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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

1 / 35

My research areas / Contributions1. Stratified False Discovery Rate (FDR)

I Finding: per-stratum local FDR maximizes power controlling overall FDRI Improved power in protein domain predictionI Identified protein classes with problematic statistics

2. Genetic and other factors controlling the placebo responseI Collaboration with Otsuka PharmaceuticalI Mixed-effects modeling of longitudinal response in drug trials for

Schizophrenia, Bipolar Disorder, and Major Depressive DisorderI Genome-wide association study for individual-specific placebo response

3. Kinship and FST for arbitrary population structuresI Motivation: world-wide human population structureI Generalized definitions and modelsI Novel bias calculations validated by simulationsI Novel non-parametric estimator with greatly improved accuracy

2 / 35

My research areas / Contributions1. Stratified False Discovery Rate (FDR)

I Finding: per-stratum local FDR maximizes power controlling overall FDRI Improved power in protein domain predictionI Identified protein classes with problematic statistics

2. Genetic and other factors controlling the placebo responseI Collaboration with Otsuka PharmaceuticalI Mixed-effects modeling of longitudinal response in drug trials for

Schizophrenia, Bipolar Disorder, and Major Depressive DisorderI Genome-wide association study for individual-specific placebo response

3. Kinship and FST for arbitrary population structuresI Motivation: world-wide human population structureI Generalized definitions and modelsI Novel bias calculations validated by simulationsI Novel non-parametric estimator with greatly improved accuracy

2 / 35

My research areas / Contributions1. Stratified False Discovery Rate (FDR)

I Finding: per-stratum local FDR maximizes power controlling overall FDRI Improved power in protein domain predictionI Identified protein classes with problematic statistics

2. Genetic and other factors controlling the placebo responseI Collaboration with Otsuka PharmaceuticalI Mixed-effects modeling of longitudinal response in drug trials for

Schizophrenia, Bipolar Disorder, and Major Depressive DisorderI Genome-wide association study for individual-specific placebo response

3. Kinship and FST for arbitrary population structuresI Motivation: world-wide human population structureI Generalized definitions and modelsI Novel bias calculations validated by simulationsI Novel non-parametric estimator with greatly improved accuracy

2 / 35

Why study relatedness?

Human geneticsis fascinating!

Pop. structureconfoundsassociationstudies (GWAS)

Heritability ofcomplex traits

Animal and plantbreeding

3 / 35

Why study relatedness?

Human geneticsis fascinating!

Pop. structureconfoundsassociationstudies (GWAS)

Heritability ofcomplex traits

Animal and plantbreeding

3 / 35

Why study relatedness?

Human geneticsis fascinating!

Pop. structureconfoundsassociationstudies (GWAS)

Heritability ofcomplex traits

Animal and plantbreeding

3 / 35

Why study relatedness?

Human geneticsis fascinating!

Pop. structureconfoundsassociationstudies (GWAS)

Heritability ofcomplex traits

Animal and plantbreeding

3 / 35

Single Nucleotide Polymorphism (SNP) data

⇒

Genotype xijCC 0CT 1TT 2

⇒

Individuals

Loci

0 2 2 1 1 0 10 2 1 0 12 ...

X

4 / 35

Single Nucleotide Polymorphism (SNP) data

⇒

Genotype xijCC 0CT 1TT 2

⇒

Individuals

Loci

0 2 2 1 1 0 10 2 1 0 12 ...

X

4 / 35

Single Nucleotide Polymorphism (SNP) data

⇒

Genotype xijCC 0CT 1TT 2

⇒

Individuals

Loci

0 2 2 1 1 0 10 2 1 0 12 ...

X4 / 35

Hardy-Weinberg Equillibrium (HWE): Binomial draws

xij = genotype at locus i for individual j .

pTi = frequency of reference allele at locus i , (ancestral) population T .

Under HWE:

Pr(xij = 2|pTi ) =(pTi)2,

Pr(xij = 1|pTi ) = 2pTi(1− pTi

),

Pr(xij = 0|pTi ) =(1− pTi

)2.

HWE not valid under population structure!

5 / 35

Hardy-Weinberg Equillibrium (HWE): Binomial draws

xij = genotype at locus i for individual j .

pTi = frequency of reference allele at locus i , (ancestral) population T .

Under HWE:

Pr(xij = 2|pTi ) =(pTi)2,

Pr(xij = 1|pTi ) = 2pTi(1− pTi

),

Pr(xij = 0|pTi ) =(1− pTi

)2.

