sommer (version 1.5)

mmer: Mixed Model Equations in R

Description

This function is the core of the package and solves the mixed model equations proposed by Henderson (1975). It has been implemented to work with incidence matrices and variance covariance matrices for each random effect. In the details we will explain the methods implemented by this function. Currently 3 methods are supported; "EMMA" efficient mixed model association (Kang et al. 2008), "AI" average information (Gilmour et al. 1995; Lee et al. 2015), and "EM" expectation maximization (Searle 1993; Bernardo 2010). The EMMA method is implemented when only one variance component other than the error variance component (Var(e)) is estimated, is based on optimizing the likelihood function (see details). On the other hand when more than one variance component needs to be estimated the "AI" and "EM"" methods should be used. The package provides kernels to estimate additive (A.mat), dominant (D.mat), and epistatic (E.mat) relationship matrices that have been shown to increase prediction accuracy. The package provides flexibility to fit other genetic models such as full and half diallel models as well, see hdm function information to see how to fit those models using sommer. The core algorithm in our mixed model solver is the Direct Average Information proposed by Lee et al. (2015) which surpasses in performance the MME-based Average Information (Gilmour et al. 1995) when dense covariance structures are present (GBLUP and GWAS case). When these matrices are sparse (no covariance structures), the MME-based is more effective and we recommend the user to use lme4 for such cases of NO covariance structures.

Finally, feel free to get in touch with me if you have any questions or suggestion at:

[e-mail]

I'll be glad to help or answer any question. We have spend a valuable amount of time deveoping this package. Please cite us in your publication. Type 'citation("sommer")' to know how to cite it.

Usage

mmer(y, X=NULL, Z=NULL, W=NULL, R=NULL, method="AI", REML=TRUE, 
     iters=40, draw=FALSE, init=NULL, n.PC=0, P3D=TRUE,
     models="additive", ploidy=2, min.MAF = 0.05, silent=FALSE, 
     family=NULL, constraint=TRUE, sherman=FALSE, MTG2=FALSE,
     Fishers=FALSE, gss=TRUE, forced=NULL, full.rank=TRUE)

Arguments

y

a numeric vector for the response variable

X

an incidence matrix for fixed effects related to environmental effects or experimental design. This has to be provided as a matrix, NOT in a list structure.

Z

incidence matrices and var-cov matrices for random effects. This works for ONE OR MORE random effects. THIS NEEDS TO BE PROVIDED AS A 2-LEVEL LIST STRUCTURE. For example: . ETA <- list( A=list(Z=Z1, K=K1), B=list(Z=Z2, K=K2), C=li

W

an incidence matrix for extra fixed effects and only to be used if GWAS is desired and markers will be treated as fixed effects according to Yu et al. (2006) for diploids, and Rosyara et al (2016) for polyploids. Theoretically X and W are both fixed effec

R

a matrix for variance-covariance structures for the residuals, i.e. for longitudinal data. if not passed is assumed an identity matrix. THIS PART STILLS IN DEVELOPMENT, NOT FUNCTIONAL YET, it is plan to be implemented in version 1.6.

method

this refers to the method or algorithm to be used for estimating variance components. The package currently is supported by 3 algorithms; "EMMA" efficient mixed model association (Kang et al. 2008), "AI" average information (Gilmour et al. 1995; Lee et al

REML

a TRUE/FALSE value indicating if restricted maximum likelihood should be used instead of ML. The default is TRUE.

iters

a scalar value indicating how many iterations have to be performed if the EM or AI algorithms are selected. There is no rule of tumb for the number of iterations. The default value is 50 iterations or EM steps, but usually will take less than that stoppin

draw

a TRUE/FALSE value indicating if a plot of updated values for the variance components and the log-likelihood should be drawn or not during the optimization process. The default is FALSE. It's been set to FALSE because is less the computation time when the

init

an vector of initial values for the EM or AI algorithms. If not provided the program uses a starting values the variance(y)/#random.eff which are usually good starting values.

n.PC

when the user performs GWAS this refers to the number of principal components to include as fixed effects for Q + K model. Default is 0 (equals K model).

