estimating genetic variation within...
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Estimating genetic variation within families
Peter M. VisscherQueensland Institute of Medical
ResearchBrisbane, Australia
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Overview
• Estimation of genetic parameters• Variation in identity• Applications
– mean and variance of genome-wide IBD sharing for sibpairs
– estimation of heritability of height– genome partitioning of genetic variation
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Estimation of genetic parameters
• Model– expected covariance between relatives
• Genetics• Environment
• Data– correlation/regression of observations between
relatives• Statistical method
– ANOVA– regression– maximum likelihood– Bayesian analysis
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[Galton, 1889] 4
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The height vs. pea debate
(early 1900s)
Do quantitative traits have the same hereditary and evolutionary properties as discrete characters?
Biometricians Mendelians
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RA Fisher (1918). Transactions of the Royal Societyof Edinburgh52: 399-433.
m-a m+d m+a
Trait
m-a m+d m+a
Trait
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Genetic covariance between relatives
covG(yi,yj) = aijσA2 + dijσD
2
a = additive coefficient of relationship= 2 * coefficient of kinship (= E(π))
d = coefficient of fraternity= Prob(2 alleles are IBD)
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Examples (no inbreeding)
Relatives a d
MZ twins 1 1Parent-offspring ½ 0Fullsibs ½ ¼Double first cousins ¼ 1/16
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Controversy/confounding:nature vs nurture
• Is observed resemblance between relatives genetic or environmental?– MZ & DZ twins (shared environment)– Fullsibs (dominance & shared environment)
• Estimation and statistical inference– Different models with many parameters may
fit data equally well
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Total mole count for MZ and DZ twins
0
100
200
300
400
0 100 200 300 400
Twin 2
Twin
1
0
100
200
300
400
0 100 200 300 400
Twin 2
Twin
1
MZ twins - 153 pairs, r = 0.94 DZ twins - 199 pairs, r = 0.60
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Sources of variation in Queensland school test results of 16-year olds
12%78%
10%
Additivegenetic
Sharedenvironment
Non-sharedenvironment
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An unbiased approach
Estimate genetic variance within
families
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Actual or realised genetic relationship
= proportion of genome shared IBD (πa)
• Varies around the expectation– Apart from parent-offspring and MZ twins
• Can be estimated using marker data
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x
1/4 1/4 1/4 1/414
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IDENTITY BY DESCENTSib 1
Sib 2
4/16 = 1/4 sibs share BOTH parental alleles IBD = 2
8/16 = 1/2 sibs share ONE parental allele IBD = 1
4/16 = 1/4 sibs share NO parental alleles IBD = 015
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Single locus
Relatives E(πa) var(πa)
Fullsibs ½ 1/8Halfsibs ¼ 1/16
Double 1st cousins ¼ 3/32
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Several notations
IBD Probability Actual
IBD0 k0 0 or 1IBD1 k1 0 or 1IBD2 k2 0 or 1
Σ=1 Σ=1
πa = ½k1 + k2 = R = 2θπd = k2 = ∆xy
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[e.g., LW Chapter 7; Weir and Hill 2011, Genetics Research]
Realisationsk0 k1 k2
1 0 00 1 00 0 1
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n multiple unlinked loci
Relatives E(πa) var(πa)
Fullsibs ½ 1/8n
Halfsibs ¼ 1/16n
Double 1st cousins ¼ 3/32n
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Loci are on chromosomes
• Segregation of large chromosome segments within families– increasing variance of IBD sharing
• Independent segregation of chromosomes– decreasing variance of IBD sharing
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Theoretical SD of πa
Relatives 1 chrom (1 M) genome (35 M)
Fullsibs 0.217 0.038Halfsibs 0.154 0.027Double 1st cousins 0.173 0.030
[Stam 1980; Hill 1993; Guo 1996; Hill & Weir 2011]20
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Fullsibs: genome-wide (Total length L Morgan)
var(πa) ≈ 1/(16L) – 1/(3L2)
var(πd) ≈ 5/(64L) – 1/(3L2)
var(πd)/ var(πa) ≈ 1.3 if L = 35
[Stam 1980; Hill 1993; Guo 1996]
Genome-wide variance depends more on total genome length than on the number of chromosomes
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Fullsibs: Correlation additive and dominance relationships
r(πa, πd) = σ(πa) / σ(πd) ≈ [1/(16L) / (5/(64L))]0.5 = 0.89.
