Spectral Graph Theory and Ancestry in Genomewide Association Studies

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Spectral Graph Theory and Ancestry in Genome-wide
Association Studies

• Kathryn Roeder
• Coauthors Diana Luca, Ann Lee,
• Bernie Devlin Bert Klei

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Find Genetic Variants Associated with Disease

• Data
• Cases
• Controls
• Several unrecorded ethnic groups
• ½ million genetic markers (SNPs)
• Analysis Logistic regression
• Test each variant for association

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Data

• X counts the number of minor alleles at each
  marker across genome.
• Y takes on value 1 for affected and 0 for
  unaffected individuals.

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Population Stratification
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Distribution of the O blood type in native
populations around the world.
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Spurious Associations

• Common genetic approach to control spurious
  findings
• Restrict to common continental ancestry
• Statistical approaches using G
• Match cases and controls by ancestry

G2
C
Y
G1
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High Dimensional Data

• N
• Information content
• Each SNP offers related information about
  ancestry
• Information per SNP is minimal
• Over genome it counts up

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Clusters of subpopulations

• Individuals from a common subpopulation are
  correlated and form clusters.

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Principal component map

• Let X be a N x L matrix, centered and scaled by
  allele frequency.
• Project XXt U?Ut into D-dimensional space
  defined by the D largest eigenvalues.
• Ancestry of ith individual is (ui1,ui2,,uiD)

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PCA and Multidimensional Scaling
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Representation of Ancestry (Patterson et al.)
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Eigenstrat (Price et al, Nature Genetics 2006)

• Modeling ancestry using Eigenstrat
• Find eigenvectors U1, U2, , U10
• Regress out eigenvectors to remove confounding of
  ancestry

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Alternative to Regression is Matching
Pair Match vs Full Match
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Distance Between Individuals

• Match cases and controls that are similar in
  ancestry
• Distance Euclidean distance between
  eigenvectors, scaled by eigenvalues
• Use eigenvalues ?1, ?2, to weight importance of
  axes

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Determining D ( dimensions)

• Test for structure equivalent to testing for
  large eigenvalues. (Patterson et al. 2006)
• Standardize distribution based on
• N subjects
• L independent SNPs
• Tracy Widom distribution describes distribution
  of the largest eigenvalue. Get a P-value for each
  dimension tested. (Johnstone 1991).
• Weakness distribution depends on L which must be
  estimated. MOM estimator assumes null.

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Association data

• Cases
• Americans of European descent
• Controls
• Northern Germans
• Southern Germans

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Starting Data (D17)
PO or FS similarities
Duplicates or MZ-twins
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Northern/Southern Germans
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After Eliminating outliers and unmatchables
(D2)
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Performance good, requires care

• Outliers cases (or controls) that are unusual
  at many SNPs
• High influence on eigenmap, failure to discover
  key dimensions of ancestry
• Unmatchables cases far from controls or vice
  versa.
• Caution must remove outliers and unmatchables

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Multi-ethnic example.

• Cases and controls descended from
• Europe
• Africa
• Asia
• Expect hierarchical structure
• Multi-continent
• Differences within continent

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3 Major Ethnic Groups (D2)
African Descent
European Descent
Asian Descent
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European sub-sample D4
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Problems

• Approach is too sensitive to outliers
• Not able to discover hierarchical structure
  without performing several iterations of
  analysis.
• Tests for number of dimensions are not reliable
  when data are heterogeneous.
• Estimate of independent dimensions (L) is poor
  when data are heterogenious.
• Try a new eigenmap.

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Can spectral graph theory help?

• For N subjects, define a graph G (V,E)
• V vertices
• E edges
• Wij weights between vertices (i,j)
• large value for tight connection
• 0 for no connection
• Must be positive and symmetric
• Di degree of the vertex

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Defining a Weight matrix

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Goal

• Want to embed a weighted graph into D-dimensional
  space so that similar vertices are close.
• A map that meets this requirement is the
  Laplacian eigenmap of G.
• Based on top non-trivial eigenvectors

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Spectral embedding and clustering

• Close connection between locality-preserving
  spectral embeddings, spectral clustering and
  cuts.
• Min Cut finds outliers and clumps
• Other weighted cuts find major splits and
  clusters

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Spectral Embedding Laplacian

• Choices for embedding.
• Degree matrix
• D diag(D1,,DN)
• Embedding based on decomposing
• W
• Laplacian L D-W
• Normalized Laplacian D-1/2LD-1/2

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Cuts
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Choices of eigenmaps

• W
• Emphasizes outliers
• PCA analysis similar since
• W 0-thresholded version of K
• Normalized Laplacian D-1/2(D-W)D-1/2
• Finds major clusters
• Robust to outliers

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Eigen-gap Heuristic

• Tracy-Widom theory for eigenvalues of a
  covariance matrix no longer applies.
• The eigen-gap heuristic says,
• find the largest k such that
• ?k ?k1
• is significantly greater than 0.
• Find the distribution of this statistic by
  simulation.
• Distribution depends on N, not L. Easy to obtain.

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Approach

• Find eigenvectors and eigenvalues of normalized
  Laplacian
• Use square-root-threshold weights
• Use eigengaps to determine dimension

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Example with hierarchical structure(3 major
groups, 3 subgroups in 1)
Groups 2 and 3 Black Group 1
clusters Red
Blue Green
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POPRES data

• Worldwide sample
• 6000 people worldwide
• 500,000 SNPs
• Ancestry (only used to validate)
• Country of birth
• Country of birth of parents

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Eigenvectors for POPRES
Africa Europe China Mexico India
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Eigenvectors for POPRES
Africa Europe China Mexico India
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Eigenvectors for POPRES
Africa Europe China Mexico India
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Eigenvectors for POPRES
Africa Europe China Mexico India
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Clustering

• Distance calculated based on D eigenvectors.
• Hierarchical clustering (Wards algorithm) at
  each step the union of every possible cluster
  pair is considered. The clusters whose fusion
  results in the minimum information loss are
  combined.
• Start with k2
• Test for population structure
• If one cluster is homogeneous, set aside the
  cluster and continue
• Repeat this process until all clusters are
  homogeneous
• K homogeneous.
• Find matches and unmatchable observations

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Overall Clustering Results

• 4 Non-European clusters
• Africa
• China
• Mexico
• India
• 8 European clusters

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Clusters of European Ancestry

• Clusters generate a map of history and geography
  of Europe
• Portugal, Spain
• Italy
• Italy
• Switzerland (France)
• Ireland, England, UK, (France)
• France, Belgium, (Switz, Italian)
• Netherlands, Belgium, Germany, (UK)
• Germany, Poland, Hungary, Romania
• Other nationalities with small samples fit too
• Slovakia, Slovenia, Kosovo, Montenegro, Czech,
  Croatia

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Comparison to principal components

• Using effective number of independent SNPs,
  find D3
• missing structure in Europeans.
• Using actual number of SNPs, D27
• pick up a lot of noise
• Clustering finds 127 clusters
• Unhelpful for interpretation.

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Summary

• The traditional eigenmap based on principal
  components has bad properties.
• Normalized Laplacian leads to a better spectral
  embedding that helps to find more meaningful
  ancestry vectors.
• Resulting eigenvectors emphasize major clusters
  and are robust to outliers.
• Final clustering corresponds to true ancestry
  remarkably well.
• Matching cases and controls from homogeneous
  clusters provides appropriate strata for
  conditional logistic analysis.