$y = X\beta + Zu_1 + Zu_2 + Zu_S + \epsilon$

We estimate variance components for $GCA_1$, $GCA_2$ and $SCA$ and use them to estimate heritability. Additionally BLUPs for GCA and SCA effects can be used to predict crosses. ```{r} data(DT_cornhybrids) DT <- DT_cornhybrids DTi <- DTi_cornhybrids GT <- GT_cornhybrids modFD <- lmebreed(Yield~Location + (1|GCA1)+(1|GCA2)+(1|SCA), verbose = FALSE,data=DT) vc <- VarCorr(modFD); print(vc,comp=c("Variance")) Vgca <- vc$GCA1 + vc$GCA2 Vsca <- vc$SCA Ve <- attr(vc, "sc")^2 Va = 4*Vgca Vd = 4*Vsca Vg <- Va + Vd (H2 <- Vg / (Vg + (Ve)) ) (h2 <- Va / (Vg + (Ve)) ) ``` Don't worry too much about the `h2` value, the data was simulated to be mainly dominance variance, therefore the `Va` was simulated extremely small leading to such value of narrow sense `h2`. In the second data set we show a small half diallel with 7 parents crossed in one direction. There are n(n-1)/2 possible crosses; 7(6)/2 = 21 unique crosses. Parents appear as males or females indistictly. Each with two replications in a CRD. For a half diallel design a single GCA variance component for both males and females can be estimated and an SCA as well ($\sigma^2_GCA$ and $\sigma^2_SCA$ respectively), and BLUPs for GCA and SCA of the parents can be extracted. We will show how to do so using the `overlay()` function. The specific model here is: $y = X\beta + Zu_g + Zu_s + \epsilon$ ```{r} data("DT_halfdiallel") DT <- DT_halfdiallel head(DT) DT$femalef <- as.factor(DT$female) DT$malef <- as.factor(DT$male) DT$genof <- as.factor(DT$geno) # overlay matrix to be added to the addmat argument Z <- with(DT, overlay(femalef,malef) ) # create inital values for incidence matrix but irrelevant # since these will be replaced by admat argument fema <- (rep(colnames(Z), nrow(DT)))[1:nrow(DT)] #### model using overlay without relationship matrix modh <- lmebreed(sugar ~ (1|genof) + (1|fema), addmat = list(fema=Z), verbose = FALSE, data=DT) vc <- VarCorr(modh); print(vc,comp=c("Variance")) ve <- attr(vc, "sc")^2;ve ``` Notice how the `overlay()` argument makes the overlap of incidence matrices possible making sure that male and female are joint into a single random effect. ### 4) Genomic selection: predicting mendelian sampling In this section we will use wheat data from CIMMYT to show how genomic selection is performed. This is the case of prediction of specific individuals within a population. It basically uses a similar model of the form:

$y = X\beta + Zu + \epsilon$

and takes advantage of the variance covariance matrix for the genotype effect known as the additive relationship matrix (A) and calculated using the `A.mat` function to establish connections among all individuals and predict the BLUPs for individuals that were not measured. The prediction accuracy depends on several factors such as the heritability ($h^2$), training population used (TP), size of TP, etc. ```{r} # data(DT_wheat) # DT <- DT_wheat # GT <- GT_wheat[,1:200] # colnames(DT) <- paste0("X",1:ncol(DT)) # DT <- as.data.frame(DT);DT$line <- as.factor(rownames(DT)) # # select environment 1 # rownames(GT) <- rownames(DT) # K <- A.mat(GT) # additive relationship matrix # colnames(K) <- rownames(K) <- rownames(DT) # # GBLUP pedigree-based approach # set.seed(12345) # y.trn <- DT # vv <- sample(rownames(DT),round(nrow(DT)/5)) # y.trn[vv,"X1"] <- NA # head(y.trn) # ## GBLUP # K <- K + diag(1e-4, ncol(K), ncol(K) ) # ans <- lmebreed(X1 ~ (1|line), # relmat = list(line=K), # control = lmerControl( # check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore" # ), verbose = FALSE, # data=y.trn) # vc <- VarCorr(ans); print(vc,comp=c("Variance")) # # # take a extended dataset and fit a dummy model # # just to get required matrices # y.tst <- y.trn; y.tst$X1 <- imputev(y.tst$X1) # ans2 <- update(ans, # start = getME(ans, "theta"), # data = y.tst, # control = lmerControl(check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore", # optCtrl = list(maxeval= 1), # calc.derivs = FALSE)) # # compute predictive ability # cor(ranef(ans2)$line[vv,],DT[vv,"X1"], use="complete") # # # other approach # # mme <- getMME(ans2, vc=vc, recordsToKeep = which(!is.na(y.trn$X1))) # # cor(mme$bu[vv,],DT[vv,"X1"], use="complete") # # ## rrBLUP # M <- tcrossprod(GT) # xx <- with(y.trn, redmm(x=line, M=M, nPC=100, returnLam = TRUE)) # custom <- (rep(colnames(Z), nrow(DT)))[1:nrow(DT)] # ansRRBLUP <- lmebreed(X1 ~ (1|custom), verbose = FALSE, # addmat = list(custom=Z), # data=y.trn) # re <- ranef(ansRRBLUP)$custom # u = tcrossprod(xx$Lam, t(as.matrix( re[colnames(xx$Lam),] ) )) # cor(u[vv,],DT[vv,"X1"], use="complete") ``` ### 5) Indirect genetic effects General variance structures can be used to fit indirect genetic effects. Here, we use an example dataset to show how we can fit the variance and covariance components between two or more different random effects. We now fit the indirect genetic effects model with covariance between DGE and IGE. On top of that we can include a relationship matrix for the two random effects that are being forced to co-vary ```{r} # data(DT_ige) # DT <- DT_ige # A_ige <- A_ige + diag(1e-4, ncol(A_ige), ncol(A_ige) ) # # Define 2 dummy variables to make a fake covariance # # for two different random effects # DT$fn <- DT$nn <- 1 # # Create the incidence matrix for the first random effect # Zf <- Matrix::sparse.model.matrix( ~ focal-1, data=DT ) # colnames(Zf) <- gsub("focal","", colnames(Zf)) # # Create the incidence matrix for the second random effect # Zn <- Matrix::sparse.model.matrix( ~ neighbour-1, data=DT ) # colnames(Zn) <- gsub("neighbour","", colnames(Zn)) # # Make inital values for incidence matrix but irrelevant # # since these will be replaced by the addmat argument # both <- (rep(colnames(Zf), nrow(DT)))[1:nrow(DT)] # # Fit the model # modIGE <- lmebreed(trait ~ block + (0+fn+nn|both), # addmat = list(both=list(Zf,Zn)), # relmat = list(both=A_ige), # verbose = FALSE, data = DT) # vc <- VarCorr(modIGE); print(vc,comp=c("Variance")) # blups <- ranef(modIGE) # pairs(blups$both) # cov2cor(vc$both) ``` ### 6) Genomic selection: single cross prediction When doing prediction of single cross performance the phenotype can be dissected in three main components, the general combining abilities (GCA) and specific combining abilities (SCA). This can be expressed with the same model analyzed in the diallel experiment mentioned before:

$y = X\beta + Zu_1 + Zu_2 + Zu_S + \epsilon$

with:

$u_1$ ~ N(0, $K_1$$\sigma^2_u1$) $u_2$ ~ N(0, $K_2$$\sigma^2_u2$) $u_s$ ~ N(0, $K_3$$\sigma^2_us$)

And we can specify the K matrices. The main difference between this model and the full and half diallel designs is the fact that this model will include variance covariance structures in each of the three random effects (GCA1, GCA2 and SCA) to be able to predict the crosses that have not ocurred yet. We will use the data published by Technow et al. (2015) to show how to do prediction of single crosses. ```{r} # data(DT_technow) # DT <- DT_technow # Md <- (Md_technow*2) - 1 # Mf <- (Mf_technow*2) - 1 # Ad <- A.mat(Md) # Af <- A.mat(Mf) # Ad <- Ad + diag(1e-4, ncol(Ad), ncol(Ad)) # Af <- Af + diag(1e-4, ncol(Af), ncol(Af)) # # simulate some missing hybrids to predict # y.trn <- DT # vv1 <- which(!is.na(DT$GY)) # vv2 <- sample(DT[vv1,"hy"], 100) # y.trn[which(y.trn$hy %in% vv2),"GY"] <- NA # ans2 <- lmebreed(GY ~ (1|dent) + (1|flint), # relmat = list(dent=Ad, # flint=Af), # verbose = FALSE, data=y.trn) # vc <- VarCorr(ans2); print(vc,comp=c("Variance")) # # # take a extended dataset and fit a dummy model # # just to get required matrices # y.tst <- y.trn; y.tst$GY <- imputev(y.tst$GY) # ans2p <- update(ans2, # start = getME(ans2, "theta"), # data = y.tst, # control = lmerControl(check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore", # optCtrl = list(maxeval= 1), # calc.derivs = FALSE)) # # re <- ranef(ans2p) # # Pdent <- as.matrix(re$dent[,1,drop=FALSE]) %*% Matrix(1, ncol=nrow(re$flint), nrow=1) # Pflint <- as.matrix(re$flint[,1,drop=FALSE]) %*% Matrix(1, ncol=nrow(re$dent), nrow=1) # P <- Pdent + t(Pflint); colnames(P) <- rownames(re$flint) # # preds <- real <- numeric() # for(iHyb in vv2){ # parents <- strsplit(iHyb,":")[[1]] # preds[iHyb] <- P[which(rownames(P) %in% parents),which(colnames(P) %in% parents)] # real[iHyb] <- DT[which(DT$hy == iHyb),"GY"] # } # plot(preds, real) # cor(preds, real) ``` In the previous model we only used the GCA effects (GCA1 and GCA2) for practicity, altough it's been shown that the SCA effect doesn't actually help that much in increasing prediction accuracy, but does increase a lot the computation intensity required since the variance covariance matrix for SCA is the kronecker product of the variance covariance matrices for the GCA effects, resulting in a 10578 x 10578 matrix that increases in a very intensive manner the computation required. A model without covariance structures would show that the SCA variance component is insignificant compared to the GCA effects. This is why including the third random effect doesn't increase the prediction accuracy. ### 8) Spatial modeling: using the 2-dimensional spline We will use the CPdata to show the use of 2-dimensional splines for accomodating spatial effects in field experiments. In early generation variety trials the availability of seed is low, which makes the use of unreplicated designs a neccesity more than anything else. Experimental designs such as augmented designs and partially-replicated (p-rep) designs are becoming ever more common these days. In order to do a good job modeling the spatial trends happening in the field, special covariance structures have been proposed to accomodate such spatial trends (i.e. autoregressive residuals; ar1). Unfortunately, some of these covariance structures make the modeling rather unstable. More recently, other research groups have proposed the use of 2-dimensional splines to overcome such issues and have a more robust modeling of the spatial terms (Lee et al. 2013; Rodríguez-Álvarez et al. 2018). In this example we assume an unreplicated population where row and range information is available which allows us to fit a 2 dimensional spline model. ```{r} data(DT_cpdata) DT <- DT_cpdata # add the units column DT$units <- as.factor(1:nrow(DT)) # get spatial incidence matrix Zs <- with(DT, tps(Row, Col))$All rownames(Zs) <- DT$units # reduce the matrix to its PCs Z = with(DT, redmm(x=units, M=Zs, nPC=100)) # create dummy variable spatial <- (rep(colnames(Z), nrow(DT)))[1:nrow(DT)] # fit model mix1 <- lmebreed(Yield~ (1|Rowf) + (1|Colf) + (1|spatial), addmat =list(spatial=Z), control = lmerControl( check.nobs.vs.nlev = "ignore", check.nobs.vs.rankZ = "ignore", check.nobs.vs.nRE="ignore" ), verbose = FALSE, data=DT) vc <- VarCorr(mix1); print(vc,comp=c("Variance")) ``` Notice that the job is done by the `spl2Da()` function that takes the `Row` and `Col` information to fit a spatial kernel. ### 9) Multivariate genetic models and genetic correlations Sometimes is important to estimate genetic variance-covariance among traits--multi-reponse models are very useful for such a task. Let see an example with 2 traits (`color`, `Yield`) and a single random effect (genotype; `id`) although multiple effects can be modeled as well. We need to use a variance covariance structure for the random effect to be able to obtain the genetic covariance among traits. ```{r} # data(DT_cpdata) # DT <- DT_cpdata # GT <- GT_cpdata # MP <- MP_cpdata # #### create the variance-covariance matrix # A <- A.mat(GT) # additive