HWE not valid under population structure!

5 / 35

Hardy-Weinberg Equillibrium (HWE): Binomial draws

xij = genotype at locus i for individual j .

pTi = frequency of reference allele at locus i , (ancestral) population T .

Under HWE:

Pr(xij = 2|pTi ) =(pTi)2,

Pr(xij = 1|pTi ) = 2pTi(1− pTi

),

Pr(xij = 0|pTi ) =(1− pTi

)2.

HWE not valid under population structure!

5 / 35

Goal: measure dependence structure of genotype matrix columns

IndividualsLo

ci0 2 2 1 1 0 10 2 1 0 12 ...

X

High-dimensional binomial data

Population structure⇒ dependence between individuals (columns)

Linkage disequilibrium⇒ dependence between loci (rows)

6 / 35

Goal: measure dependence structure of genotype matrix columns

IndividualsLo

ci0 2 2 1 1 0 10 2 1 0 12 ...

X

High-dimensional binomial data

Population structure⇒ dependence between individuals (columns)

Linkage disequilibrium⇒ dependence between loci (rows)

6 / 35

Goal: measure dependence structure of genotype matrix columns

IndividualsLo

ci0 2 2 1 1 0 10 2 1 0 12 ...

X

High-dimensional binomial data

Population structure⇒ dependence between individuals (columns)

Linkage disequilibrium⇒ dependence between loci (rows)

6 / 35

Model parametersIBD(T ): “Identical By Descent” for ancestral population T — shared coin flips

f Tj : Inbreeding coefficientPr. that the two alleles at a random locus of individual j are IBD(T )

Var(xij |T ) = 2pTi(1− pTi

)(1 + f Tj )

ϕTjk : Kinship coefficient

Pr. that two alleles, one at random from each of individuals j and k , at onerandom locus are IBD(T )

Cov(xij , xik |T ) = 4pTi(1− pTi

)ϕTjk

FST: Fixation indexPr. that two random alleles in a subpopulation at a random locus are IBD(T )

7 / 35

Model parametersIBD(T ): “Identical By Descent” for ancestral population T — shared coin flips

f Tj : Inbreeding coefficientPr. that the two alleles at a random locus of individual j are IBD(T )

Var(xij |T ) = 2pTi(1− pTi

)(1 + f Tj )

ϕTjk : Kinship coefficient

Pr. that two alleles, one at random from each of individuals j and k , at onerandom locus are IBD(T )

Cov(xij , xik |T ) = 4pTi(1− pTi

)ϕTjk

FST: Fixation indexPr. that two random alleles in a subpopulation at a random locus are IBD(T )

7 / 35

Model parametersIBD(T ): “Identical By Descent” for ancestral population T — shared coin flips

f Tj : Inbreeding coefficientPr. that the two alleles at a random locus of individual j are IBD(T )

Var(xij |T ) = 2pTi(1− pTi

)(1 + f Tj )

ϕTjk : Kinship coefficient

Pr. that two alleles, one at random from each of individuals j and k , at onerandom locus are IBD(T )

Cov(xij , xik |T ) = 4pTi(1− pTi

)ϕTjk

FST: Fixation indexPr. that two random alleles in a subpopulation at a random locus are IBD(T )

7 / 35

Model parametersIBD(T ): “Identical By Descent” for ancestral population T — shared coin flips

f Tj : Inbreeding coefficientPr. that the two alleles at a random locus of individual j are IBD(T )

Var(xij |T ) = 2pTi(1− pTi

)(1 + f Tj )

ϕTjk : Kinship coefficient

Pr. that two alleles, one at random from each of individuals j and k , at onerandom locus are IBD(T )

Cov(xij , xik |T ) = 4pTi(1− pTi

)ϕTjk

FST: Fixation indexPr. that two random alleles in a subpopulation at a random locus are IBD(T )

7 / 35

Existing approaches

1. FST estimationI For independent subpopulations only!I Weir-Cockerham (WC) estimator (1984) — 15K citations!I “Hudson” pairwise estimator (2013) tweaks WCI BayeScan (2008) — 1.2K citations

2. Kinship estimationI “Standard” kinship estimator (1950s)

I Used by most modern GWAS approaches that control for population structure(PCA, LMM, adj. χ2; top paper 6K citations)

I GCTA heritability estimation (2 papers: 4K citations)I Our novel finding: accuracy requires unstructured population

