Human genetic variation, Fst and Lewontin's fallacy in pictures

In an earlier post European genetic substructure, I displayed the following graphic, illustrating the genetic clustering of human populations.




Figure: The three clusters shown above are European (top, green + red), Nigerian (light blue) and E. Asian (purple + blue).

The figure seems to contradict an often stated observation about human genetic diversity, which has become known among experts as Lewontin's fallacy: genetic variation between two random individuals in a given population accounts for 80% or more of the total variation within the entire human population. Therefore, according to the fallacy, any classification of humans into groups ("races") based on genetic information is impossible. ("More variation within groups than between groups.")

To understand this statement better, consider the F statistic of population genetics, introduced by Sewall Wright:

Fst = 1 - Dw / Db

Db and Dw represent the average number of pairwise differences between two individuals sampled from different populations (Db = "difference between") or the same population (Dw = "difference within"). Even in the most widely separated human populations Fst < .2 so Dw / Db > .8 (roughly). This may not sound like very much genetic diversity, but it is more than in many other animal species. See here for recent high statistics Fst values by nationality.

Dw / Db > .8 means that the average genetic distance measured in number of base pair differences between two members of a group (e.g., two randomly selected Europeans) is at least 80 percent of the average distance between distant groups (e.g., Europeans and Asians or Africans). In other words, if two individuals from very distant groups (e.g., a Japanese and a Nigerian) have on average N base pair differences, then two from the same group (e.g., two Nigerians or two Japanese) will on average have roughly .8 N base pair differences.

How can the Fst result ("more variation within groups than between groups") be consistent with the clusters shown in the figure? I've had to explain this on numerous occasions, always with great difficulty because the explanation requires a little mathematics. In order to make the point more accessible, I've created the figures below, which show two population clusters, each represented by an ellipsoid (blob). The different figures depict the same pair of objects, just viewed from different angles.

The blobs are constructed and arranged so that the average distance between two points (individuals) within the same cluster is almost as big as the average distance between two points (individuals) in different clusters. This is easy to achieve if the ellipsoids are big and flat (like pancakes) and placed close to each other along the flat directions. The figure is meant to show how one can have small Fst, as in humans, yet easily resolved clusters. The direction in which the gap between the clusters appears is one of the principal components in the space of human genetic variation, as recently found by bioinformaticists. The figure at the top of this post plots individuals as points in the space generated by the two largest principal components extracted from the combination of data from HapMap and from large statistics sampling of Europeans. Exhibited this way, isolated clusters ("races") are readily apparent.

The real space of genetic variation has many more than 3 dimensions, so it can't be easily visualized. But some aspects of the figures below still apply: there will be particular directions of variation over which different populations are more or less identical (orthogonal to the principal component; i.e. along the flat directions of each pancake), and there will be directions in which different populations differ radically and have little or no overlap. Note, however, that we are specifically referring to genetic variation, which may or may not translate into phenotypic variation.







Related posts: "no scientific basis for race" , metric on the space of genomes.

The existence of this clustering has been known for 40 years.

East Asian genetic substructure

Below are some results on East and Southeast Asian genetic substructure. As you can see, Koreans are (sort of) midway between Chinese and Japanese. It will almost certainly be possible to differentiate between different regional origins based on DNA, once larger statistics studies become available. (See European results.)

Figures: Each point is an individual, and the axes are two principal components in the space of genetic variation. Colors correspond to individuals of different Asian ancestry.






Thanks to Chao Tian of UC Davis for sending me an early draft of the paper.

Analysis of East Asia Genetic Substructure: Population Differentiation and PCA Clusters Correlate with Geographic Distribution.

C. Tian1, R. Kosoy1, A. Lee2, P. Gregersen2, J. Belmont2, M. Seldin1

1) Rowe Program Human Genetics, Univ California Sch Medicine, Davis, CA; 2) North Shore-LIJ Res Inst, Manhasset, NY, Baylor Col Med., Houston TX.

