ll l l 0.4 inarbitrarypopulationstructures 0
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Transcript of ll l l 0.4 inarbitrarypopulationstructures 0
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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 [email protected]
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
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Ju_hoan_South
Ju_hoan_North
Taa_West
Taa_East
Taa_North
NaroGui
Hoan
Xuun
GanaTshwa
Khomani
Nama
Haiom
Kgalagadi
Shua
Khwe
Mbuti
Biaka
BantuSA
Tswana
Damara
Himba
Wambo
YorubaEsan
Mende
BantuKenya
Luhya
MandenkaGambian
Luo
Dinka
Kikuyu
Hadza
Sandawe
Masai
Datog
Somali
OromoJew_Ethiopian
Saharawi
MoroccanMozabite
Algerian Tunisian
Libyan Egyptian
Yemeni
Jew_Libyan
Jew_TunisianJew_Moroccan
Jew_Yemenite
Saudi
BedouinBBedouinA
PalestinianJordanian
SyrianLebanese_Muslim
Lebanese_Christian
Jew_Turkish
Assyrian
Druze
Lebanese
Jew_iraqi
Iranian_Bandari
Iranian
Jew_Iranian
Turkish
Jew_Ashkenazi
CypriotMaltese
Canary_Islander
Italian_SouthSicilian
Romanian
FrenchN
SpanishSW Greek
Italian_North
SpanishNE
FrenchS
Sardinian
Basque
Orcadian
English
Icelandic
Norwegian
Irish
Irish_UlsterScottish
Shetlandic
GermanSorb
Polish
Czech
Croatian
Hungarian
AlbanianBulgarian
Mordovian
FinnishRussian
Estonian
Ukrainian
BelarusianLithuanian
Armenian
Jew_GeorgianGeorgian
Abkhasian
Adygei
North_Ossetian
Chechen
Lezgin
KumykBalkar
Nogai
Makrani
BrahuiBalochi
Jew_Cochin
Tajik
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Sindhi
Burusho
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Punjabi
Vishwabrahmin
Lodhi
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Turkmen Uzbek
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Mansi
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Tuvinian
Nganasan
Dolgan
Yakut
Xibo
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OroqenUlchi
Even
Yukagir
ItelmenKoryak
Chukchi
Eskimo
AleutAleut_Tlingit
Kusunda
Burmese
CambodianThai
Kalmyk
Ami
Atayal
Malay
Kinh
Vietnamese
Dai
Lahu
NaxiYi
TujiaHan
MiaoShe
Tu
Mongola
Daur
JapaneseKorean
Chipewyan
CreeAlgonquin
Ojibwa
Pima
Mixe
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Zapotec
Mayan
Inga
Kaqchikel
Cabecar
Piapoco
Karitiana
Surui
Bolivian
Aymara
Quechua
Guarani
Chilote
Australian
Nasoi
Papuan
Ata
Baining_Malasait
Baining_Marabu
Bajo
Buka
Dusun
Ilocano
Kankanaey
Kol_New_Britain
Kove
Kuot_Kabil
Kuot_LamalauaLavongaiLebbo Madak
MamusiMamusi_Paleabu
Mangseng
Manus
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Mengen
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Nakanai_Bileki
Nakanai_Loso
Nailik
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Saposa
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Tagalog
Teop
Tigak
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Visayan
Subpopulations
SAfricaMAfricaNAfricaMiddleEastEuropeCaucasusSAsiaNAsiaEAsiaAmericasOceania
2,922 indivs. from 244 locs. — 593,124 loci — SNP chipLazaridis et al. (2014), (2016); Skoglund et al. (2016)
9 / 35
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YemeniJew_Libyan
Jew_TunisianJew_MoroccanJew_Yemenite
SaudiBedouinBBedouinA
PalestinianJordanian
SyrianLebanese_Muslim
Lebanese_ChristianJew_Turkish
AssyrianDruze
LebaneseJew_iraqi
Iranian_BandariIranian
Jew_IranianTurkish
Jew_AshkenaziCypriotMaltese
Canary_IslanderItalian_South
SicilianRomanian
FrenchNSpanishSW
GreekItalian_North
SpanishNEFrenchS
SardinianBasque
OrcadianEnglish
IcelandicNorwegian
IrishIrish_Ulster
ScottishShetlandic
GermanSorb
PolishCzech
CroatianHungarian
AlbanianBulgarian
MordovianFinnish
RussianEstonian
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00.
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3
Kin
ship
Indi
vidu
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Our new kinshipestimatesGenotypes from “Human Origins”(Lazaridis et al. 2014, 2016;Skoglund et al. 2016)
Edited from Ephert [CC BY-SA 3.0], viaWikimedia Commons
*Inbreeding coeffs. on diagonal
10 / 35
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−0.0
50
0.05
0.1
0.15
Kin
ship
Indi
vidu
als
Standard kinshipestimatesGenotypes from “Human Origins”(Lazaridis et al. 2014, 2016;Skoglund et al. 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.
10.
2
Kin
ship
Indi
vidu
als
12 / 35
Population-level inbreeding increases with distance from Africa
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Inbr
eedi
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oeff.
13 / 35
Differentiation (FST) previously underestimated
0.0 0.1 0.2 0.3 0.4 0.5
020
40
Inbreeding Coefficient
Den
sity
SAfricaMAfricaNAfricaMiddleEastEuropeCaucasusSAsiaNAsiaEAsiaAmericasOceania
Subpopulations
SAfricaMAfricaNAfricaMiddleEastEuropeCaucasusSAsiaNAsiaEAsiaAmericasOceania
FST estimates
BayeScanWCHudsonKNew
14 / 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
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
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STU
CDX
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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
00.
10.
20.
30.
4
Kin
ship
Indi
vidu
als
0.0
0.5
1.0
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.
10.
2
Kin
ship
Indi
vidu
als
20 / 35
FST in the independent subpopulation model
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le fr
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ncy
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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Alle
le fr
eque
ncy
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 − FIS1− 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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Alle
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Median diff. SNP in Human Origins (rs2650044; given MAF ≥ 10%).
F̂WCST ≈ 0.0961 using Weir-Cockerham estimator and K = 244.
24 / 35
FST measures population structure / differentiation
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Alle
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ncy
Median diff. SNP in Human Origins (rs2650044; given MAF ≥ 10%).
F̂WCST ≈ 0.0961 using Weir-Cockerham estimator and K = 244.
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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
vidu
als
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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
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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).
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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
(for independent individuals ϕ̄T = 12n(1 + FST). ⇒ nEff ≈ 4 in Human Origins!)
Therefore, naive estimators that use p̂Ti (next) are not consistent!
28 / 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
(for independent individuals ϕ̄T = 12n(1 + FST). ⇒ nEff ≈ 4 in Human Origins!)
Therefore, naive estimators that use p̂Ti (next) are not consistent!
28 / 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
(for independent individuals ϕ̄T = 12n(1 + FST). ⇒ nEff ≈ 4 in Human Origins!)
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
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
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 =1
n − 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
) .
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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 =1
n − 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
) .
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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
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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
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