(a minority of closely-related individuals)

8 / 35

Existing approaches

1. FST estimationI For independent subpopulations only!I Weir-Cockerham (WC) estimator (1984) — 15K citations!I “Hudson” pairwise estimator (2013) tweaks WCI BayeScan (2008) — 1.2K citations

2. Kinship estimationI “Standard” kinship estimator (1950s)

I Used by most modern GWAS approaches that control for population structure(PCA, LMM, adj. χ2; top paper 6K citations)

I GCTA heritability estimation (2 papers: 4K citations)I Our novel finding: accuracy requires unstructured population

(a minority of closely-related individuals)

8 / 35

Dataset: Human Origins (Lazaridis et al. 2014, 2016)

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Khomani

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Biaka

BantuSA

BantuKenya

Luhya

YorubaEsan

MandenkaGambian

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Luo

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Datog

Somali

Jew_Ethiopian

Saharawi

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Algerian Tunisian

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Yemeni

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SyrianLebanese_Muslim

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Jew_Turkish

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Lebanese

Jew_iraqi

Iranian_Bandari

Iranian

Jew_Iranian

TurkishItalian_SouthSicilian

Maltese

SpanishSW

Canary_Islander

SpanishNE

FrenchS

Italian_North

Sardinian

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FrenchN

Greek

BulgarianCroatian

English

Scottish

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Icelandic

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Cypriot

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Turkmen

ChechenAbkhasian

Georgian

Adygei

Armenian

BalochiBrahui

Makrani

Kalash

Pathan

Sindhi

Burusho

Punjabi

Hazara

Jew_Cochin

Gujarati Bengali

Uzbek

Mansi

Kusunda

Uygur

Altaian

Dolgan

Even

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Kyrgyz

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Tubalar Tuvinian

ThaiCambodian

Ami

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Dai

Han

Japanese

Kinh

Korean

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Miao

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Naxi She

Tujia

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Subpopulations

SAfricaNAfricaMiddleEastEuropeCaucasusSAsiaNAsiaEAsiaAmrOc

2,066 indivs. from 163 locs. — 595,911 loci — SNP chip

9 / 35

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Edited from Ephert [CC BY-SA 3.0], viaWikimedia Commons

*Inbreeding coeffs. on diagonal

10 / 35

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YukagirItelmenKoryak

ChukchiEskimo

AleutAleut_Tlingit

KusundaCambodian

ThaiAmi

AtayalKinh

DaiLahuNaxi

YiTujiaHan

MiaoShe

TuMongola

KalmykDaur

JapaneseKorean

PimaMixe

MixtecZapotec

MayanPiapoco

SuruiKaritianaBolivian

QuechuaAustralian

NasoiPapuan

SAfrica NAfrica MiddleEast Europe Caucasus SAsia NAsia EAsia Amr Oc

SA

fric

aN

Afr

ica

Mid

dleE

ast

Eur

ope

Cau

casu

sS

Asi

aN

Asi

aE

Asi

aA

mr

Oc

−0.0

50

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Standard kinshipestimatesGenotypes from “Human Origins”(Lazaridis et al. 2014, 2016)

Edited from Ephert [CC BY-SA 3.0], viaWikimedia Commons

11 / 35

Only our new estimator is accurate in simulations

True KinshipA New estimateB Standard est.C

00.

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12 / 35

Population-level inbreeding increases with distance from Africa

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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

14 / 35

Only our new method estimates generalized FST accurately

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Wei

r−C

ocke

rham

Hud

sonK

Bay

eSca

n

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ndar

dK

insh

ip

New

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Wei

r−C

ocke

rham

Hud

sonK

Bay

eSca

n

Sta

ndar

dK

insh

ip

New

− −−

−

−

−

True FST

FSTindep limit

Estimates−

FS

T e

stim

ate

FST estimator

15 / 35

Recently-admixed populations

African-Americans

Baharian et al. (2016)

Hispanics

Moreno-Estrada et al. (2013)16 / 35

Admixed siblings from different populations?

Lucy and Maria, UK

Ochoa brothers, MX

High Admixture LD:

Moreno-Estrada et al. (2013)

Solution: treat every individual as its own population!

17 / 35

Admixed siblings from different populations?

Lucy and Maria, UK

Ochoa brothers, MX

High Admixture LD:

Moreno-Estrada et al. (2013)

Solution: treat every individual as its own population!

