Statistical Analysis of Genetic and Phenotypic Data for Breeders: - - PowerPoint PPT Presentation

Statistical Analysis of Genetic and Phenotypic Data for Breeders: Hands on Practical Sessions (BayesB)

Paulino Pérez 1 perpdgo@gmail.com José Crossa 2 j.crossa@CGIAR.org

1ColPos-México 2CIMMyT-México

June, 2015.

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Contents

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BayesB model and others

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BayesB model

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Examples

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BayesB model and others

BayesB model and others

Several analytical approaches for genome-based prediction of breeding values, yi = x′

iβ + ei, i = 1, ..., n

(1) Approaches differ respect to the assumptions about markers effects, Ridge regression-GBLUP: All the marker effects are normally distributed and all the markers have identical variances. BayesA and BayesB model introduced in GS by Meuwissen et al., 2001. BayesA: Markers are assumed to have different variances and are modeled following a χ−2 distribution. BayesB: Variance of some markers equal to zero with probability π and variance of the rest of the markers follows an inverted scaled χ−2 distribution with ν degrees of freedom and scale parameter s.

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BayesB model and others

Advantages and disadvantages of BayesB

Advantages Simulation studies have shown that BayesB provided more accurate predictions of genetic values than BayesA and Ridge regression. It can be used effectively for variable selection. Disadvantages It requires the use of computer intensive MCMC techniques. Huge computation times in computer simulations of genomic selection breeding schemes.

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BayesB model

BayesB model

BayesB uses a mixture distribution with a mass at zero, such that the (conditional) prior distribution of marker effects is given by βj|σ2

j , π =

with probability π N(0, σ2

j )

with probability 1 − π (2) The prior assigned to σ2

j , j = 1, ..., p is the same for all the markers, i.e. a

scaled inverted chi squared distribution χ−2(df, S). The model can be implemented using the Metropolis algorithm.

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BayesB model −3 −2 −1 1 2 3 0.0 0.1 0.2 0.3 0.4 0.5 β p(β) p(β) = γ1 2πexp(−1 (2σβ

2)β2) +(1 − γ)δ(0)

Figure 1: Example of prior for βj, j = 1, ..., p.

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Examples

Examples

Data for n = 599 wheat lines evaluated in 4 environments, wheat improvement program, CIMMyT. The dataset includes p = 1279 molecular markers (xij, i = 1, ..., n, j = 1, ..., p) (coded as 0,1). The pedigree information is also available. Lets load the dataset in R,

1

Load R

2

Install BGLR package (if not yet installed), this package is not yet available in CRAN, so you have to install it from https://r-forge.r-project.org/R/?group_id=1525

install.packages("BGLR", repos="http://R-Forge.R-project.org")

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Load the package

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Load the data

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Examples

R code

rm(list=ls()) setwd(tempdir()) library(BGLR) data(wheat) n<-599 # should be <= 599 p<-300 # should be <= than 1279=ncol(X) nQTL<-30 # should be <= than p X<-wheat.X[1:n,1:p] ## Centering and standarization for(i in 1:p) { X[,i]<-(X[,i]-mean(X[,i]))/sd(X[,i]) } # Simulation b0<-rep(0,p) whichQTL<-sample(1:p,size=nQTL,replace=FALSE) b0[whichQTL]<-rnorm(length(whichQTL), sd=sqrt(1/length(whichQTL))) signal<-as.vector(X%*%b0) error<-rnorm(n=n,sd=sqrt(0.5)) y<-signal +error

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Examples

Continued...

nIter=500; burnIn=400; thin=10; saveAt=’’; S0=NULL; weights=NULL; R2=0.5; ETA<-list(list(X=X,model="BayesB",probIn=0.05)) fit_BB=BGLR(y=y,ETA=ETA,nIter=nIter,burnIn=burnIn, thin=thin,saveAt=saveAt,df0=5,S0=S0, weights=weights,R2=R2) plot(fit_BB$yHat,y)

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Examples

Continued...

Notes: In this model the proportion of markers with zero effects should be fixed by the user. BayesCπ offers the possibility of estimating this proportion from the sample. In BGLR the proportion of makers whose effect is effectively different from zero are indicated using the parameter probIn, in this case probIn = 0.05, that means that at most 1279 × 0.05 ≈ 64 markers are different from zero. hyper parameter for χ2(σ2

j |df, S), j = 1, ..., p are fixed using heritability

rules, see de los Campos et al., (2012) for more details. In this case h2 = 0.5 (in the code R2=0.5), but you can change it according to your knowledge of heritabilities. Also it is possible to set this hyperparameters directly.

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Examples

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−2 −1 1 2 3 −4 −2 2 4 fit_BB$yHat y

#Preditions > fit_BB$yHat #Error variance > fit_BB$varE [1] 0.5973773 #Marker effects > fit_BB$ETA[[1]]$b #Variance for markers > fit_BB$ETA[[1]]$varB

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Examples

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400 600 800 1000 1200 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 Index fit_BB$ETA[[1]]$b CIMMYT, México-SAGPDB Statistical Analysis of Genetic and Phenotypic Data for Breeders:Hands on Practical 12/15

Examples

Continued...

The GEBVs can be obtained easily in R,

#GEVBs #option 1 X%*%fit_BB$ETA[[1]]$b #option 2 fmL$yHat-fit_BB$mu

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Examples

Questions?

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Examples

References

de los Campos G., J. Hickey, R. Pong-Wong, H. D. Daetwyler and M.P .L. Calus,. 2012. Whole Genome Regression and Prediction Methods Applied to Plant and Animal Breeding, Genetics. Meuwissen THE, Hayes BJ, Goddard ME. 2001. Prediction of total genetic values using genome wide dense marker maps, Genetics 157: 1819-1829.

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