relationship matrix # A <- A + diag(1e-4, ncol(A), ncol(A)) # #### look at the data and fit the model # head(DT) # DT2 <- stackTrait(data=DT, traits = c("Yield","color")) # head(DT2$long) # # mix1 <- lmebreed(valueS~ (0+trait|id), # relmat=list(id=A), # control = lmerControl( # check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore" # ), verbose = FALSE, # data=DT2$long) # vc <- VarCorr(mix1); print(vc,comp=c("Variance")) ``` Now you can extract the BLUPs using `ranef(ans.m)`. Also, genetic correlations and heritabilities can be calculated easily. ```{r} # cov2cor(vc$id) ``` ## SECTION 2: Special topics in Quantitative genetics ### 1) Partitioned model The partitioned model was popularized by () to show that marker effects can be obtained by fitting a GBLUP model to reduce the computational burden and then recover them by creating some special matrices MM' for GBLUP and M'(M'M)- to recover marker effects. Here we show a very easy example using the DT_cpdata: ```{r} # data("DT_cpdata") # DT <- as.data.frame(DT_cpdata) # M <- GT_cpdata # # ################ # # PARTITIONED GBLUP MODEL # ################ # # MMT <-tcrossprod(M) ## MM' = additive relationship matrix # MMTinv<-solve(MMT) ## inverse # MTMMTinv<-t(M)%*%MMTinv # M' %*% (M'M)- # # mix.part <- lmebreed(color ~ (1|id), # relmat = list(id=MMT), # control = lmerControl( # check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore" # ), verbose = FALSE, # data=DT) # # #convert BLUPs to marker effects me=M'(M'M)- u # re <- ranef(mix.part)$id # me.part<-MTMMTinv[,rownames(re)]%*%matrix(re[,1],ncol=1) # plot(me.part) ``` As can be seen, these two models are equivalent with the exception that the partitioned model is more computationally efficient. ### 2) UDU' decomposition Lee and Van der Warf (2015) proposed a decomposition of the relationship matrix A=UDU' together with a transformation of the response and fixed effects Uy = Ux + UZ + e, to fit a model where the phenotypic variance matrix V is a diagonal because the relationship matrix is the diagonal matrix D from the decomposition that can be inverted easily and make multitrait models more feasible. ```{r} # # data("DT_wheat") # rownames(GT_wheat) <- rownames(DT_wheat) # G <- A.mat(GT_wheat) # Y <- data.frame(DT_wheat) # # # make the decomposition # UD<-eigen(G) # get the decomposition: G = UDU' # U<-UD$vectors # D<-diag(UD$values)# This will be our new 'relationship-matrix' # rownames(D) <- colnames(D) <- rownames(G) # X<-model.matrix(~1, data=Y) # here: only one fixed effect (intercept) # UX<-t(U)%*%X # premultiply X and y by U' # UY <- t(U) %*% as.matrix(Y) # multivariate # # # dataset for decomposed model # DTd<-data.frame(id = rownames(G) ,UY, UX =UX[,1]) # DTd$id<-as.character(DTd$id) # head(DTd) # # modeld <- lmebreed(X1~ UX + (1|id), # relmat=list(id=D), # control = lmerControl( # check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore" # ), verbose = FALSE, # data=DTd) # vc <- VarCorr(modeld); print(vc,comp=c("Variance")) # # # dataset for normal model # DTn<-data.frame(id = rownames(G) , DT_wheat) # DTn$id<-as.character(DTn$id) # # modeln <- lmebreed(X1~ (1|id), # relmat=list(id=G), # control = lmerControl( # check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore" # ), verbose = FALSE, # data=DTn) # # ## compare regular and transformed blups # red <- ranef(modeld)$id # ren <- ranef(modeln)$id # plot(x=(solve(t(U)))%*% red[colnames(D),], # y=ren[colnames(D),], # xlab="UDU blup", ylab="blup") # ``` As can be seen, the two models are equivalent. Despite the fact that lme4breeding doesn't take a great advantage of this trick because it was built for dense matrices using the Armadillo library. Other software may be better using this trick. ### 3) Mating designs Estimating variance components has been a topic of interest for the breeding community