17 / 35

Admixed siblings from different populations?

Lucy and Maria, UK

Ochoa brothers, MX

High Admixture LD:

Moreno-Estrada et al. (2013)

Solution: treat every individual as its own population!

17 / 35

Admixed siblings from different populations?

Lucy and Maria, UK

Ochoa brothers, MX

High Admixture LD:

Moreno-Estrada et al. (2013)

Solution: treat every individual as its own population!17 / 35

Dataset: 1000 Genomes Project (2013)

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ACB

ESN

GWD

LWK

MSLYRI

GBR

FIN

IBS

TSI

CEU

BEB

GIH

ITU

PJL

STU

CDX

CHB

JPT

KHV

CHS

CLM

MXL

PEL

PUR

Subpopulations

SAfricaEuropeSAsiaEAsiaAmr

2,504 indivs. from 26 locs. — 20,417,698 loci (asc. in YRI) — WGS trios, etc.

18 / 35

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Individuals

Anc

estr

y fr

ac.

x

PU

RC

LMP

EL

MX

L

Pop

ulat

ion

x

AF

RA

MR

EU

R

Anc

estr

y

Kinship driven byadmixture in Hispanics

Our new kinship estimates

Genotypes from the 1000 Genomes Project (2013)

19 / 35

Comparison of population structures in simulation

T

S1

f TS1

S2

f TS2

...SK

f TSK

Indep. Subpops.T

S1 S2 ... SK

A1 A2 ... An

Admixture

00.

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ship

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20 / 35

FST in the independent subpopulation model

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Illustration.

FST =Var(pSi∣∣T)

pTi(1− pTi

) .Here FST = proportion of variance explained by pop. structure

21 / 35

FST in the independent subpopulation model

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Illustration.

FST =Var(pSi∣∣T)

pTi(1− pTi

) .Here FST = proportion of variance explained by pop. structure

21 / 35

Wright’s FST

T = Total, S = Subpopulation, I = Individual.

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.

22 / 35

Our generalized FSTNeed new “local” subpopulations Lj (separates total from local inbreeding):(

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 =1n

n∑j=1

Pr(xij = 1|T ) = 2pTi(1− pTi

)(1− FST) .

23 / 35

Our generalized FSTNeed new “local” subpopulations Lj (separates total from local inbreeding):(

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 =1n

n∑j=1

Pr(xij = 1|T ) = 2pTi(1− pTi

)(1− FST) .

23 / 35

Our generalized FSTNeed new “local” subpopulations Lj (separates total from local inbreeding):(

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 =1n

n∑j=1

Pr(xij = 1|T ) = 2pTi(1− pTi

)(1− FST) .

23 / 35

FST measures population structure / differentiation

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Median diff. SNP in Human Origins (rs7626601; given MAF ≥ 10%).

F̂WCST ≈ 0.0712 using Weir-Cockerham estimator and K = 163.

24 / 35

FST measures population structure / differentiation

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Median diff. SNP in Human Origins (rs7626601; given MAF ≥ 10%).

F̂WCST ≈ 0.0712 using Weir-Cockerham estimator and K = 163.

24 / 35

Comparison of population structures in simulation

T

S1

f TS1

S2

f TS2

...SK

f TSK

Indep. Subpops.T

S1 S2 ... SK

A1 A2 ... An

Admixture

00.

10.

2

Kin

ship

Indi

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als

25 / 35

Our admixture simulation (R package ‘bnpsd’ on CRAN)Intermediate subpop. diff.

FS

T

01

A

0.0

0.2

Intermediate subpop. spread

dens

ity

B

Admixture proportions

ance

stry

01

C

Discrete subpop. approx.

ance

stry

01

D

position in 1D geography

26 / 35

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).

27 / 35

Problem: common estimators not consistent under structure

Estimate of ancestral allele frequency:

p̂Ti =12

n∑j=1

wjxij

Variance asymptotically non-zero under population structure:

Var(p̂Ti∣∣T) = pTi

(1− pTi

)ϕ̄T

Therefore, naive estimators that use p̂Ti (next) are not consistent!

28 / 35

Bias in standard kinship estimator

ϕ̂T ,stdjk =

m∑i=1

(xij − 2p̂Ti

) (xik − 2p̂Ti

)4

m∑i=1

p̂Ti(1− p̂Ti

) , p̂Ti =12

n∑j=1

wjxij .