Accounting for genetic substructure within European populations has been important in reducing type 1 errors in genetic studies of complex disease. As efforts to understand complex genetic disease are expanded to other continental populations an understanding of genetic substructure within these continents will be useful in design and execution of association tests. In this study, population differentiation(Fst) and Principal Components Analyses(PCA) are examined using >200K genotypes from multiple populations of East Asian ancestry(total 298 subjects). The population groups included those from the Human Genome Diversity Panel[Cambodian(CAMB), Yi, Daur, Mongolian(MGL), Lahu, Dai, Hezhen, Miaozu, Naxi, Oroqen, She, Tu, Tujia, Naxi, and Xibo], HapMap(CHB and JPT), and East Asian or East Asian American subjects of Vietnamese(VIET), Korean(KOR), Filipino(FIL) and Chinese ancestry. Paired Fst(Wei and Cockerham) showed close relationships between CHB and several large East Asian population groups(CHB/KOR, 0.0019; CHB/JPT, 00651; CHB/VIET, 0.0065) with larger separation with FIL(CHB/FIL, 0.014). Low levels of differentiation were also observed between DAI and VIET(0.0045) and between VIET and CAMB(0.0062). Similarly, small Fst's were observed among different presumed Han Chinese populations originating in different regions of mainland of China and Taiwan(Fst < 0.0025 with CHB). For PCA, the first two PC's showed a pattern of relationships that closely followed the geographic distribution of the different East Asian populations. For example, the four "corner" groups were JPT, FIL, CAMB and MGL with the CHB forming the center group, and KOR was between CHB and JPT. Other small ethnic groups were also in rough geographic correlation with their putative origins. These studies have also enabled the selection of a subset of East Asian substructure ancestry informative markers(EASTASAIMS) that may be useful for future genetic association studies in reducing type 1 errors and in identifying homogeneous groups.

Related posts: "no scientific basis for race" , metric on the space of genomes

More genetic substructure

Via Genetic Future, two more principal components analyses of SNP data. Click the genetics label for related posts.

European, Nigerian and East Asian samples from HapMap and 100 African-American samples clustered based on data from ~600,000 genetic markers. The bend toward the East Asian cluster is probably due to Native American admixture.

Alkes L. Price, Nick Patterson, et al. (2008). Effects of cis and trans Genetic Ancestry on Gene Expression in African Americans, PLoS Genetics, 4 (12) DOI: 10.1371/journal.pgen.1000294.





Resolving Finns and Swedes. Jakkula et al. (2008) The Genome-wide Patterns of Variation Expose Significant Substructure in a Founder Population, The American Journal of Human Genetics, 83 (6), 787-794 DOI: 10.1016/j.ajhg.2008.11.005.

European genetic substructure

Figure: Each point is an individual, and the axes are two principal components in the space of genetic variation. Colors correspond to individuals of different European ancestry.

The figure above is from the Nature paper: European Journal of Human Genetics (2008) 16, 1413–1429; doi:10.1038/ejhg.2008.210

Abstract: An investigation into fine-scale European population structure was carried out using high-density genetic variation on nearly 6000 individuals originating from across Europe. The individuals were collected as control samples and were genotyped with more than 300 000 SNPs in genome-wide association studies using the Illumina Infinium platform. A major East–West gradient from Russian (Moscow) samples to Spanish samples was identified as the first principal component (PC) of the genetic diversity. The second PC identified a North–South gradient from Norway and Sweden to Romania and Spain. ...

Some interesting points:

1) Significant East-West and North-South substructure is apparent already from the figure. The resolution of the study is sufficiently high that Swedes and Norwegians can be distinguished with 90 percent accuracy (Table 4). Crime scene forensics will never be the same -- "the Swede did it!" ;-)

In conclusion, we have shown that using PCA techniques it is possible to detect fine-level genetic variation in European samples. The genetic and geographic distances between samples are highly correlated, resulting in a striking concordance between the scatter plot of the first two components from a PCA of European samples and a geographic map of sample origins. We have shown how this information can be used to predict the origin of unknown samples in a rapid, precise and robust manner, and that this prediction can be performed without requiring access to the individual genotype data on the original samples of known origin. ...

2) Genetic distances between population clusters are roughly as follows: the distance between two neighboring western European populations is of order one in units of standard deviations and the distance to the Russian cluster is several times larger than that -- say, 3 or 4. From HapMap data, the distance from Russian to Chinese and Japanese clusters is about 18, and the distance of southern Europeans to the Nigerian cluster is about 19. The chance of mis-identifying a European as an African or E. Asian is exponentially small! (Table 5)

...The distance measure is a measure of the distance in standard deviations from a sample to the center of the closest matching population.

...For the other HapMap populations, the classification procedure assigned 100% of the YRI [Yoruban = Nigerian] samples to France, and almost 100% of the CHB and JPT [Chinese and Japanese] samples to Russia. However, the distribution of the distance measure for the four populations was quite different. For the CEU [HapMap European] samples, the median and 95% CI of the distance measure were 0.41 (0.11–1.01), whereas for the YRI, CHB and JPT populations, the median and 95% CIs were 19.3 (18.0–20.6), 17.7 (15.9–19.3) and 18.0 (15.4–19.6), respectively.

...The Yoruban [Nigerian] and Asian samples were identified as belonging to the countries on the south and east edges, respectively, of the European cluster, and the distance measure clearly indicates that they do not fit well into any of the proposed populations. ...

