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Modeling Genetic Heterogenity using Random-effects Interactions

For further details see de los Campos et al., (JABES, 2015), Veturi et al. (Genetics, 2019), and this example. For a similar application involving GxE see Lopez-Cruz et al. (G3, 2015), and for one involving multi-breed and GxE see Yao et al. (J. D. Sci., 2017).

Clustering and displaying PCs

  ## Clustering using 5-PCs and kmeans
   library(BGLR)
   data(wheat)
   X=scale(wheat.X)/sqrt(ncol(wheat.X))
   y=wheat.Y[,1]
   PC=svd(X,nu=2,nv=0)$u
   group=kmeans(x=PC,centers=2,nstart=100)$cluster
   plot(PC,col=group*2,cex=.5)

Preparing inputs

  X0=X # for main effects
  X1=X; X1[group==2,]=0 #interactions
  X2=X; X2[group==1,]=0 #interactions
  Z2=as.matrix(as.integer(group==2)) # a dummy variable for group 2

Next we will fit 3 models: (1) a model assuming that effects are homogeneous across clusters, (2) a statified analysis, and (3) a model with interactions.

(1) Homogeneous effects model

 fm0=BGLR( y=y,ETA=list( 
 		  int=list(X=Z2, model='FIXED'),
 		  list(X=X0,model='BRR') 
 	      ),
	  nIter=6000,burnIn=1000,saveAt='m0_')

(2) Stratified Analysis

 fm1=BGLR( y=y[group==1],
 	   ETA=list( list(X=X0[group==1,],model='BRR') ),
	   nIter=6000,burnIn=1000, saveAt='m1_')
	   
 fm2=BGLR( y=y[group==2],
 	   ETA=list( list(X=X0[group==2,],model='BRR') ),
	   nIter=6000,burnIn=1000, saveAt='m2_')

(3) Interaction model

  fm12=BGLR(y=y,ETA=list(
  		  int=list(X=Z2, model='FIXED'),
                  main=list(X=X0,model='BRR'),
                  int1=list(X=X1,model='BRR'),
                  int2=list(X=X2,model='BRR')
		),
	     nIter=6000,burnIn=1000,groups=group, saveAt='m12_')

  varU1=fm12$ETA[[2]]$varB+fm12$ETA[[3]]$varB
  varU2=fm12$ETA[[2]]$varB+fm12$ETA[[4]]$varB
  varE1=fm12$varE[1] 
  varE2=fm12$varE[2]

  h2_1=varU1/(varU1+varE1)
  h2_2=varU2/(varU2+varE2)
  
  fm12$ETA[[2]]$varB/sqrt(varU1*varU2) #correlation of effects

NOTES:

  • The above example uses a Gaussian prior, for information on other priors implemented in BGLR see Bayesian Alphabet
  • Within Gaussian contexts, the models described above can also be fitted using kinship matrices or marker-derived PCs, for this see GBLUP
  • Random effects interactions can also be used to deal with GxE

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