P3D

when the user performs GWAS, P3D=TRUE means that the variance components are estimated by REML only once, without any markers in the model. When P3D=FALSE, variance components are estimated by REML for each marker separately. The default is the first case

models

The model to be used in GWAS. The default is the additive model which applies for diploids and polyploids but the model can be a vector with all possible models, i.e. "additive","1-dom-alt","1-dom-ref","2-dom-alt","2-dom-ref" models are supported for poly

ploidy

A numeric value indicating the ploidy level of the organism. The default is 2 which means diploid but higher ploidy levels are supported. This should only be modified if you are performing GWAS in polyploids.

min.MAF

when the user performs GWAS min.MAF is a scalar value between 0-1 indicating what is theminor allele frequency to be allowed for a marker during a GWAS analysis when providing the matrix of markers W. In general is known that results for markers with alle

silent

a TRUE/FALSE value indicating if the function should draw the progress bar and poems (see poe function) while working or should not be displayed. The default is FALSE, which means is not silent and will display

family

a family object to specify the distribution of the response variable. The program will only use the link function to transform the response. For details see family help page. The argument would look somethin

constraint

a TRUE/FALSE value indicating if the program should use the boundary constraint when one or more variance component is close to the zero boundary. The default is TRUE but needs to be used carefully. It works ideally when few variance components are close

sherman

a TRUE/FALSE value indicating if Sherman-Morrison-Woodbury formula (Seber, 2003, p. 467) should be used when estimating variance components. This will perform faster when a mixed model with no covariance structures is fitted (only AI algorithm). The defau

MTG2

a TRUE/FALSE value indicating if an eigen decomposition for the additive relationship matrix should be performed or not. This is based on Lee (2015). The limitations of this method are: 1) can only be applied to one relationship matrix 2) The

Fishers

a TRUE/FALSE value indicating if the program should calculate at the final step and return the inverse of the Fishers Information Matrix.

gss

a TRUE/FALSE value indicating if a genomic selection is being fitted just for using certain constraints. When is FALSE the program can make some EM steps to find initial values for variance components when the starting values are to far from the real valu

forced

a vector of numeric values for variance components including error if the user wants to force the values of the variance components. On the meantime only works for forcing all of them and not a subset of them. The default is NULL, meaning that variance co

full.rank

a TRUE/FALSE value indicating if the program should investigate X'X to be full rank to avoid problems when solving the linear system. By default this is TRUE which will display a message in the console to le the user know if the X is full rank or not. and

Value

• If all parameters are correctly indicated the program will return a list with the following information: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],In addition, we have included a couple of random poems from Latin American writers to help the scientist (an me haha) remember from time to time that life is more than analyzing data. You can always silence this feature by setting the argument silent=TRUE, which will avoid the program to display the poems. If you want to contribute with a poem, phrase or short citation for future versions of sommer, feel free to send it to me to:

  [e-mail]

  Please share your ideas and code, future generations of scientists can be better if we are not greedy sharing our knowledge. Feel free to use my code for your own software! good luck with your analysis.

Details

The package has been developed to provide R users with code to understand how most common algorithms in mixed model analysis work related to genetics field, but also allowing to perform their real analysis. This package allows the user to calculate the variance components for a mixed model with the advantage of specifying the variance-covariance structure of the random effects. This program focuses in the mixed model of the form:

.

............................. y = Xb + Zu + e ........ with distributions:

.

y ~ N[Xb, var(Zu+e)] ......where;

.

b ~ N[b.hat, 0] ............zero variance because is a fixed term

u ~ N[0, G] ....... where G is equal to:

.

|K1*sigma2(u1)......................0...........................0.........| |.............0.............K2*sigma2(u2).......................0.........| = G

|......................................................................................|

|.............0....................0.........................Ki*sigma2(ui)...|

.

for the i.th random effects, allowing the user to specify the variance covariance structures in the K matrices and

.

e ~ N[0, R] .....................where: I*sigma(e) = R

.

also Var(y) = Var(Zu+e) = ZGZ+R = V which is the phenotypic variance

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The functions in the sommer packages allow the user to specify the incidence matrices with their respective variance-covariance matrix in a 2 level list structure. For example imagine a mixed model with three random effects:

.

fixed = only intercept...................................b ~ N[b.hat, 0]

random = GCA1 + GCA2 + SCA.................u ~ N[0, G]

.

then G takes the form:

.

|K1*sigma2(gca1).....................0..........................0.........| |.............0.............K2*sigma2(gca2).....................0.........| = G

|.............0....................0......................K3*sigma2(sca)...|

.

This mixed model would be specified in the mmer function as:

.

X1 <- matrix(1,length(y),1) incidence matrix for intercept only

ETA <- list(gca1=list(Z=Z1, K=K1), gca2=list(Z=Z2, K=K2), sca=list(Z=Z3, K=K3)) for 3 random effects

.

where Z1, Z2, Z3 are incidence matrices for GCA1, GCA2, SCA respectively created using the model.matrix function and K1, K2, K3 are their var-cov matrices. Now the fitted model will be typed as:

.

ans <- mmer(y=y, X=X1, Z=ETA)

or

ans <- mmer2(y~1, random= ~ gca1 + gca2 + sca, G=list(gca1=K1, gca2=K2, sca=K3), data=yourdata)

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FOR DETAILS ON HOW THE "AI", EM" AND "EMMA" ALGORITHMS WORK PLEASE REFER TO AI , EM AND EMMA

In addition, the package contains a very nice function to plot genetic maps with numeric variable or traits next to the LGs, see the map.plot2 function to see how easy can be done. The package contains other functions:

transp function transform a vector of colors in transparent colors.

fdr calculates the false discovery rate for a vector of p-values.

