GENOMIC SELECTION WORKSHOP: Hands on Practical Sessions …genomics.cimmyt.org/SLU/Slides Paulino/4....

GENOMIC SELECTION WORKSHOP:Hands on Practical Sessions (BL)

Paulino Pérez 1

José Crossa 2

1ColPos-México 2CIMMyT-México

September, 2014.

SLU,Sweden GENOMIC SELECTION WORKSHOP:Hands on Practical Sessions (BL) 1/29

Contents

1 General comments

2 LASSO

3 Application examples

4 Extension of BL to include infinitesimal effect


General comments

General comments

The regression linear model is given by,

yi = µ+

p∑j=1

xijβj + ei , (1)

where ei ∼ N(0, σ2e), i = 1, ...,n.

The key Idea is obtain estimates for β and then obtain GEBVs. β can beobtained using penalized regression methods, for example ridge regression(G-BLUP). Now we review another penalized regression method calledLASSO=Least Angle and Shrinkage Operator.


LASSO

LASSO

In LASSO estimates for β are obtained by minimizing the augmented sum ofsquares:

minβ

{(y −

∑Xjβj)

′(y −∑

Xjβj) + λ∑|βj |

}, (2)

where λ ≥ 0 is a regularization parameter that controls the trade-offs betweengoodness of fit (measured with sum of squares of error, SCE) and modelcomplexity (measured with

∑β2

j )

Notes:1 The value for λ can be fixed by using cross-validation methods.2 Some of the entries in β take the value of 0, so LASSO can be useful as

a variable selection method.


LASSO

Continued...

Problems with LASSO:1 At most, n entries in β can be different from 0. This is problematic in GS,

where usually n << p (curse of dimensionality).2 It can be difficult to select the value for λ.3 It is difficult to obtain estimates for σ2

e .4 It is difficult to obtain confidence intervals for βj , j = 1, ...,p.

Alternatives:

Bayesian estimation methods ...


LASSO

Bayesian LASSO


LASSO

Continued...


LASSO

Continue...


LASSO

Continued...

−4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

β

Den

sity

func

tion

Figure 1: Prior in BL and in BRR

In ridge regression,

p(βj |σ2β) = N(βj |0, σ2

β), j = 1, ...,p

In LASSO

p(βj |σ2e , λ) = DE(βj |0, λ/σ2

e)


LASSO

Join posterior distribution of model unknowns

The join distribution for model unknowns is given by:

p(β, σ2e , µ|data) =

n∏i=1

N(yi |µ+∑

xijβj , σ2e)×

p∏j=1

p(βj |ω)×p(σ2e)×p(µ)×p(λ2),

(3)

where p(µ) ∝ 1, p(σ2e) = χ−2(σ2

e |df ,S) and p(λ2) = Gamma(λ2|rate, shape).

This model can be implemented using MCMC methods, for more detail seePark and Casella, 2008; de los Campos et al. (2009).

The model is implemented in the package BLR.


Application examples Example 1: Barley dataset

Contents

1 General comments

2 LASSO

3 Application examplesExample 1: Barley datasetExample 2: Wheat dataset (CIMMyT)




Example 1: Barley dataset

This example comes from Xi and Xu (2008).

DH population with n = 145 lines, each line tested in 25 environments. Theresponse variable is grain yield. We have p = 127 MM covering 7chromosomes.

BL model fitted using the BLR package in R with B = 20,000 iterations, burnin = 10,000, thin=10.



0 20 40 60 80 100 120

−1

01

23

B=20,000, burnin=10,000, a=b=0.1

j

β j

Figure 2: Point estimates for βSLU,Sweden GENOMIC SELECTION WORKSHOP:Hands on Practical Sessions (BL) 13/29


−0.5 0.0 0.5 1.0 1.5 2.00.

00.

40.

8

β2

0 1 2 3

0.0

0.2

0.4

0.6

0.8

β12

−0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.4

0.8

β13

−1.5 −1.0 −0.5 0.0 0.5

0.0

0.4

0.8

1.2

β27

−0.5 0.0 0.5 1.0 1.5

0.0

0.4

0.8

1.2

β32

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5 β34

Figure 3: Posterior distributions for β’sSLU,Sweden GENOMIC SELECTION WORKSHOP:Hands on Practical Sessions (BL) 14/29


−1.5 −1.0 −0.5 0.0 0.50.

00.

40.

81.

2 β37

−1.5 −1.0 −0.5 0.0

0.0

0.5

1.0

1.5 β43

−1.5 −1.0 −0.5 0.0 0.5

0.0

0.4

0.8

1.2

β95

−0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.4

0.8

β101

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.4

0.8

β102

Figure 4: Posterior distributions for β’sSLU,Sweden GENOMIC SELECTION WORKSHOP:Hands on Practical Sessions (BL) 15/29

Application examples Example 2: Wheat dataset (CIMMyT)

Contents

1 General comments

2 LASSO

3 Application examplesExample 1: Barley datasetExample 2: Wheat dataset (CIMMyT)




Example 2: Wheat dataset (CIMMyT)

Data for n = 599 wheat lines evaluated in 4 environments, wheatimprovement program, CIMMyT. The dataset includes p = 1279 molecularmarkers (xij , i = 1, ...,n, j = 1, ...,p) (coded as 0,1). The pedigree informationis also available.

