect to a specific trait which has been under selection, at their extremes, northern and southern Europeans differ in height by approximately one standard deviation (Turchin et al., 2012; supplementary data). These height differences are substantially genetically determined (Turchin et al., 2012). Yet average European interpopulation SNP Fst values are trivial at 0.001 to 0.01 (Tian et al., 2009).
A-2. Measures of genetic differentiation based on fixation (e.g., FST and ΦST) are often poor measures of true genetic differentiation. As Bird et al. (2011) remind us: "Using fixation indices will systematically underestimate genetic differentiation, especially when using highly polymorphic markers such as microsatellites (Hedrick 1999)." This is because maximum Fst values are limited by heterozygosity. To provide an example of this underestimation, Long and Kittles (2003) found a between-population microsatellite Fst of 0.11 based on a sample of human populations; when they added chimpanzees to the set of human populations, the between-population Fst rose only to 0.18. There were several reasons for the results, one of which was that the maximum possible Fst value, given the markers used, was well below the theoretical maximum of 1 both in the case of the human comparisons and in the case of the human and primate comparisons. The take away is that, as Mountain and Risch (2004) noted after citing this example in relation to their discussion of genetic contributions to phenotypic differences among ethnic and racial groups, “a low Fst estimate implies little about the degree to which genes contribute to between-group differences.”
A-3. The above noted, with caution genetic variability as indexed by Fst and Fst analogs can be and often is used to index the expected variation in quantitative traits owing to genetic drift (neutral variation) (Leinonen et al., 2013). That is, researchers sometimes make comparisons between Fst and an index of heritable quantitative trait variability called Qst. This comparison is made to help evaluate if the quantitative trait variation found between populations is larger or smaller than would be expected based on neutral variation (i.e., if the traits under question were under selection). Whitlock (2008) explains the measure Qst:
The calculation of QST for a trait requires two quantities: the additive genetic variance of the trait within a population (V A, within) and the genetic variance among populations (V G, among). For diploids, QST is calculated as
Qst = V G, among / (V G, among + 2V A, within)
For haploids, the same equation applies, but without the '2' in the denominator. [That '2' for the diploid case comes from the fact that the quantitative genetic variance among populations is proportional to two times FST (Wright 1951).]
What is particularly relevant to the present discussion is Whitlock's last statement, since we are interested in predicting quantitative genetic variance from Fst, not comparing Qst to Fst. Amongst diploid populations, the predicted quantitative genetic trait variance is equal to 2FST/(1 + FST) (Leinonen et al., 2013). The 2 in the equation comes from the fact that roughly half of the genetic variation within diploid populations is within individuals. (Note: 68: Cole et al. (1986), alternatively, note: "Wright (1943, 1951) showed, using a model of additive gene effects at a single locus, that variation among populations in the value of a selectively neutral quantitative character is, in expectation, ơ2B = 2Fst ơ20 , where ơ20 is the genetic variance expected under panmixia with the same gene frequencies, and FST is the correlation among uniting gametes relative to the total population.")
A-4. The point immediately above is often missed even by respected population geneticists, so it is worth elaborating on. It is not infrequently erroneously claimed that, in context to human races, 85 to 95% of the total genetic variance is between individuals within populations (e.g., Barbujani and Colonna, 2010). This is incorrect since among diploids a large chunk of the total genetic variance is captured within individuals (Harpending, 2002; Sarich and Miele, 2004). The between population variance values given by Fst and Fst analogs is out of the summed variance between populations, between individuals within populations, and within individuals (e.g., Weir and Cockerham (1984)). Roughly half of the total variance for diploid populations is expected to be intra-individual. As such, the ratio of genetic variance between populations to that between individuals and populations (but not within individuals) is Fst/(Fst+1/2(1-Fst)), which is mathematically equivalent to 2FST/(1 + FST), the predicted amount of quantitative genetic trait variance owing to drift. Nishiyama et al. (2012) give an example which illustrates the flaw in deducing from low between population Fst values that the overwhelming portion of variance is between individuals. The authors decomposed the SNP genetic variance for various Japanese populations into inter-subpopulational, inter-individual, and intra-individual variance. They found that between 96.7 and 99.6% of the variance was located within individuals. When intra-individual variance was partitioned out, roughly the same percent of genetic variance was located between individuals and between subpopulations as between individuals and within subpopulations. The decomposition is shown in Table 4.11 below.