Using β(πa on πd) = 1
Difficult but not impossible to disentangle additive and dominance variance
NB Practical 22
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SummaryAdditive and dominance (fullsibs)
SD(πa) SD(πd)
Single locus 0.354 0.433One chromsome (1M) 0.217 0.247Whole genome (35M) 0.038 0.043
Predicted correlation 0.89(genome-wide πa and πd)
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Application (1)Aim: estimate genetic variance from actual
relationships between fullsib pairs
• Two cohorts of Australian twin families
Adolescent AdultFamilies 500 1512Individuals 1201 3804Sibpairs with genotypes 950 3451Markers per individual 211-791 201-1717Average marker spacing 6 cM 5 cM
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Application (1)
• Phenotype = height
Number of sibpairs with phenotypesand genotypes
Adolescent cohort 931Adult cohort 2444Combined 3375
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Mean IBD sharing across the genome for the jth sib pair was based on IBD estimated from Merlin every
centimorgan and averaged at all 3491 points
3491/ˆˆ3491
1)()( ∑
=
=i
ijaja ππ
3491/ˆ3491
1)(2)( ∑
=
=i
ijjd pπ
additive
dominance
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And for the cth chromosome of length lc cM
c
l
i
cc lc
ijaja/ˆˆ
1)()( ∑
=
= ππ
c
l
iij
c lpc
jd/ˆ
1)(2)( ∑
=
=π
additive
dominance
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Mean and SD of genome-wide additive relationships
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Mean and SD of genome-wide dominance relationships
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Empirical and theoretical SD of additive relationshipscorrelation = 0.98 (n = 4401)
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Empirical and theoretical SD of dominance relationshipscorrelation = 0.98 (n = 4401)
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Additive and dominance relationships correlation = 0.91 (n= 4401)
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Phenotypes
After adjustment for sex and age:σp = 7.7 cm σp = 6.9 cm 33
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Phenotypic correlation between siblings
Raw After age & sex
Adolescents 0.33 0.40Adults 0.24 0.39
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Models
C= Family effectA = Genome-wide additive geneticE = Residual
Full model C + A + EReduced model C + E
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Estimation
• Maximum Likelihood variance components
• Likelihood-ratio-test (LRT) to calculate P-values for hypothesesH0: A = 0H1: A > 0
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Estimates: null model (CE)
Cohort Family effect (C)
Adolescent 0.40 (0.34 – 0.45)Adult 0.39 (0.36 – 0.43)Combined 0.39 (0.36 – 0.42)
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Estimates: full model (ACE)
Cohort C A P
Adolescent 0 0.80 0.0869Adult 0 0.80 0.0009Combined 0 0.80 0.0003
►All family resemblance due to additive genetic variation
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Sampling variances are large
Cohort A (95% CI)
Adolescent 0.80 (0.00 – 0.90)Adult 0.80 (0.43 – 0.86)Combined 0.80 (0.46 – 0.85)
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F+A more accurately estimated
Cohort C+A (95% CI)
Adolescent 0.80 (0.36 – 0.90)Adult 0.80 (0.61 – 0.86)Combined 0.80 (0.62 – 0.85)
►Prediction of MZ correlation from fullsibs!
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Power and SE of estimates
• True parameters (t)• Sample size (n)• Variance in genome-wide IBD sharing (var(π))
NCP = nh4var(π)(1+t2) / (1-t2)2
[ ]))var()(1(/)1()ˆvar( 2222 πntth +−≈
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• Aims– Estimate genetic variance from genome-wide
IBD in larger sample– Partition genetic variance to individual
chromosomes• using chromosome-wide coefficients of relationship
– Test hypotheses about the distribution of genetic variance in the genome
Application (2)Genome partitioning of additive
genetic variance for height
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Sample # Sibpairs Sib Correlation
AU 5952 0.43US 3996 0.50NL 1266 0.45
Total 11,214 0.46
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Realised relationshipsMean 0.499Range 0.31 – 0.64SD 0.036
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Estimates from genome-wide additive and dominance coefficients
ACE modelHeritability 0.86 (0.49 – 0.95) P<0.0001Family 0.03 (0.00 – 0.03) P=0.38
ADCE modelAdditive component 0.70Dominance component 0.16 (P=0.35)
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y = 1.006x + 0.0001R2 = 0.9715
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.00 0.02 0.04 0.06 0.08 0.10 0.12
From single chromosome analyses
From
com
bine
d ch
rom
osom
e an
alys
is
Estimates of chromosomal heritabilities
No epistasis?
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222120
19
18
17
16
15
14
13
12
11 10
9
8
7
6
5
4
3
2
1
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
50 100 150 200 250 300
Length of chromosome (cM)
Estim
ate
of h
erita
bilit
y
WLS analysis: P<0.001; intercept NS
Longer chromosomes explain more additive genetic variance: ~0.03 per 100 cM
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19
9
7
3
17
14
4
8
15
12
18
1
2016
21
2213
1011
62 5
y = 0.9623xR2 = 0.2813
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Heritability AU
Her
itabi
lity
USA
Estimates are consistent across countries
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-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
00 1 2 3 4 5 6 7 8
Number of chromosomal additive genetic variance components
Scal
ed A
ICStepwise analyses: at least 6 chromosomes are needed to explain the additive genetic variance
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Hypothesis test
Model h2 c2 df LRT
Full (22 chrom.) 0.92 0.00 22Genome-wide 0.86 0.03 1 19.2
Additive genetic variance in proportion to lenght not rejected
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17
612
18
15
9
34
8
14 5
10 13
17
19
216
2111
20220.00
0.02
0.04
0.06
0.08
0.10
0.12
0 1 2 3 4
Number of publications with LOD > 1.9
Estim
ate
of h
erita
bilit
y
Data consistent with published QTL results
Rank test: P=0.002
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Conclusions
• Empirical variation in genome-wide IBD sharing follows theoretical predictions
• Genetic variance can be estimated from genome-wide IBD within families– results for height consistent with estimates from
between-relative comparisons– no assumptions about nature/nurture causes of family
resemblance• Genetic variance can be partitioned onto
chromosomes53
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Conclusions
• With large sample sizes it will become possible to estimate – dominance variance– epistatic variance– genome-wide parent-of-origin variance
– genetic relative risk to disease
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Genetic architecture for height
• Additive genetic variance• No QTL of large effects• Chromosomes explain ~10% of genetic
variance• Consequences for genome-wide
association
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Other applications: breeding programmes
• Exploit variance in genome-wide IBD by using the realised A-matrix– large increase in accuracy of selection if
• variance in identity is large• family size is large
• “Genomic Selection”
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Using the realised A-matrix: Reliability of EBV for an unphenotyped individual from n-1 phenotyped relatives(a simulation study)
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