introduction to animal breeding with examples of (non-)gaussian traits
DESCRIPTION
Talk at INLA group meeting (http://www.r-inla.org) at NTNU, Department of Mathematical Sciences (http://www.ntnu.no/imf), Trondheim, NorwayTRANSCRIPT
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Introduction to Animal Breeding withExamples of (Non-)Gaussian Traits
Gregor Gorjanc
University of Ljubljana, Biotechnical Faculty, Department of Animal Science, Slovenia
INLA for Animal Breeders “Project"Trondheim, Norway30th August 2010
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Thank you for the invitation to NTNU!!!
My department ...
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Table of Contents
1. Animal breeding crash course
2. Categorical trait example
3. Survival analysis example
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1. Animal Breeding Crash Course
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Introduction
I Animal breeding= mixture(animal science, genetics, statistics, . . . )
I Many species (cattle, chicken, pig, sheep, goat, horse, dog,salmon, shrimp, honeybee, . . . )
I Many (complex) traits:I production (milk, meat, eggs, . . . )I reproduction (no. of offspring, insemination success, . . . )I conformation (body height, width, . . . )I health & longevityI . . .
I Genetic evaluation - to enhance selective breeding
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Selective BreedingI Measure phenotype in candidates and select those with the
most favourable values (= "mass” selection)I Selected candidates will bred the next (better) generation
I . . . , but phenotype is not transmitted to the next generation
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Decomposition of Phenotypic Value
Genotype Environment
Phenotype
P = G + E + G × E
I Genetic evaluation = inference of genotypic value given thedata and postulated model (= “BLUP” selection)
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Postulated Model and DataI Postulated model
P = G + E + G × E = A + D + I + . . .
I A - additive (breeding) valueI D - dominanceI I - epistasis
I DataI phenotypes on various relatives (pedigree)
I own performance testI progeny testI (half-)sib testI . . .
I recently also genotype marker data
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Evaluation via Pedigree based Mixed ModelsI Not so standard example - “maternal animal model”
y|b, c, ad , am,R ∼ N (Xb + Zcc + Zadad + Zamam,R)
R = Iσ2e
b ∼ const.c|C ∼ N (0,C)
C = Iσ2c
a =(aT
d , aTm)T |G ∼ N (0,G)
G = G0 ⊗ A,G0 =
(σ2
adσad ,am
sym. σ2am
)data: y (phenotypes), X,Z∗(“covariates”), A (pedigree)
parameters: b, c, a (means)σ2
c , σ2ad, σad ,am , σ
2am , σ
2e (variances)
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Inference (for Gaussian models)I “Standard”
I means - solve Mixed Model (Normal) Equations (MME∗)Henderson (1949+)
I SE of means (needed for accuracies) - inversion of LHS orsome approximation
I variances - maximize Restricted Likelihood (REML)Patterson & Thompson (1971)
I “Powerfull/Popular/Fancy/. . . ” - McMC
I ∗MME
LHS =
XTR−1X XTR−1Zc XTR−1Za
ZTc R−1Zc + C−1 ZT
c R−1ZaZT
a R−1Za + G−1 ⊗ A−1
sym.
RHS =
((XTR−1y
)T,(ZT
c R−1y)T,(ZT
a R−1y)T)T
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Graphical Model View of Pedigree Model
A−1 =(T−1)TW−1T−1
= (I− 1/2P)TW−1(I− 1/2P)
Wi ,i = 1− 1/4(1 + F f (i)
)− 1/4
(1 + F m(i)
)σ2
a
af (i) am(i)
ai
i = 1 : nI
Wi ,i
1/2 1/2
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Genetic GroupsI Different means in founders (usually due to different origin)
= sort of hierarchical centering for pedigree model
. . .
a|G ∼ N (ZaQa0,G)
a0 ∼ const.. . .
after some "massage"
LHS =
. . . . . . . . . 0
. . . . . . 0ZT
a R−1Za + G−1 ⊗ A−1i ,i G−1 ⊗ A−1
i ,gsym. G−1 ⊗ A−1
g ,g
i − individuals, g − genetic groups
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Genetic Groups - Graphical Model ViewI Unknown (phantom) parents are represented with (few!)
