Inference on population structure using multilocus genotype data STRUCTURE V2'1 Pritchard, J'K', and

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Inference on population structure using
multi-locus genotype dataSTRUCTURE V2.1
Pritchard, J.K., and Wen, W. (2004)
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Model based cluster analysis

• We assume some statistical distribution on each
  individual

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The mixture of normal distributions model (a
very simple case)
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N(0.098,0.67) y N(5.098,1.69)
o o o o o o o
o o ooo o oo o
o
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the f.d.p are
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or the mixture (each individual follows the
distribution)
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f(y) 0.7 N(0.098, 0.67) 0.3 N(5.098, 1.69)
o o o o o o o
o o ooo o oo o o
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Membership probability

• Pyi ? ?1 Pyi ? ?2 1
• i 1,2,,n (individuals)
• if Pyi ? ?1 gt Pyi ? ?2 then yi ? ?1

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Two variables X1, X2
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Mixture of three Normal bivariate
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with f.d.p
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Where
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Inference on population structure using
multi-locus genotype dataSTRUCTURE V2.1
Pritchard, J.K., and Wen, W. (2004)

• Pritchard, Stephens, and Donnelly (2000)
• Falush, Stephens, and Pritchard (2003)

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Main objective

• Assign individuals to populations on the bases of
  their genotypes, while simultaneously estimating
  population allele frequencies

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Other objectives

• Begin with a set of predefined populations and to
  classify individuals of unknown origin
• Identify the extent of admixture of individuals
• Infer the origin of particular loci in the
  sampled individuals

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Structure is a Model Based method of clustering

• (we must be assumptions about a lot of parameters
  and distributions)

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Four basic models

• Model without admixture
• each individual is assumed to originate in one
  (only one) of K populations
• Model with admixture
• each individual is assumed to have inherited some
  proportion of its ancestry from each of K
  populations

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Four basic models

• Linkage model
• Chunks of chromosomes as derived as intact
  units from one or another K population and all
  allele copies on the same chunk derive from the
  same population.
• The model consider the derived correlations in
  ancestry

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Four basic models

• F model
• The populations all diverged from a common
  ancestral population at the same time, but allows
  that the populations may have experienced
  different amounts of drift since the divergence
  event

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Assumptions

• Our main modeling assumptions are
  Hardy-Weinberg equilibrium within populations and
  complete linkage equilibrium between loci within
  populations
• Loosely speaking, the idea here is that the
  model accounts for the presence oh HWD or LD by
  introducing population structure and attempts to
  find populations groupings that (as far as
  possible) are not in disequilibrium

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Data

• Consider a sample of N individuals each one
  genotyped at L loci
• Assume that the individuals represent a mixture
  of K unobserved populations (K unknown)
• If diploid, we have an N2L data matrix X
• If n-ploid X is N
• where Jl is the number of alleles at the lth
  locus

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l 1 l l l
L j1 j2 j1 j2 j1
j2
X is N2L
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Example
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Model without admixture

• each individual is assumed to originate in one
  (only one) of K populations

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P-matrix (allele frequencies by population)
l 1 l l l
L j1 j2 j1 j2 j1
j2
pklj is the frequency of the jth allele, at the
lth loci, at the kth population k1,2,,K
l1,2,,L j1,2 (diploid)
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z-vector (membership of the ith individual to
kth population)

• If the ith individual is a member of the kth
  population then z(i) k

• P(z(i) k) is the membership
• probability

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Model with admixture

• each individual is assumed to have inherited some
  proportion of its ancestry from each of K
  populations

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P-matrix is equal to the above model
Q-matrix (proportion of the genome of
the ith individual inherited from the kth
population)
i1,2,,N k1,2,,K
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Z-matrix
zl(i,j) is equal to k if the jth allele at the
lth loci at the ith individual was originated
from the kth population k1,2,,K l1,2,,L
j1,2 (diploid)
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F Model

• The populations all diverged from a common
  ancestral population at the same time, but allows
  that the populations may have experienced
  different amounts of drift since the divergence
  event

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Ancestral population (diploid)
Conditional on PA
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F-model
Fk is the drift rate of the kth population, and
it is associated to the Wrights Fst
Fk
pklj pAlj
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Interpreting FST

• Can range from 0 (no genetic differentiation) to
  1 (fixation of alternative alleles).
• Wrights Guidelines
• 0 - 0.05, little differentiation
• 0.05 0.15, moderate
• 0.15 0.25, great
• gt 0.25, very great

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The Dirichlet distribution

• The probability density of the Dirichlet
  distribution for variables
  p (p1, p2,, pn)
• with parameters u (u1,u2,,un)
• is defined by
• The parameters ui can be interpreted as prior
  observation counts''

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Kullback-Leiber

• The Kullback-Leiber divergence is a non-negative
  value and equals 0 only when the two
  distributions are identical. The Kullback-Leiber
  divergence is a measure of the discriminative
  power between the probability distributions of
  the two classes