A novel approach has been presented to enhance the accuracy of prediction in genomic selection and the accuracy of identifying QTL under collinearity in GWAS. BLUP assumes marker effects are randomly assigned from a normal distribution with homogeneous variance, while Bayesian alphabet models assume marker effect variances are heterogeneous. The proposed method SREML is between BLUP and Bayesian alphabet models. Unlike the assumption of BLUP, the novel method assumes a large group of markers have small effects and a small number of markers have extra effects than the remaining markers. This assumption allows we obtain additional information from the data to improve the model performance and obtain the markers with large genetic effects simultaneously. The simu- lation study in genomic selection shows the proposed method outperforms other methods when we used model 3.2.1 to generate phenotypic values.
As discussed above, the procedure of SREML is stochastic and it will be impressed that this algorithm is similar to Bayesian approaches. However, SREML is distinguishing from Bayesian approaches because we don’t expect it will converge to a specific value. The result from each iteration is stochastic even through it is based on the result from the previous iteration. The reason is every element of the u vector is randomly sampled from a Bernoulli distribution with the posterior probability of success ρ, which involves a random variable π. And the value of π is randomly sampled from a Beta distribution and its parameters are calculated by taking advantage of the data information. It implies that the number of nonzero-effect markers is automatically obtained based on the data itself, but not an artificial setting. The unknown parameters are then estimated by the REML
function. Compared with Bayesian approaches, the cluster label vector u of SREML is the only random variable sampled from a distribution. Variance components are obtained according to the current cluster labels. Vector u will never converge. Therefore, the values of variance components vary from iteration to iteration. In addition, all unknown parameters are sampled from distributions in Bayesian approaches. Sometimes, the rate of convergence can be slow, which requires large iteration numbers. However, combining the stochastic step and the REML step in SREML facilitates to obtain good parameter estimates within the comparatively small iteration numbers.
The other contribution of this study is to make the algorithm to be computationally tractable. For each iteration, all markers need to be scanned to update the indicator variable ui. It leads to the computational burden because the inverse and determinant of updated
phenotypic variance Vineed to be calculated for every marker. Due to it, Woodbury matrix
identities and the matrix determinant lemma are applied in the algorithm to reduce the computational efforts.
One application of GWAS is to identify QTL which are markers with large genetic effects. Nowadays, millions of SNPs can be produced as our genetic information, but only few of them are associated with quantitative traits. Therefore, a large amount of informa- tion is useless and time-consuming for GWAS. Selecting informative markers can achieve accuracy enhancement and marker interpretability, where SREML, LASSO, and BayesB make contributions on excluding redundant markers. Based on the results above, SREML outperforms LASSO and BayesB under collinearity condition in GWAS. LASSO tends to select unimportant markers than the other two methods, and LASSO and BayesB tends to select fewer of correlated markers than SREML. Since the genotypic values are highly
correlated, the posterior probabilities of ui and uj tends to be close to each other if their
current labels are the same. Consequently, for this case, SREML is superior to LASSO and BayesB.
3.A
Appendix of Chapter 3
A The R code for SREML
SREML<−function ( z , y , w, i t e r n u m ) { n<−ncol ( z ) m<−nrow( z ) x<−matrix ( 1 , n , 1 ) m1<−m∗w m2<−m∗(1−w)
