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Inbr

eedi

ng c

oeff.

Relatedness and differentiationin arbitrary population structures

Alejandro Ochoa, John D. Storey Lab, Princeton University7 DrAlexOchoa � viiia.org/research/ R ochoa@princeton.edu

Why study relatedness?

Human geneticsis fascinating!

Search fordisease-causinggenetic variants

Heritability ofcomplex traits

Animal and plantbreeding

The kinship coefficient for siblings: 14 on average

Visscher et al. (2006)

The inbreeding coefficient in populations

Measurements relativeto a reference pop.:

Inbreeding = 0 in thelocal population

Inbreeding ≥ 0 relativeto a distant ancestralpopulation

Better measured usingcovariance

Recently admixed populations

African-Americans

Baharian et al. (2016)

Hispanics

Moreno-Estrada et al. (2013)

Kinship model for genotypes

symbol meaningT ref ancestral populationi locus index

j , k individual indexespTi ref allele frequencyxij genotype (num ref alleles)ϕTjk kinship of j , k

f Tj inbreeding of j

Statistical model:

E[xij |T ] = 2pTi ,

Var(xij |T ) = 2pTi(1− pTi

)(1 + f Tj ),

Cov(xij , xik |T ) = 4pTi(1− pTi

)ϕTjk .

(Wright 1921, 1951; Malécot 1948; Jacquard 1970).

We developed a new kinship estimation framework that works forarbitrary population structures!

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LebaneseJew_iraqi

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Jew_IranianTurkishMaltese

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SicilianFrenchN

SpanishSWSpanishNE

FrenchSItalian_North

SardinianBasque

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EnglishIcelandic

NorwegianCzech

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AlbanianGreek

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UkrainianEstonianRussianFinnish

MordovianRomanian

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ship

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New kinshipestimates

Genotypes from “Human Origins” (Lazaridis

et al. 2014, 2016)

Edited from Ephert [CC BY-SA 3.0], via

Wikimedia Commons

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Standard kinshipestimates

Genotypes from “Human Origins” (Lazaridis

et al. 2014, 2016)

Edited from Ephert [CC BY-SA 3.0], via

Wikimedia Commons

Population-level inbreeding

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0.2

0.3

0.4

Inbr

eedi

ng c

oeff.

Differentiation (FST) previously underestimated

0.0 0.1 0.2 0.3 0.4 0.5

020

40

Inbreeding Coefficient

Den

sity

Subpopulations

SAfricaNAfricaMiddleEastEuropeCaucasusSAsiaNAsiaEAsiaAmrOc

FST estimates

WCBayeScanHudsonKNew

Generalized FST: reduction in mean heterozygosity from ancestral population.(Prev. FST: proportion of variation between independent subpopulations).

00.

10.

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Kin

ship

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0.5

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Individuals

Anc

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Kinship driven byadmixture in Hispanics

New kinship estimates

Genotypes from the 1000 Genomes Project (2012)

Improved relatedness has repercussions across genetics!

Easy to measureroutinely

Search fordisease-causinggenetic variants

Heritability ofcomplex traits

Animal and plantbreeding

Acknowledgments

John D. StoreyAndrew BassIrineo CabrerosWei HaoRiley Skeen-Gaar

Neo Christopher ChungUniversity of Warsaw

Funding:National Institutes of HealthOtsuka Pharmaceuticals

Lewis-Sigler Institute for Integrative Genomics

Wright’s FST

Total inbreeding: FIT =1|S |∑j∈S

f Tj ,

Local inbreeding: FIS =1|S |∑j∈S

f Sj ,

Structural inbreeding: FST =FIT − FIS

1− FIS.

The generalized FST

Need the new concept of local subpopulations Lj (separates total from localinbreeding): (

1− f Tj)

=(1− f

Ljj

)(1− f TLj

).

Generalized FST: applicable to arbitrary population structures, equals previousdefinition for non-overlapping subpopulations:

FST =n∑

j=1

wj fTLj.

Mean heterozygosity in a structured population:

H̄i = 2pTi(1− pTi

)(1− FST) .

Bias in FST estimators for independent subpopulations

Previous estimators are biased for n dependent subpopulations even when eachsubpopulation is infintely large (known AFs πij):

p̂Ti =1n

n∑j=1

πij , σ̂2i =

1n − 1

n∑j=1

(πij − p̂Ti

)2, θ̄T =

1n2

n∑j=1

n∑k=1

θTjk ,

F̂ indepST =

m∑i=1

σ̂2i

m∑i=1

p̂Ti(1− p̂Ti

)+ 1

nσ̂2i

a.s.−−−→m→∞

FST − 1n−1

(nθ̄T − FST

)1− 1

n−1

(nθ̄T − FST

) .

Bias in standard kinship estimator

Estimator has a distorted bias (varies for every pair of individuals j , k):

p̂Ti =12

n∑j=1

wjxij , ϕ̄Tj =

n∑k ′=1

wk ′ϕTjk ′ , ϕ̄T =

n∑j ′=1

n∑k ′=1

wj ′wk ′ϕTj ′k ′

ϕ̂T ,stdjk =

m∑i=1

(xij − 2p̂Ti

) (xik − 2p̂Ti

)4

m∑i=1

p̂Ti(1− p̂Ti

) a.s.−−−→m→∞

ϕTjk − ϕ̄T

j − ϕ̄Tk + ϕ̄T

1− ϕ̄T.

Standard ancestral variance estimate also downwardly biased:

E[p̂Ti(1− p̂Ti

)∣∣T ] = pTi(1− pTi

) (1− ϕ̄T

),

New estimator: two stepsStep 1: “pre-adjusted” kinship estimator with uniform bias.

ϕ̂T ,preadjjk =

m∑i=1

(xij − 1)(xik − 1)− 1

4m∑i=1

p̂Ti(1− p̂Ti

) + 1 a.s.−−−→m→∞

ϕTjk − ϕ̄T

1− ϕ̄T,

Step 2: Estimate minimum kinship, use to unbias “step 1” estimates.

ϕ̂T ,preadjmin

a.s.−−−→m→∞

− ϕ̄T

1− ϕ̄T, ϕ̂T ,new

jk =ϕ̂T ,preadjjk − ϕ̂T ,preadj

min

1− ϕ̂T ,preadjmin

a.s.−−−→m→∞

ϕTjk

f̂ T ,newj = 2ϕ̂T ,new

jj − 1 a.s.−−−→m→∞

f Tj , F̂ newST =

n∑j=1

wj f̂T ,newj

a.s.−−−→m→∞

FST.