genetic group(s) - “graphical parent(s)”I Algorithm to set up A−1 directly available!!!I Hierarchical prior can be put on genetic groups for
stability/shrinkage
σ2a
af (i) am(i)
ai
i = 1 : nI
Wi ,i
1/2 1/2
a0g(i)
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Multi-trait = multi-variate
y =(yT
1 , yT2)T, X = . . .
y| . . . ∼ N (Xb + Zcc + Zadad + Zamam,R)
R = R0 ⊗ I,R0 =
(σ2
e1 σe1,e2
sym. σ2e2
)c|C ∼ N (0,C)
C = C0 ⊗ I,C0 =
(σ2
c1 σc1,c2
sym. σ2c2
)ad , am|G ∼ N (0,G)
G = G0 ⊗ A,G0 =
σ2
ad1σad1 ,ad2
σad1,am1σad1 ,am2
σ2ad2
σad2 ,am1σad2 ,am2
σ2am1
σam1 ,am2
sym. σ2am2
I there are now 16 variance components!!!
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Non-Gaussian TraitsI Categorical (health status, calving ease score, . . . )
I threshold model = (ordered) probit model, cumulative linkmodel, . . .
I multinomial categories mostly treated separately as binarytraits
I Counts (no. of offspring, . . . )I Poisson, but rarely used - replacements: threshold and/or
Gaussian model
I Time (longevity)I survival (Weibull & Cox) models
I MixturesI Gaussian componentsI zero-inflated (no. of black spots in sheep skin -> wool, cure
model - bivariate threshold model)
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2. Categorical Trait Example(Calving ease score)
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Calving Ease ScoreI Of great economical importance!!!I We can not measure calving difficulty -> subjective score
I 1 = no problemI 2 = easyI 3 = difficultI 4 = mechanical help or ceasearean
I Reasons for difficult calving?I sex (male calfs bigger)I number of calfs - data usually omittedI parity (more problems with the 1st calving)I age (especially in the 1st parity; younger cows more problems)I season?I environment (= herd, herd-year)I . . .
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Calving Ease Score III Reasons for difficult calving - genetics?
I morphology of calfI “direct” genetic effect or “sire/bull” effectI genes expressed in calfI “origin” of genes - father and mother of a calf
I morphology of cows’ pelvic areaI “maternal” genetic effectI genes expressed in cowI “origin” of genes - father and mother of a cow
I Negative genetic correlationI larger animals (↑direct effect -> bad) have
larger pelvic area (↓maternal effect -> good)
I Parity specific genetic effects - 1st vs. 2nd+
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Threshold Model(Wright, . . . , Gianola & Foulley, Sorensen, . . . )
l|b, c, ad , am,R ∼ N (Xb + Zcc + Zadad + Zamam,R)
Pr (yi = k|µi , t) = Pr (tk−1 < li < tk |µi , t)
= Φ
(tk − µi
σ
)− Φ
(tk−1 − µi
σ
). . .
I Model σ as well to improve model fit? log (σ) = . . .I Methods: approx. EM-REML, Laplace approx., McMC
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Approximative (Gaussian) Model - Example(joint work with Marija Špehar - Croatia)
I Dataset: ~150k phenotypes, ~200k animals, 10 dataset samplesI Homogenization of variance by region and period of recording -
scale problems?I Bi-variate (1st & 2nd+ parity) maternal animal model with
heterogenous (by sex within parity class) residual varianceI 18 variance components - with VCE-6 program
I herd-year interaction (3) -> better with autoregressive prior?σ2
h1, σ2
h2+, σh1,h2+
I permanent effect of a cow (repeated records) (1)σ2
c2+
I direct & maternal genetic effect (10)σ2
ad1, σ2
ad2+, σad1 ,ad2+
, . . . σ2am2+
I residual (4)σ2
em1, σ2
ef1, σ2
em2+, σ2
ef2+
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Approximative (Gaussian) Model - ExampleI Residual variancesσ2
em1= 0.295, σ2
ef1= 0.204, σ2
em2+= 0.228, σ2
ef2+= 0.162
I Ratios and correlations (1st vs. 2nd+)Herd-year Direct Maternal Perm.
