Population stratification using a statistical model on hypergraphs

Alexei Vazquez

Phys. Rev. E 77, 066106 – Published 10 June 2008

Abstract

Population stratification is a problem encountered in several areas of natural science, engineering, and public health. We tackle this problem by mapping a population and its element attributes onto a hypergraph, a natural extension of the concept of graph or network to encode associations among any number of elements. On this hypergraph, we construct a statistical model reflecting our intuition about how the element attributes can emerge from a postulated population structure. Finally, we introduce the concept of stratification representativeness as a mean to identify the simplest stratification already containing most of the information about the population structure. We demonstrate the power of this framework stratifying an animal and a human population based on phenotypic and genotypic properties, respectively.

Received 10 December 2007

DOI:https://doi.org/10.1103/PhysRevE.77.066106

Authors & Affiliations

Alexei Vazquez

The Simons Center for Systems Biology Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA

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Vol. 77, Iss. 6 — June 2008

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Images

Figure 1
(Color online) Hypergraph: (a) A hypergraph with three edges. Each edge is represented by a circle and its composed by the nodes within the circle. (b) Bipartite graph representation of the hypergraph in (a), the squares representing the hypergraph edges. (c) and (d) Nearest-neighbor mapping of a graph onto a hypergraph. The graph in (c) is mapped onto the hypergraph in (d), where each hyperedge represents a set of nearest neighbors of a node in graph (c), as indicated by the enumeration.Reuse & Permissions
Figure 2
Stratification according to phenotypic attributes: (a) A list of animals is given together with certain attributes characterizing them. The complete data set is available from Ref. [6]. Except for the attribute “legs,” 1 and 0 indicate possession or not, respectively, of the corresponding attribute. The problem consists in determining the optimal stratification of the animal population based on the provided attributes. (b) Hypergraph representing the zoo data. Each line corresponds with an edge, whose elements are specified within the right column. (c) ML stratification for the case of eight groups.Reuse & Permissions
Figure 3
Mapping genotypic information into a hypergraph: (a) A population of individuals, labeled by $1, 2, 3, \dots$ , is given together with their genotype for specific DNA positions within. These positions have been selected because they exhibit significant variation across the human population, referred to as single nucleotide polymorphisms (SNPs), and are labeled using the standard SNP notation: rsNUMBER. The letters A, C, G, and T represent nucleotides, and two letters are reported because each DNA position appears in two different chromosome copies. (b) Hypergraph representing the genotypic data. Each line corresponds with an edge, whose elements are specified within the right column.Reuse & Permissions
Figure 4
Test example: The best choice of $n_{g}$ and normalized mutual information $I$ between the predicted population optimal stratification and the original stratification as a function of degree $K$ . These results are obtained computing the optimal stratification for $n_{g} = 1, \dots, 20$ using the EM algorithm with one initial condition. The optimal $n_{g}$ was obtained using the AIC (solid circles), the representativeness criterion (open squares), and assuming it equal to 4 (solid triangles). In (a) and (b) the case study hypergraphs have $n = 100$ nodes divided into four groups of equal size and $m = 10$ edges, while in (c) and (d) the number of edges is doubled to $m = 20$ . In (e) and (f) the case study hypergraphs are similar to those in (a) and (b) except that the groups have different sizes: 40, 30, 20, and 10 nodes.Reuse & Permissions
Figure 5
Representativeness plot: Representativeness as a function of the number of groups for the (a) zoo and (b) MDM4 problems. Different symbols indicate different numerical accuracies of the numerical algorithm to find the ML stratification. The arrow indicates the number of groups maximizing the representativeness. The different symbols indicate different numbers of initial conditions for the EM algorithm, from 100 (open symbols) to 10 000 (solid symbols).Reuse & Permissions
Figure 6
Stratification according to genotypic attributes: A population of 90 Caucasians is studied, focusing on SNPs within the MDM4 gene, as reported by the HapMap project [9]. SNPs with no variation within this particular subpopulation have been excluded. A, C, G, and T represent the different nucleotides, and NN represents data that are not available. The specific SNPs under consideration are indicated by the bottom labels, using the standard SNP notation. The figure shows the ML stratification for the case of five (left) and eight (right) groups, the latter corresponding with the best choice of $n_{g}$ according to the representativeness criterion [Fig. 5b]. The vertical lines in between indicate the SNPs at which the adjacent groups differ significantly.Reuse & Permissions

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