for a long time. Here we show how to calculate additive and dominance variance using the North Carolina Design I (Nested design) and North Carolina Design II (Factorial design) using the classical Expected Mean Squares method and the REML methods from lme4breeding and how these two are equivalent. #### North Carolina Design I (Nested design) ```{r} data(DT_expdesigns) DT <- DT_expdesigns$car1 DT <- aggregate(yield~set+male+female+rep, data=DT, FUN = mean) DT$setf <- as.factor(DT$set) DT$repf <- as.factor(DT$rep) DT$malef <- as.factor(DT$male) DT$femalef <- as.factor(DT$female) #levelplot(yield~male*female|set, data=DT, main="NC design I") ############################## ## Expected Mean Square method ############################## mix1 <- lm(yield~ setf + setf:repf + femalef:malef:setf + malef:setf, data=DT) MS <- anova(mix1); MS ms1 <- MS["setf:malef","Mean Sq"] ms2 <- MS["setf:femalef:malef","Mean Sq"] mse <- MS["Residuals","Mean Sq"] nrep=2 nfem=2 Vfm <- (ms2-mse)/nrep Vm <- (ms1-ms2)/(nrep*nfem) ## Calculate Va and Vd Va=4*Vm # assuming no inbreeding (4/(1+F)) Vd=4*(Vfm-Vm) # assuming no inbreeding(4/(1+F)^2) Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg ############################## ## REML method ############################## mix2 <- lmebreed(yield~ setf + setf:repf + (1|femalef:malef:setf) + (1|malef:setf), verbose = FALSE, data=DT) vc <- VarCorr(mix2); print(vc,comp=c("Variance")) Vfm <- vc$`femalef:malef:setf` Vm <- vc$`malef:setf` ## Calculate Va and Vd Va=4*Vm # assuming no inbreeding (4/(1+F)) Vd=4*(Vfm-Vm) # assuming no inbreeding(4/(1+F)^2) Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg ``` As can be seen the REML method is easier than manipulating the MS and we arrive to the same results. #### North Carolina Design II (Factorial design) ```{r} DT <- DT_expdesigns$car2 DT <- aggregate(yield~set+male+female+rep, data=DT, FUN = mean) DT$setf <- as.factor(DT$set) DT$repf <- as.factor(DT$rep) DT$malef <- as.factor(DT$male) DT$femalef <- as.factor(DT$female) #levelplot(yield~male*female|set, data=DT, main="NC desing II") head(DT) N=with(DT,table(female, male, set)) nmale=length(which(N[1,,1] > 0)) nfemale=length(which(N[,1,1] > 0)) nrep=table(N[,,1]) nrep=as.numeric(names(nrep[which(names(nrep) !=0)])) ############################## ## Expected Mean Square method ############################## mix1 <- lm(yield~ setf + setf:repf + femalef:malef:setf + malef:setf + femalef:setf, data=DT) MS <- anova(mix1); MS ms1 <- MS["setf:malef","Mean Sq"] ms2 <- MS["setf:femalef","Mean Sq"] ms3 <- MS["setf:femalef:malef","Mean Sq"] mse <- MS["Residuals","Mean Sq"] nrep=length(unique(DT$rep)) nfem=length(unique(DT$female)) nmal=length(unique(DT$male)) Vfm <- (ms3-mse)/nrep; Vf <- (ms2-ms3)/(nrep*nmale); Vm <- (ms1-ms3)/(nrep*nfemale); Va=4*Vm; # assuming no inbreeding (4/(1+F)) Va=4*Vf; # assuming no inbreeding (4/(1+F)) Vd=4*(Vfm); # assuming no inbreeding(4/(1+F)^2) Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg ############################## ## REML method ############################## mix2 <- lmebreed(yield~ setf + setf:repf + (1|femalef:malef:setf) + (1|malef:setf) + (1|femalef:setf), verbose = FALSE, data=DT) vc <- VarCorr(mix2); print(vc,comp=c("Variance")) Vfm <- vc$`femalef:malef:setf` Vm <- vc$`malef:setf` Vf <- vc$`femalef:setf` Va=4*Vm; # assuming no inbreeding (4/(1+F)) Va=4*Vf; # assuming no inbreeding (4/(1+F)) Vd=4*(Vfm); # assuming no inbreeding(4/(1+F)^2) Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg ``` As can be seen, the REML method is easier than manipulating the MS and we arrive to the same results. ### 4) GWAS by GBLUP Gualdron-Duarte et al. (2014) and Bernal-Rubio et al. (2016) proved that in (SingleStep)GBLUP or RRBLUP/SNP-BLUP, dividing the estimate of the marker effect by its standard error is mathematically equivalent to fixed