Of course, most of the variance was still “inter-individual” in the sense of inter- plus intra-individual (i.e., intrapopulational). In the same way, of course, most diversity, in general, is “inter-racial” in the sense of inter-racial plus inter individual plus intra-individual. It just was not mostly inter-individual in the sense of exclusively between individuals. Does this matter? Well, it casts the oft-referenced genetic variance ratios in a different light. And it is relevant if one's argument is that phenotypic differences between individuals between groups can not be substantially congenitally conditioned because there is "too little" between group genetic variation (relative to that between individuals within groups).
Table 4.11. Total genetic variance partitioned into variance between subpopulations, among individuals within subpopulations, and within individuals in a Japanese sample
Between Among individuals Within
Population within subpopulations Individuals
Amami vs. Mainland Variance component 0.03 0.02 2.13
Relative proportion (5) 1.2(%) 1.1(%) 97.7(%)
Okinawa vs. Mainland Variance component 0.04 0.03 2.12
Relative proportion (5) 1.9(%) 1.4(%) 96.7(%)
Amami vs. Okinawa Variance component <0.01 <0.01 2.2
Relative proportion (5) 0.2(%) 0.2(%) 99.6(%)
(Based on Table 4 in Nishiyama et al. (2012).)
A-5. Getting back to the main point, if we wish to estimate expected quantitative genetic trait variation it is often advised to avoid using low mutation rate genetic markers such as microsatellites, which, as discussed, have high Hs values and thus necessarily exhibit low Fst values. It is often advised instead to use SNPs, both because these markers do not tend to have very high Hs values and because SNP variation codes for typical quantitative trait variation (Edelaar and Björklund, 2011). Another way to look at this is to consider that the magnitude of (fixation index estimated) genetic differentiation varies by the class of loci analyzed, with part of this variation being attributable to loci variation in Hs (Jakobsson et al., 2013); for example, for humans, continental microsatellite, SNP, and mtDNA Fst values are typically around, respectively, 0.05, 0.12, and 0.20. Were one to try to infer the magnitude of genetically conditioned phenotypic variation from typical indices of fixation (e.g., Fst values), it would make sense to use the class of loci that most likely underpins the relevant trait variation. For example, since variation in single-nucleotide polymorphisms (SNPs) explains variation in many interesting polygenetic traits such height and intelligence (for example: Yang et al., 2010; Davies et al., 2011), it would make more sense to attempt to infer magnitudes of genetic differentiation in these traits from SNP Fst values than from microsatellite or mtDNA ones.
Now, these five considerations set up the problem for the “too little variance” argument, with its implicit premise that the ratio of genetically mediated phenotypic variability in socially significant traits is roughly concordant with the ratio of average genetic variability. As will be seen, the argument lends itself to the opposite of the conclusion drawn by biological race antagonists. This is for the following reasons:
B-1. The magnitude of average genetic differentiation depends on the biological divisions in question. It makes no sense to argue that differences between regional biological races (e.g., Europeans and West Africans) cannot be genetically conditioned on the account of supposedly small differences between continental races (e.g., Caucasoids and Negroids). The magnitudes of the genetic differentiation in SNPs between some regional races are shown below in Table 4.10.
Table 4.10. Intercontinental autosomal genetic distance based on SNPs for 1000 Genomes (below diagonal) and HapMap3 (above diagonal)
YRI (Yoruba) CHB (Chinese) CEU (European)
YRI (Yoruba) 0.183 0.156
CHB (Chinese) 0.161 0.11
CEU (European) 0.139 0.106
(From: Bhatia et al. (2013), Table 2. Based on recommended ratio to average method.)
