Animal Breeding Applications of PedigreeBased Mixed Model

Gregor Gorjanc

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

House sparrow "lunch” meetingTrondheim, 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 use2.1 variations in the phenotype model - likelihood (data sampling

model)2.2 variations in the genetic model - prior

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)

Pedigree based Mixed ModelsI Standard example - “animal model”

y|b, c, a,R ∼ N (Xb + Zcc + Zaa,R)

R = Iσ2e

b ∼ const.c|C ∼ N (0,C)

C = Iσ2c

a|G ∼ N (0,G)

G = Aσ2a

data: y (phenotypes), X,Z∗(“covariates”), A (pedigree)

parameters: b, c, a (means)σ2

c , σ2a , σ

2e (variances)

Multi-trait (= Multi-variate) ExampleI analyze many traits together - to estimate correlations and to

cover for possible selection on one trait

y =(yT

1 , yT2)T, X = . . .

y| . . . ∼ N (Xb + Zcc + Zaa,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

)a|G ∼ N (0,G)

G = G0 ⊗ A,G0 =

(σ2

ad1σad1 ,ad2

sym. σ2ad2

)I there are now 9 variance components!!!

Phenotype Model “Variation” - Non-GaussianI 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)

Statistical ViewI Specify the model

y|µ,θ ∼ Some − Distribution (µ,θ)

g (µ) = η = Xb + Zcc + Zaab ∼ const.

c|C ∼ N (0,C)

C = Iσ2c

a|G ∼ N (0,G)

G = Aσ2a

I Find some inference (black-box) engine (BUGS, JAGS, INLA,. . . ) to do the job of finding parameter estimates ;)

I Some multi-variate combinations have already beenimplemented

Genotype Model “Variation”

I Standard - additive genetic modelI Some extensions

I maternal/paternal modelI dominanceI epistasisI social interactions - competitive effectsI genotype x environment interactionI genetic variation of environmental variationI genotype marker data -> genomic selectionI . . .

I Proper data structure and the amount of data is essential!!!

Maternal/Paternal ModelI Phenotype (say growth) of animal influenced by:

I genes for growth “in animal”I genes for milk “in mother”

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

)

Social Interactions - Competitive EffectsI Phenotype of animal influenced by:

I genes for that trait “in animal”I genes for competition (for resources) “in companions”

yi = µ+ . . .+ adi +∑

j∈c(i)

asj + ei

c (i) a set of companions of individual iy|b, c, ad , as ,R ∼ N (Xb + Zcc + Zadad + Zasas ,R)

R = Iσ2e

b ∼ const.c|C ∼ N (0,C)

C = Iσ2c

a =(aT

d , aTs)T |G ∼ N (0,G)

G = G0 ⊗ A,G0 =

(σ2

adσad ,as

sym. σ2as

)

Thank you!