d e l t a<−sample ( c ( rep ( −1 ,m1 ) , rep ( 1 ,m2 ) ) ) p r e d<−NULL d e l t<−NULL max. p r e d<− −1e 1 0 max. d e l t a<−d e l t a max. i t e r<− 0 max. PRESS<−0 f o r ( i t e r i n 1 : i t e r n u m ) {
z1<−z [ which ( d e l t a ==−1),] z2<−z [ which ( d e l t a== 1 ) , ] k1<−crossprod ( z1 ) k2<−crossprod ( z2 ) kk<−l i s t ( k1 , k2 ) p<−length ( kk )
f i t<−mixed ( x , y=y , kk ) #REML method t o e s t i m a t e unknown p a r a m e t e r s f n 0<− − f i t [ [ 1 ] ] $ f n v1<− f i t [ [ 2 ] ] $v1 v2<− f i t [ [ 2 ] ] $v2 ve<− f i t [ [ 2 ] ] $ve parm<−c ( v1 , v2 , ve ) v<−diag ( n ) ∗parm [ p+1] g<−matrix ( 0 , n , n ) vv<−0 f o r ( k i n 1 : p ) { vv<−vv+parm [ k ] g<−g+kk [ [ k ] ] ∗vv } v<−v+g v i<−solve ( v ) H<−g%∗%v i
b l u p 0<−y−x%∗%f i t [ [ 1 ] ] $beta b l u p<−H%∗%b l u p 0
PRESS<−sum ( ( ( blup0−b l u p ) /(1−diag (H) ) ) ˆ 2 ) SS<−sum( ( blup0−mean( b l u p 0 ) ) ˆ 2 ) PRED<−1−PRESS/SS i f (PRED>max. p r e d ) { max. p r e d<−PRED max. d e l t a<−d e l t a max. i t e r<−i t e r max. PRESS<−PRESS max. parm<−parm
max. beta<− f i t [ [ 1 ] ] $beta } b<−b l u p 0 bb<−t ( b )%∗%v i%∗%b xx<−t ( x )%∗%v i%∗%x dv<−u n l i s t ( d e t e r m i n a n t ( v ) ) dv<−dv [ 1 ] a l p h a<−length ( which ( d e l t a ==−1)) beta<−m−a l p h a
w<−rbeta ( 1 , a l p h a+m, beta +0.1∗m)
d e l t<−rbind ( d e l t , c ( i t e r , d e l t a ) ) f o r ( i i n 1 :m) { zk<−as . matrix ( z [ i , ] ) z z<−t ( zk )%∗%v i%∗%zk bz<−t ( b )%∗%v i%∗%zk zb<−t ( bz ) xz<−t ( x )%∗%v i%∗%zk zx<−t ( xz ) xHx<−xx−xz%∗%solve ( zz −1/ ( d e l t a [ i ] ∗v2 ) )%∗%zx bHb<−bb−bz%∗%solve ( zz −1/ ( d e l t a [ i ] ∗v2 ) )%∗%zb z v z<−abs(− d e l t a [ i ] ∗v2∗ z z +1) dx<−u n l i s t ( d e t e r m i n a n t ( xHx ) ) dx<−dx [ 1 ] dv2<−log ( z v z ) f n<− −0.5∗ ( dv+dv2+dx+bHb ) u<−rbinom ( 1 , 1 , w/ (w+(1−w) ∗exp ( d e l t a [ i ] ∗ ( fn0−f n ) ) ) ) d e l t a [ i ]<−1−2∗u } } } mixed<−function ( x , y , kk ) {
l o g l i k e<−function ( parm ) { v<−diag ( n ) ∗parm [ p+1] vv<−0 f o r ( k i n 1 : p ) { vv<−vv+parm [ k ] v<−v+kk [ [ k ] ] ∗vv } v i<−solve ( v ) xx<−t ( x )%∗%v i%∗%x xy<−t ( x )%∗%v i%∗%y b<−solve ( xx , xy ) d1<−u n l i s t ( d e t e r m i n a n t ( v ) ) d1<−d1 [ 1 ] d2<−u n l i s t ( d e t e r m i n a n t ( xx ) ) d2<−d2 [ 1 ] r<−y−x%∗%b q<−t ( r )%∗%v i%∗%r l o g l i k e<− −0.5∗ ( d1+d2+q ) return(− l o g l i k e ) } f i x e d<−function ( parm ) {
v<−diag ( n ) ∗parm [ p+1] g<−matrix ( 0 , n , n ) vv<−0 f o r ( k i n 1 : p ) { vv<−vv+parm [ k ] g<−g+kk [ [ k ] ] ∗vv } v<−v+g v i<−solve ( v ) xx<−t ( x )%∗%v i%∗%x xy<−t ( x )%∗%v i%∗%y covb<−solve ( xx ) beta<−solve ( xx , xy )
yhat<−g%∗%v i%∗%( y−x%∗%beta ) yobs<−y−x%∗%beta
r 2<−cor ( yobs , yhat ) ˆ 2
r e s u l t<−l i s t ( beta , covb , r 2 ) return ( r e s u l t ) } l o g l i k e 0<−function ( x , y ) { xx<−t ( x )%∗%x xy<−t ( x )%∗%y
b<−solve ( xx , xy ) r<−y−x%∗%b s 2<−drop ( t ( r )%∗%( r ) ) / ( n−ncol ( x ) ) v<−diag ( n ) ∗ s 2 v i<−diag ( n ) / s 2 d1<−u n l i s t ( d e t e r m i n a n t ( v ) ) d1<−d1 [ 1 ] d2<−u n l i s t ( d e t e r m i n a n t ( xx ) ) d2<−d2 [ 1 ] q<−t ( r )%∗%v i%∗%r l o g l i k e<− −0.5∗ ( d1+d2+q ) return(− l o g l i k e ) } n<−length ( y ) p<−length ( kk ) f n 0<−l o g l i k e 0 ( x , y ) parm0<−rep ( 1 , p+1)
r e s u l t<−optim ( par=parm0 , f n=l o g l i k e , h e s s i a n = TRUE, method=”L−BFGS−B” , lower=1e −5 ,upper=1e5 )
parm<−r e s u l t $par
conv<−r e s u l t $ c o n v e r g e n c e f n<−r e s u l t $ v a l u e
l r t<−2∗ ( fn0−f n ) h e s s<−r e s u l t $ h e s s i a n covp<− solve ( h e s s ) bb<−f i x e d ( parm ) beta<−bb [ [ 1 ] ] covb<−bb [ [ 2 ] ] r 2<−bb [ [ 3 ] ]
f i x e d<−data . frame ( conv , fn0 , fn , l r t , beta , covb , r 2 ) v1<−parm [ 1 ] v2<−parm [ 2 ] ve<−parm [ 3 ] parm<−data . frame ( v1 , v2 , ve ) r e s u l t<−l i s t ( f i x e d , parm ) return ( r e s u l t ) } B Supplemental tables
Table 3.10 Variance components estimated by SREML and BLUP in Scenario II. Method σγ2 σS2 σL2 σ2e
SREML - 8.8391 104.9971 1.4338 BLUP 22.2774 - - 1.4377
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