B-2. The magnitude of SNP differentiation, as indexed by Fst, is not small, even between continental races, according to population genetic and social scientific standards. The median continental race SNP Fst value is said to be around 0.12 (Li et al., 2008; Campbell and Tishkoff, 2008; Elhaik, 2012; Bhatia et al., 2013), with the estimated magnitudes varying somewhat due to the choice of specific loci, the method of aggregation employed, the Fst estimators used, and so on; see Bhatia et al. (2013). With regards to population genetic standards, the difference would be moderate by Sewall Wright’s (1978) not infrequently cited scale. By this:
0 to 0.05 indicates little genetic differentiation; 0.05 to 0.15 indicates moderate genetic differentiation, 0.15 to 0.25 indicates great genetic differentiation, 0.25 indicate very great genetic differentiation.
Now, with regards to social scientific standards, if we naively treat our Fst statistic as indexing the proportion of the total genetic variance lying between groups we can interpret it in terms of eta-squared. A between group variance of 0.12 would be moderate. Alternatively, treating the SNP Fst = 0.12 as an index of between group variance, one can convert the value into standardized differences, a metric in which, in the social sciences, group comparisons are often made. The formula is shown below:
Given the law of total variance:
z = 2(sqrt((a/w)))
z = between group standardized difference; a = ratio of variance between to within populations;
w = variance within populations.
If one assumes normality and equal variances, a 12% between-population variation is equivalent to a d-value of ~0.74, which is typically said to be “medium” to "large." For illustration, the relationships between percent variance between populations and various statistics are shown in Table 4.12 (from Cohen (1988)).
Table 4.12. Interpreting and comparing effect sizes in the social sciences
Size of Effect Cohen's f [1] %Variance (eta-squared) [2] Cohen's d [3] Pearson's r
Small 0.1 1 0.2 0.1
Medium 0.25 6 0.51 0.25
Large 0.4 14 0.81 0.38
[1] Cohen's f = Square Root of eta-squared / (1-eta-squared). [2] Eta-squared is interpretable as the variance that lies between groups relative to the total variance (Cahan and Gamliel, 2011). [3] Cohen's d is the mean difference between populations divided by the pooled standard deviation.
Of course, as noted, contrary to what is sometimes stated (e.g., Fish (2013)), unlike eta-squared, Fst values are rarely out of 1 in practice. For example, in their Table 1 and 2, Xu et al. (2008); give expected heterozygosity (Hs) values for Japanese (JPT), Chinese (CHB), Uyghurs (UIG), Europeans (CEU), and Yorubi (YRI) based on 20177 SNPs. The average Hs came out to about 0.30, meaning that the maximum possible SNP Fst value – the value that would be found if populations had no alleles in common – would be 0.7. Treating Fst as something akin to eta-squared is therefore problematic.
B-3. But all of this neglects the points made in A-3 and A-4. As said, our Fst value is relative to total variance, which includes the irrelevant, in this context, intra-individual variance. Instead of the expected (simply owing to neutral variation) between group quantitative genetic variation being proportional to Fst, it is proportional to roughly 2(Fst). Solving Fst= V G, among / (V G, among + 2V A, within) for a between group quantitative genetic variance value (V G, among), when Fst = 0.12, gives us V G, among = 0.22. This value can then be inputted back into the equation given in B.-2. When done so, we find that more than a little quantitative genetic variance is expected to lie between groups.
Can the “too little variance” argument be salvaged? It cannot. To avoid a racial-hereditarian conclusion, it must be discarded — but how? One could, citing the points made in A-2, argue that, in general, there is little correlation between average genetic variability and genetically mediated phenotypic variability. But this is not the case at least with regard to many classes of character differences. Relethford (2009), for example, notes:
Several studies have looked at estimates of FST based on the global craniometric dataset originally collected by Howells (1973, 1989, 1995, 1996)… Using an average heritability of 0.55, Relethford (1994, 2002) found that estimates of FST based on all 57 traits ranged from 0.11 to 0.14 depending on the number of geographic regions sampled. These FST values are similar in magnitude to those estimated in a number of studies of classical genetic markers and DNA markers.