Figure: The three clusters shown above are European (top, green + red), Nigerian (light blue) and E. Asian (purple + blue).


See additional discussion at gnxp (the modified figure is from Razib), Dienekes

Related posts: "no scientific basis for race" , metric on the space of genomes

Pleiotropy, g, and specific learning abilities

This came out earlier in the year but I just noticed it. See also Myths, Sisyphus and g.

Summary for those wishing to follow science but who can't do math: consider many pairs of individuals and ask to what extent similarity in genotype is related to similarity in phenotype (g score, specific ability score, height, weight, etc.). From this analysis one can estimate the extent to which genes influence phenotype (heritability), and to what extent the same genes are influencing two different traits (e.g., height and weight, or reading ability and general cognitive ability g). The bivariate method is described here in more detail.

"These results indicate that genes related to diverse neurocognitive processes have general rather than specific effects."

DNA Evidence for Strong Genome-Wide Pleiotropy of Cognitive and Learning Abilities  (DOI 10.1007/s10519-013-9594-x Behavior Genetics)

Abstract: Very different neurocognitive processes appear to be involved in cognitive abilities such as verbal and non-verbal ability as compared to learning abilities taught in schools such as reading and mathematics. However, twin studies that compare similarity for monozygotic and dizygotic twins suggest that the same genes are largely responsible for genetic influence on these diverse aspects of cognitive function. It is now possible to test this evidence for strong pleiotropy using DNA alone from samples of unrelated individuals. Here we used this new method with 1.7 million DNA markers for a sample of 2,500 unrelated children at age 12 to investigate for the first time the extent of pleiotropy between general cognitive ability (aka intelligence) and learning abilities (reading, mathematics and language skills). We also compared these DNA results to results from twin analyses using the same sample and measures. The DNA-based method revealed strong genome-wide pleiotropy: Genetic correlations were greater than 0.70 between general cognitive ability and language, reading, and mathematics, results that were highly similar to twin study estimates of genetic correlations. These results indicate that genes related to diverse neurocognitive processes have general rather than specific effects. 

[GCTA] ... The bivariate method extends the univariate model by relating the pairwise genetic similarity matrix to a phenotypic covariance matrix between traits 1 and 2 (Lee et al. 2012). The eight principal components described earlier were used as covariates in our bivariate GCTA analyses; as mentioned in the previous section, all phenotypes were age- and sex-regressed prior to analysis.

Twin modelling. The twin design and model-fitting is discussed elsewhere (Plomin et al. 2013a). We fit a bivariate Cholesky decomposition using OpenMx (Boker et al. 2011), which provided a direct comparison with the bivariate GCTA. The correlated factor solution is the least restricted model allowing variables to correlate with one another via genetic, shared environment, and non-shared environment.

... Table 1 shows GCTA-estimated genetic correlations (and standard errors, SE) between ‘g’ and learning abilities for more than 2,238 12-year-old UK twins (randomly selecting only one member of each twin pair to control for potential confounds, such as birth order) based on 1.7 million SNPs measured from the Affymetrix 6.0 GeneChip or imputed from HapMap 2,3 and WTCCC controls (Trzaskowski et al. 2013). Genetic correlations are significant and substantial for all three comparisons—between ‘g’ and language (0.81), mathematics (0.74), and reading (0.89). The GCTA-estimated genetic correlations between ‘g’ and learning abilities are similar in magnitude to the GCTA-estimated genetic correlation between height and weight (0.76). In addition, Table 1 includes bivariate results for ‘g’ versus height and ‘g’ versus weight as ‘negative controls’; their phenotypic correlations are both 0.07. As expected, these comparisons yielded negligible and nonsignificant genetic correlations (−0.03 and −0.06, respectively).

... A more novel question, and central to the present paper, is why, as we have shown here, bivariate genetic correlations estimated by GCTA are as great as twin study estimates. The likely reason is that attenuation of the estimated additive genetic variance due to imperfect linkage disequilibrium between causal variants and genotyped SNPs applies to both the additive genetic variance of the two traits and to their additive genetic covariance by the same proportion. Thus, the GCTA estimate of the genetic correlation is unbiased because it is derived from the ratio between genetic covariance and the genetic variances of the two traits.

Are generalist genes all in the mind (cognition) or are they in the brain as well? That is, genetic correlations between cognitive and learning abilities might be epiphenomenal in the sense that multiple genetically independent brain mechanisms could affect each ability, creating genetic correlations among abilities. However, the genetic principles of pleiotropy (each gene affects many traits) and polygenicity (many genes affect each trait) lead us to predict that generalist genes have their effects further upstream, creating genetic correlations among brain structures and functions, a prediction that supports a network view of brain structure and function.