regression EMMAX GWAS, even if markers are estimated as random effects in GBLUP and as fixed effects in EMMAX. That way fitting a GBLUP model is enough to perform GWAS for additive and on-additive effects. Let us use the DT_cpdata dataset to explore the GWAS by GBLUP method ```{r} data(DT_cpdata) DT <- DT_cpdata GT <- GT_cpdata#[,1:200] MP <- MP_cpdata M<- GT n <- nrow(DT) # to be used for degrees of freedom k <- 1 # to be used for degrees of freedom (number of levels in fixed effects) ``` Instead of fitting the RRBLUP/SNP-BLUP model we can fit a GBLUP model which is less computationally demanding and recover marker effects and their standard errors from the genotype effects. ```{r} # ########################### # #### GWAS by GBLUP approach # ########################### # MMT <-tcrossprod(M) ## MM' = additive relationship matrix # MMT <- MMT + diag(1e-4, ncol(MMT), ncol(MMT) ) # MMTinv<-solve( MMT ) ## inverse # MTMMTinv<-t(M)%*%MMTinv # M' %*% (M'M)- # # mix.part <- lmebreed(color ~ (1|id) + (1|Rowf) + (1|Colf), # relmat = list(id=MMT), # control = lmerControl( # check.nobs.vs.nlev = "ignore", # check.nobs.vs.rankZ = "ignore", # check.nobs.vs.nRE="ignore" # ), verbose = FALSE, # data=DT) # vc <- VarCorr(mix.part); print(vc,comp=c("Variance")) # mme <- getMME(object=mix.part) # #convert BLUPs to marker effects me=M'(M'M)- u # re <- ranef(mix.part)$id # a.from.g<-MTMMTinv[,rownames(re)]%*%matrix(re[,1],ncol=1) # var.g <- kronecker(MMT[rownames(re),rownames(re)],vc$id) - # mme$Ci[rownames(re),rownames(re) ] # var.a.from.g <- t(M)%*%MMTinv[,rownames(re)]%*% (var.g) %*% t(MMTinv[,rownames(re)])%*%M # se.a.from.g <- sqrt(diag(var.a.from.g)) # t.stat.from.g <- a.from.g/se.a.from.g # t-statistic # pvalGBLUP <- dt(t.stat.from.g,df=n-k-1) # -log10(pval) ``` Now we can look at the p-values coming from the 3 approaches to indeed show that results are equivalent. ```{r} # plot(-log(pvalGBLUP), main="GWAS by GBLUP") ``` ## Literature Giovanny Covarrubias-Pazaran (2024). lme4breeding: enabling genetic evaluation in the age of genomic data. To be submitted to Bioinformatics. Bates Douglas, Maechler Martin, Bolker Ben, Walker Steve. 2015. Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1-48. Bernardo Rex. 2010. Breeding for quantitative traits in plants. Second edition. Stemma Press. 390 pp. Gilmour et al. 1995. Average Information REML: An efficient algorithm for variance parameter estimation in linear mixed models. Biometrics 51(4):1440-1450. Henderson C.R. 1975. Best Linear Unbiased Estimation and Prediction under a Selection Model. Biometrics vol. 31(2):423-447. Kang et al. 2008. Efficient control of population structure in model organism association mapping. Genetics 178:1709-1723. Lee, D.-J., Durban, M., and Eilers, P.H.C. (2013). Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases. Computational Statistics and Data Analysis, 61, 22 - 37. Lee et al. 2015. MTG2: An efficient algorithm for multivariate linear mixed model analysis based on genomic information. Cold Spring Harbor. doi: http://dx.doi.org/10.1101/027201. Maier et al. 2015. Joint analysis of psychiatric disorders increases accuracy of risk prediction for schizophrenia, bipolar disorder, and major depressive disorder. Am J Hum Genet; 96(2):283-294. Rodriguez-Alvarez, Maria Xose, et al. Correcting for spatial heterogeneity in plant breeding experiments with P-splines. Spatial Statistics 23 (2018): 52-71. Searle. 1993. Applying the EM algorithm to calculating ML and REML estimates of variance components. Paper invited for the 1993 American Statistical Association Meeting, San Francisco. Yu et al. 2006. A unified mixed-model method for association mapping that accounts for multiple levels of relatedness. Genetics 38:203-208. Tunnicliffe W. 1989. On the use of marginal likelihood in time series model estimation. JRSS 51(1):15-27.