Bias varies by j , k :

ϕ̂T ,stdjk

a.s.−−−→m→∞

ϕTjk − ϕ̄T

j − ϕ̄Tk + ϕ̄T

1− ϕ̄T.

True KinshipA New estimateB Standard estimateC Limit of standardD

00.

10.

2

Kin

ship

Indi

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als

29 / 35

Bias in standard kinship estimator

ϕ̂T ,stdjk =

m∑i=1

(xij − 2p̂Ti

) (xik − 2p̂Ti

)4

m∑i=1

p̂Ti(1− p̂Ti

) , p̂Ti =12

n∑j=1

wjxij .

Bias varies by j , k :

ϕ̂T ,stdjk

a.s.−−−→m→∞

ϕTjk − ϕ̄T

j − ϕ̄Tk + ϕ̄T

1− ϕ̄T.

True KinshipA New estimateB Standard estimateC Limit of standardD

00.

10.

2

Kin

ship

Indi

vidu

als

29 / 35

Bias in standard kinship estimator

ϕ̂T ,stdjk =

m∑i=1

(xij − 2p̂Ti

) (xik − 2p̂Ti

)4

m∑i=1

p̂Ti(1− p̂Ti

) , p̂Ti =12

n∑j=1

wjxij .

Bias varies by j , k :

ϕ̂T ,stdjk

a.s.−−−→m→∞

ϕTjk − ϕ̄T

j − ϕ̄Tk + ϕ̄T

1− ϕ̄T.

True KinshipA New estimateB Standard estimateC Limit of standardD

00.

10.

2

Kin

ship

Indi

vidu

als

29 / 35

Our new estimator (R package ‘popkin’ on CRAN)

Step 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 .

This yields consistent f̂ T ,newj , F̂ new

ST estimators!

30 / 35

Our new estimator (R package ‘popkin’ on CRAN)

Step 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 .

This yields consistent f̂ T ,newj , F̂ new

ST estimators!

30 / 35

Performance of new estimator

True KinshipA New estimateB Standard estimateC Limit of standardD

00.

10.

2

Kin

ship

Indi

vidu

als

31 / 35

Bias in FST estimators for independent subpopulationsPrevious estimator for n subpopulations, simplified for known AFs (πij):

F̂ indepST =

m∑i=1

σ̂2i

m∑i=1

p̂Ti(1− p̂Ti

)+ 1

nσ̂2i

,

p̂Ti =1n

n∑j=1

πij , σ̂2i =

1n − 1

n∑j=1

(πij − p̂Ti

)2.

Estimator is biased in dependent subpopulations:

F̂ indepST

a.s.−−−→m→∞

FST − 1n−1

(nθ̄T − FST

)1− 1

n−1

(nθ̄T − FST

) .

32 / 35

Bias in FST estimators for independent subpopulationsPrevious estimator for n subpopulations, simplified for known AFs (πij):

F̂ indepST =

m∑i=1

σ̂2i

m∑i=1

p̂Ti(1− p̂Ti

)+ 1

nσ̂2i

,

p̂Ti =1n

n∑j=1

πij , σ̂2i =

1n − 1

n∑j=1

(πij − p̂Ti

)2.

Estimator is biased in dependent subpopulations:

F̂ indepST

a.s.−−−→m→∞

FST − 1n−1

(nθ̄T − FST

)1− 1

n−1

(nθ̄T − FST

) .

32 / 35

Only our new method estimates generalized FST accurately

0.00

0.05

0.10

Ind. Subpops.A

Wei

r−C

ocke

rham

Hud

sonK

Bay

eSca

n

Sta

ndar

dK

insh

ip

New

− − −−

− AdmixtureB

Wei

r−C

ocke

rham

Hud

sonK

Bay

eSca

n

Sta

ndar

dK

insh

ip

New

− −−

−

−

−

True FST

FSTindep limit

Estimates−

FS

T e

stim

ate

FST estimator

33 / 35

The future: improved kinship has repercussions across genetics!

Accurate andefficientestimation,admixturemodeling

Associationstudies, selectiontests

Bias inheritability ofcomplex traits

Animal and plantbreeding

34 / 35

Acknowledgments

John D. StoreyAndrew BassIrineo CabrerosWei HaoRiley Skeen-Gaar

Neo Christopher ChungUniversity of Warsaw

Funding:National Institutes of HealthOtsuka Pharmaceutical

Lewis-Sigler Institute for Integrative Genomics

35 / 35