Statistical Models in Animal Breeding

Statistical Models in Animal Breeding

Gregor Gorjanc

University of Ljubljana, Biotechnical Faculty, Department of Animal Science, Slovenia

INLATrondheim, Norway1st September 2010

Thank you for the invitation to NTNU!!!

My department ...

Table of Contents

1. What is the idea of animal breeding (genetic evaluation)

2. What kind of models do we use

3. How much data do we have

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

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

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)

Postulated Model and Data

I Postulated model

P = G + E + G × E = A + D + I + . . .

I A - additive (breeding) valueI D - dominanceI I - epistasis

I DataI phenotypes (own performance, progeny, (half)-sibs

-> in fact almost all animal "management data” from teststations & farms

I pedigrees - share of genes that relatives have due to the sameorigin (identity-by-descent = IBD)

I recently also genotype marker data (identity-by-state = IBS)

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)

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

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

Animal Model GraphI 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

Multi-trait (= Multi-variate) Example

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!!!

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)

Categorical Trait Example(Calving ease score)

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 . . .

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+

Size of the DataI #records (= #calfs) ~150kI #cows ~74k, #bulls ~1kI #pedigree records (all generations + pruning)

I animal pedigree ~230k(basic set are calfs + ancestors)

ai = 1/2af (i) + 1/2am(i) + wi , var (ai) = 1/4σ2a + 1/4σ2

a + 1/2σ2a

I sire-dam pedigree ~115k(basic set are mothers and fathers of calfs + ancestors)

ai = 1/2af (i) + 1/2am(i) +��wi , var (ai) = 1/4σ2a + 1/4σ2

a +���1/2σ2

a

I two more options: sire-maternal grandsire pedigree, sirepedigree

ai = 1/2af (i) +����1/2am(i) +��wi , var (ai) = 1/4σ2a +���1/4σ2

a +���1/2σ2a

Thank you!