Lets load the dataset in R,1 Load R2 Install BGLR package (if not yet installed)3 Load the package4 Load the data



Continued...Lets assume that we want to predict the grain yield for environment 1 usingridge regression or equivalently the G-BLUP. We do not know the value forσ2

e and λ, so we can obtain estimates using the data.

We will use the function BGLR. R code below fit the BL model using Bayesianapproach with non informative priors for σ2

e , λ,

rm(list=ls())library(BGLR)data(wheat)

Y=wheat.YX=wheat.X

y=Y[,1]

setwd(’/tmp/’)

#Linear predictorETA=list(list(X=X,model="BL"))

fmL<-BGLR(y=y,ETA=ETA,nIter=10000,burnIn=5000,thin=10)

plot(fmL$yHat,Y[,1]) SLU,Sweden GENOMIC SELECTION WORKSHOP:Hands on Practical Sessions (BL) 18/29


Continued...

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−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

−2

−1

01

23

fmL$yHat

Y[,

1]

Figure shows observed vs predictedgrain yield.

Predictions y = µ+X β, and estimatesfor σ2

e , λ can be obtained easily in R

> fmL$yHat> fmL$varE[1] 0.5379243> fmL$ETA[[1]]$lambda[1] 19.19093



Continued...

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−0.10 0.00 0.10

−0.

10−

0.05

0.00

0.05

0.10

Predicted Marker effects

Bayesian LASSO

Bay

esia

n R

idge

Reg

ress

ion

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−2.0 −1.0 0.0 1.0

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0−

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1.0

Predicted Genetic Values

Bayesian LASSO

Bay

esia

n R

idge

Reg

ress

ion



Continued...

The GEBVs can be obtained easily in R,

#GEVBs

#option 1X%*%fmL$ETA[[1]]$b

#option 2fmL$yHat-fmL$mu

Excersise:

Lets assume that we want to predict the grain yield for some wheat lines.Assume that we have only the genotypic information for those lines. Write theR code for fitting a BL model.


Extension of BL to include infinitesimal effect


de los Campos et al. (2009) extended the basic BL model to include aninfinitesimal effect, that is:

yi = µ+

p∑j=1

xijβj + ui + ei , (4)

where u ∼ N(0, σ2uA) and A is the pedigree matrix.

The model can be implemented using Bayesian methods.



Example 3: Including an infinitesimal effect

In this example we continue with the analysis of the wheat dataset, and weinclude an infinitesimal effect in the model.

rm(list=ls())setwd("/tmp")library(BGLR)data(wheat) #Loads the wheat dataset

X=wheat.XA=wheat.AY=wheat.Y

y=Y[,1]

#Linear predictorETA=list(list(X=X,model="BL"),

list(K=A,model="RKHS"))

### Runs the Gibbs samplerfm<-BGLR(y=y,ETA=ETA, nIter=30000,burnIn=5000,thin=10)



0 500 1500 25000.

300.

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60Iter

σ e2

0.30 0.40 0.50 0.60

02

46

8

σe2

Den

sity

0 500 1500 2500

0.05

0.15

0.25

Iter

σ u2

0.05 0.15 0.250

24

68

10σu

2

Den

sity

Figure 5: Posterior distribution for σ2e and σ2

u



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β j

Figure 6: Marker effects



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30

Marker

h2

Figure 7: Heritability

Narrow sense heritability calculatedaccording to Xi and Xu (2008),

h2j =

Vj β2j

Vy,

where Vy is the phenotypic variance,and Vj is the sample variance ofxij ; i = 1, ...,n.



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Figure 8: Observed vs predicted values



Questions?



References

Park, T. and Casella, G. (2008).The Bayesian Lasso.Journal of the American Statistical Association, 103, 681–686.

Yi, N. y Xu, S. (2008).Bayesian Lasso for Quantitative Trait Loci Mapping.Genetics, 179, 1045–1055.

de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E.Manfredi, K. Weigel and J. Cotes. (2009).Predicting Quantitative Traits with Regression Models for DenseMolecular Markers and Pedigree.Genetics 182: 375-385.

Pérez-Rodríguez P., G. de los Campos, J. Crossa and D. Gianola. (2010).Genomic-enabled prediction based on molecular markers and pedigreeusing the BLR package in R.The plant Genome, 3(2): 106-116.


GENOMIC SELECTION WORKSHOP: Hands on Practical Sessions …genomics.cimmyt.org/SLU/Slides Paulino/4....

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