Relethford's (2009) "craniometric Fst" values approximate Qst ones — and, in this case, they also happen to be roughly equivalent to human (genetic) Fst ones. When we grouped Howells' 28 populations into six major continental races — West Eurasians, East Eurasians, Australoids, Negroids, Amerindians, and Pacific Islanders — and ran an ANOVA using the first principle component we found a (phenotypic craniometric) eta-squared of between 0.10 and 0.14 (depending on the specific method used), which is in line with Relethford's (2009) findings. Assuming a heritability of about 0.55, as Relethford does, we get a craniometric quantitative genetic variance value of around 0.22, as predicted based on the considerations in B-3. (Note 72: "The phenotypic variance between populations is lower than the quantitative genetic variance in proportion with the heritability (i.e., variance in phenotype owing to genes). When h^2 = 1, the phenotypic variance is equal to the quantitative genetic variance")
Similar results have been reported in context to dental traits (e.g., Hanihara (2008)). Taken together, the theory and evidence suggests that, between continental races, Fst values are roughly half of the size of the between group quantitative genetic ones -- at least for traits varied due to drift -- and are roughly proportional to the phenotypic variance values when the narrow heritability of the traits is modest (e.g., 0.5). Yet, as we noted above, Fst values are medium to large as judged by social scientific standards. So if one grants the premise of the "too little variance" argument, that between population genetic variance indexes variance in behavioral genetic or other social relevant traits, one is left with medium to large genetically conditioned differences. Instead of Lewontin's conclusion, one is left with the early Franz Boas's deduction:
It does not seem probable that the minds of races which show variations in their anatomical structure should act in exactly the same manner. Differences of structure must be accompanied by differences of function, physiological as well as psychological; and, as we found clear evidence of difference in structure between the races, so we must anticipate that differences in mental characteristics will be found. [...] (Boas, 1974).
Of course, one could try to argue that differentiation in the genes that underlay interesting physiological and neurological functions is trivial — but the empirical evidence speaks against such an argument. As an example of such evidence, in the context of regional (European, East Asian, West African) population differences, Wu and Zhang (2011) conclude:
[W]e find that genes involved in osteoblast development, hair follicles development, pigmentation, spermatid, nervous system and organ development, and some metabolic pathways have higher levels of population differentiation. Surprisingly, we find that Mendelian-disease genes appear to have a significant excessive of SNPs with high levels of population differentiation, possibly because the incidence and susceptibility of these diseases show differences among populations.
Another way to escape the reverse of the “too little variation” argument would be to reiterate a version of Loring Brace's (1999) argument. According to this, since each population has an equal ability to use language and to develop culture, they must not differ in behavioral traits such as intelligence. This type of argument, of course, is (logically) ridiculous when applied in context to normally distributed traits because within each and every population innumerous subpopulations exist which do differ in these traits. If these subpopulations which differ can exist then populations which exist can differ. Worse, it is already known that human populations do differ in the said traits; the debate is over “why,” not “whether.”
Ultimately, the way around the early Boas' deduction is to reiterate our point A-1. However, numerous philosophers of science and population geneticists have deemed the "too little variance" argument to be valid (for example: Kitcher, 2007; Brown and Armelago, 2001; Barbujani and Colonna, 2010), so perhaps its reversed version can not be so easily dismissed. Such an argument could never establish a genetic basis for specific differences; but, perhaps, as suggested by Boas and others, it provides probabilistic support — a baseline expectation — for the existence of some behavioral genetic differences. Whatever the case, to the extent that the "too little variance" argument is deemed to be valid it clearly fails to support the position in defense of which it has been enlisted.
