A geometric relationship of F2, F3 and F4-statistics with principal component analysis

Principal component analysis (PCA) and F-statistics sensu Patterson are two of the most widely used population genetic tools to study human genetic variation. Here, I derive explicit connections between the two approaches and show that these two methods are closely related. F-statistics have a simple geometrical interpretation in the context of PCA, and orthogonal projections are a key concept to establish this link. I show that for any pair of populations, any population that is admixed as determined by an F-3-statistic will lie inside a circle on a PCA plot. Furthermore, the F-4-statistic is closely related to an angle measurement, and will be zero if the differences between pairs of populations intersect at a right angle in PCA space. I illustrate my results on two examples, one of Western Eurasian, and one of global human diversity. In both examples, I find that the first few PCs are sufficient to approximate most F-statistics, and that PCA plots are effective at predicting F-statistics. Thus, while F-statistics are commonly understood in terms of discrete populations, the geometric perspective illustrates that they can be viewed in a framework of populations that vary in a more continuous manner.This article is part of the theme issue 'Celebrating 50 years since Lewontin's apportionment of human diversity'.

Automated summary

This article explores the relationship between principal component analysis (PCA) and F-statistics in studying human genetic variation. The author derives explicit connections between the two approaches and demonstrates that F-statistics can be interpreted geometrically in the context of PCA. The results show that PCA plots are effective at predicting F-statistics, providing a new perspective for understanding human genetic variation.