Application of Integrated Nested LaplaceApproximation to Survival Models as used in

Animal Breeding

Sara Martino1, Gregor Gorjanc2, & Ingelin Steinsland1

Norwegian University of Science and Technology1, University of Ljubljana2

Workshop in Bayesian Inference for Latent Gaussian Models . . .Zürich, Switzerland

February 2011

Outline

1. Animal breeding

2. Longevity of cows

3. INLA framework

4. Application

Animal breeding

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 - inference of unobserved/latent genotypicvalue in order to enhance selective breeding

Genetic evaluation

Phenotype decomposition(Fisher, 1918)

P = µ+ G + E + G × EG = A + D + IP w µ+ A + E

Pedigree based (mixed)model (Henderson, 1949+)

y|b, a, σ2e ∼ N

(Xb + Za, Iσ2

e)

a|A, σ2a ∼ N

(0,Aσ2

a)

Selected candidates will bredthe next (better) generation

Longevity of cows

I Breeders want high producing & robust cowsI Involuntary culling due to:

I fertility problemsI health statusI . . .

I Robust animals have better longevity = length ofproductive life (an indirect measure of ability to cope withproduction environment)

I Improved longevity has economic impact:I lowered replacement costsI less veterinary costsI more animals producing at mature levelI greater selection response

Statistical analysis of cow longevity

I Some cows are alive at the time of analysis - censored data–> survival analysis

I Need to take into account:I time varying covariatesI genetic (frailty) effectI large/huge datasets!!!

I Available software: SurvivalKit (open source, FORTRAN)I Weibull model, grouped data model, Cox modelI empirical Bayes approach (Ducrocq and Casella)

I joint posterior mode of hyperparametersI effect solutions

I now at version 6 (Ducrocq et al., 2010)

INLA framework

Let (ti , δi) be the observed time and censoring indicator. Weibullregression model is defined as:

hi(t) = h0(t) exp(ηi) = αtα−1 exp(ηi),

where ηi = xTi b is the linear predictor.

Can be casted into INLA framework (Martino et al., 2010):I Hyperparameters θ = (α)

I Latent Gaussian field l = (η,b)I Likelihood π(data|l,θ) =

∏π(datai |ηi ,θ)

INLA framework - frailty terms

As long as we keep ηi Gaussian we can add “complicated” termswithout changing the model structure:

ηi = xTi b + wT

i h + ai

b|σ2b ∼ N

(0, Iσ2

b)

h|H, σ2h ∼ N

(0,Hσ2

h)

a|A, σ2a ∼ N

(0,Aσ2

a)

INLA framework - time varying covariates

t

h(t)

disease

recoveryincreasedherd size

drying off

We assume piecewise-constant time varying covariates. These canbe included via data augmentation noting that:

ˆ t

0h(u) du =

ˆ t1

0h(u) du +

ˆ t2

t1

h(u) du + · · ·+ˆ t

tk

h(u) du

where in each interval the time varying covariate is constant.

Application - data

I A subset of data from national genetic evaluation of cowlongevity in Slovenia (a small country!!!)

I 20,330 cows (daughters of 194 bulls) from 770 herdsI Data

I age at first calvingI stage of lactation within parity (time varying)I year (time varying)I herd size change (time varying)I herd (time varying)I cowI pedigree

Data preparationI 20,330 cows –> 189,504 elementary records

id start stop event ageN parSta year herdS herd fid

---

1 1627208 0 60 0 30 1 1 4 1 1

2 1627208 60 150 0 30 2 1 4 1 1

3 1627208 150 194 0 30 3 1 4 1 1

4 1627208 194 270 0 30 3 2 4 1 1

5 1627208 270 347 0 30 4 2 4 1 1

...

16 1627208 925 1012 0 30 13 4 7 1 1

17 1627208 1012 1018 0 30 14 4 7 1 1

---

18 1628324 0 60 0 26 1 1 7 2 3

19 1628324 60 150 0 26 2 1 7 2 3

20 1628324 150 309 1 26 3 1 7 2 3

...

Model

y|η, α ∼ Weibull (η, α)log (η) = Xb + Wh + Za

b|σ2b ∼ N

(0, Iσ2

b)

covariatesh|σ2

h ∼ N(0, Iσ2

h)

herda|A, σ2

a ∼ N(0,Aσ2

a)

breeding value

Hyperparameters θ = (α, σ2h, σ

2a)

Caution: we need enough phenotypes and strcutured pedigree forproper decoupling of genetic and environmental components!!! Inparticular with non-Gaussian likelihoods.

Population structure in cattleI Only few bulls are used due to artificial inseminationI We get large groups of daughters (half-sisters) –> progeny testI Cows have few daughters

Animal & Sire model20,330 cows (sires) 0 cows

194 bulls (sires) 194 bulls (sires)σ2a

af(k) am(k)

ak

k = 1 : nI

Dk,k

1/2 1/2

bj

j = 1 : nB

hj

j = 1 : nH

ηi

σ2h

α yi

i = 1 : nY

Zi,k

Xi,j Wi,j

σ2s = 1/4σ2

aσ2s

af(k)

k = 1 : nI

I

bj

j = 1 : nB

hj

j = 1 : nH

ηi

σ2h

α yi

i = 1 : nY

Zi,k

Xi,j Wi,j

A−1 = (T−1)T D−1T−1

T−1 = I− 1/2P

INLA call from R (~30 min)## Scale times

myData$start <- myData$start / max

myData$stop <- myData$stop / max

## Specify model

model <- inla.surv(stop, event, start) ~

ageN0 + parSta + year + herdS +

f(herd, model="iid") +

f(fid, model="iid")

## Run INLA

fit <- inla(formula=model, family="weibull",

data=myData)

## For comparison with SurvivalKit

## f(..., prior="logiflat")

## control.data=list(prior="logiflat")

## To fix alpha

## control.data=list(initial=log(2), fixed=TRUE)

Results - covariates

Year

Ris

k ra

tio

−0.

9−

0.7

−0.

5−

0.3

1 2 3 4 1 2 3 4 5 6 7

0.0

0.5

1.0

1.5

Herd size change

Ris

k ra

tio

0.0

0.5

1.0

1.5

1 2 3 4 5 6 7

Stage of lactation with parity

Ris

k ra

tio

−2.

5−

2.0

−1.

5−

1.0

2 3 4 5 6 7 8 9 10 11 12 13 14

Length of productive life (days)

Haz

ard

0 200 400 600 800 1000 1200

Results - hyperparameters (SKit - lines)

α

1.70 1.75 1.80 1.85 1.90 1.95σh

20.08 0.09 0.10 0.11 0.12 0.13 0.14

σs2

0.04 0.06 0.08 0.10 0.12 0.14σa

20.2 0.3 0.4 0.5

h2 =4σ2

sσ2

s + σ2h + 1

= 0.26

To large?

Conclusion

I A long history of using latent Gaussian models in animalbreeding

I Extended INLA R-package to work with time varying covariates

I SurvivalKit & INLA give very similar point estimates

I Application (work in progress) –> continue with relationshipmatrices and larger dataset

References

I Ducrocq V. and Casella G. (1996) A Bayesian analysis of mixedsurvival models. Genet. Sel. Evol., 28:505-529

I Ducrocq V., Sölkner J. and Mészáros G. (2010) Survival Kit v6– a Software Package for Survival Analysis. In: 9th WorldCong. Genet. Appl. Livest. Prod., Leipzig, Germany

I Martino S., Akerkar R., Rue H. (2010) Approximate BayesianInference for Survival Models. Scan. J. Stat.