1st 27.545 4.548 3.548 /2nd+ 24.445 9.948 4.248 5.1Corr. 20.845 0.548 0.743 /
I Genetic correlation between direct and maternal effectDirect, 1st Direct, 2nd+
Maternal, 1st -0.490 -0.433Maternal, 2nd+ -0.377 -0.730
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A Look at my Data - StructureI Dimensions
I #records (= #calfs) ~150kI #cows ~74kI #bulls ~1kI #pedigree records (all generations + pruning)
I animal pedigree ~230k(basic set are calfs + ancestors)
I sire-dam pedigree ~115k(basic set are mothers and fathers of calfs + ancestors)
I two more options: sire-maternal grandsire pedigree, sirepedigree
I Distribution of scoresI no problem 50.3%I no problem 49.7%
I easy 43.5%I difficult 6.1%I mechanical help or ceasearean 0.1%
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A Look at my Data - Sex & Parity
I SexI females 52%I females 47%
I ParityI 1st 59%I 2nd 46%I 3rd 45%I 4th 45%I 5th 45%
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A Look at my Data - Age within Parity
20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
Age at calving
Ave
rage
sco
re
Score 1st (male)
Score 2nd...#Records
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A Look at my Data - Age within Parity & Sex
20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
Age at calving
Ave
rage
sco
re
Score 1st (male)Score 1st (female)Score 2nd...#Records
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A Look at my Data - Season
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
Season
Ave
rage
sco
re
Score#Records
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Analysis of my Data in R - Available ToolsI Bernoulli/binomial model
I glm() - package statsI glmer() - package lme4
I Laplace and adaptive Gauss-Hermite approximation (for moreeffects)
I inla()
I threshold modelI polr() - package MASSI clm() - package ordinal
I location (additive) and scale (multiplicative) modelI clmm() - package ordinal
I location (additive) and scale (multiplicative) modelI Laplace and adaptive Gauss-Hermite approximation (for one
effect)
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3. Survival Analysis Example(Longevity = Length of Productive Life)
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Model and DataI Weibull model
y|b∗,h, a, ρ ∼ Weibull (Xb∗ + Zhh + Zaa, ρ)
h (y|b∗,h, a, ρ) = ρyρ−1 exp (Xb∗ + Zhh + Zaa)
b∗ =(ρ lnλ,bT)T
b∗ ∼ const.h|γ ∼ Log − Gamma (γ, γ)
a|G ∼ N (0,G)
G = Aσ2a
I DataI ~110k cows from ~4k herds, ~40% censoringI sire-maternal grandsire pedigree with ~3k bulls
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Implementation
I Survival Kit program
I Log-Gamma prior “integrated out”
I Laplace approximation for Normal prior
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Time Independent Effect - Age at 1st Calving
Age at first calving (month)19 22 25 28 31 34 37 40 43 46 49
020
0040
0060
0080
0012
000
No.
of r
ecor
ds
1.0
1.2
1.4
1.6
1.8
Rel
ativ
e ris
k
All recordsUncensored recordsRelative riskBaseline
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Time Dependent Effect - Parity*Stage
Length of productive life (day)0 500 1000 1500 2000
020
0040
0060
0080
00N
o. o
f rec
ords
0 500 1000 1500 2000
0.00
000.
0005
0.00
100.
0015
0.00
20H
azar
d fu
nctio
n
All recordsUncensored recordsHazard function
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Thank you!
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Postulated Model and Data III Breeding value for individual
= f(parent average, phenotype deviation, progeny contribution)
b1 b2
a1 a2
y21
y22
a3y3 a4 y4
a5 y5 a6 y6
a7 a8 a9
a10y10