### Population structure math: two recent papers

Analysis of Population Structure: A Unifying Framework and Novel Methods Based on Sparse Factor Analysis
Two different approaches have become widely used in the analysis of population structure: admixture-based models and principal components analysis (PCA). In admixture-based models each individual is assumed to have inherited some proportion of its ancestry from one of several distinct populations. PCA projects the individuals into a low-dimensional subspace. On the face of it, these methods seem to have little in common. Here we show how in fact both of these methods can be viewed within a single unifying framework. This viewpoint should help practitioners to better interpret and contrast the results from these methods in real data applications. It also provides a springboard to the development of novel approaches to this problem. We introduce one such novel approach, based on sparse factor analysis, which has elements in common with both admixture-based models and PCA. As we illustrate here, in some settings sparse factor analysis may provide more interpretable results than either admixture-based models or PCA.
Theoretical Formulation of Principal Components Analysis to Detect and Correct for Population Stratification
The Eigenstrat method, based on principal components analysis (PCA), is commonly used both to quantify population relationships in population genetics and to correct for population stratification in genome-wide association studies. However, it can be difficult to make appropriate inference about population relationships from the principal component (PC) scatter plot. Here, to better understand the working mechanism of the Eigenstrat method, we consider its theoretical or “population” formulation. The eigen-equation for samples from an arbitrary number () of populations is reduced to that of a matrix of dimension , the elements of which are determined by the variance-covariance matrix for the random vector of the allele frequencies. Solving the reduced eigen-equation is numerically trivial and yields eigenvectors that are the axes of variation required for differentiating the populations. Using the reduced eigen-equation, we investigate the within-population fluctuations around the axes of variation on the PC scatter plot for simulated datasets. Specifically, we show that there exists an asymptotically stable pattern of the PC plot for large sample size. Our results provide theoretical guidance for interpreting the pattern of PC plot in terms of population relationships. For applications in genetic association tests, we demonstrate that, as a method of correcting for population stratification, regressing out the theoretical PCs corresponding to the axes of variation is equivalent to simply removing the population mean of allele counts and works as well as or better than the Eigenstrat method.