A.mat is a wrapper of the A.mat function from the rrBLUP package.

D.mat calculates the dominant relationship matrix.

E.mat calculates de epistatic relationship matrix.

score.calc is a function that can be used to calculate a -log10 p-value for a vector of BLUEs for marker effects.

Other functions such as summary, fitted, randef (notice sommer uses randef not ranef), anova, residuals, coef and plot applicable to typical linear models can also be applied to models fitted using this function which is the core of the sommer package.

References

Covarrubias-Pazaran G (2016) Genome assisted prediction of quantitative traits using the R package sommer. https://cran.rstudio.com/web/packages/sommer/

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.

Kang et al. 2008. Efficient control of population structure in model organism association mapping. Genetics 178:1709-1723.

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.

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.

Examples

####=========================================####
#### For CRAN time limitations most lines in the 
#### examples are silenced with one '#' mark, 
#### remove them and run the examples
####=========================================####

####=========================================####
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#### EXAMPLE 1
#### breeding values with 1 variance component
####=========================================####
####=========================================####

####=========================================####
#### simulate genotypic data
#### random population of 200 lines with 1000 markers
####=========================================####
M <- matrix(rep(0,200*1000),1000,200)
for (i in 1:200) {
  M[,i] <- ifelse(runif(1000)<0.5,-1,1)
}
####=========================================####
#### simulate phenotypes
####=========================================####
QTL <- 100*(1:5) #pick 5 QTL
u <- rep(0,1000) #marker effects
u[QTL] <- 1
g <- as.vector(crossprod(M,u))
h2 <- 0.5
y <- g + rnorm(200,mean=0,sd=sqrt((1-h2)/h2*var(g)))
M <- t(M)
####=========================================####
#### fit the model
####=========================================####
Z1 <- diag(length(y))
ETA <- list( list(Z=Z1, K=A.mat(M)))
ans <- mmer(y=y, Z=ETA, method="EMMA")
summary(ans)

####=========================================####
#### run the same but as GWAS 
#### just add the marker matrix in the argument W
#### markers are fixed effects
####=========================================####

#ans <- mmer(y=y, Z=ETA, W=M, method="EMMA")
#summary(ans)
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####=========================================####
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#### EXAMPLE 2
#### breeding values with 3 variance components
#### Hybrid prediction
####=========================================####
####=========================================####
data(cornHybrid)
hybrid2 <- cornHybrid$hybrid # extract cross data
A <- cornHybrid$K
y <- hybrid2$Yield
X1 <- model.matrix(~ Location, data = hybrid2);dim(X1)
Z1 <- model.matrix(~ GCA1 -1, data = hybrid2);dim(Z1)
Z2 <- model.matrix(~ GCA2 -1, data = hybrid2);dim(Z2)
Z3 <- model.matrix(~ SCA -1, data = hybrid2);dim(Z3)

####=========================================####
#### Realized IBS relationships for set of parents 1
####=========================================####
#K1 <- A[levels(hybrid2$GCA1), levels(hybrid2$GCA1)]; dim(K1) 
####=========================================####
#### Realized IBS relationships for set of parents 2
####=========================================####
#K2 <- A[levels(hybrid2$GCA2), levels(hybrid2$GCA2)]; dim(K2)
####=========================================####
#### Realized IBS relationships for cross 
#### (as the Kronecker product of K1 and K2)
####=========================================####
#S <- kronecker(K1, K2) ; dim(S)   
#rownames(S) <- colnames(S) <- levels(hybrid2$SCA)

#ETA <- list(list(Z=Z1, K=K1), list(Z=Z2, K=K2), list(Z=Z3, K=S))
#ans <- mmer(y=y, X=X1, Z=ETA)
#ans$var.comp
#summary(ans)

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####=========================================####
####=========================================####
#### EXAMPLE 3
#### COMPARE WITH MCMCglmm
####=========================================####
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####=========================================####
#### the same model run in MCMCglmm:
####=========================================####
#library(MCMCglmm)
# pro <- list(GCA1 = as(solve(K1), "sparseMatrix"), GCA2 = as(solve(K2),
#      + "sparseMatrix"), SCA = as(solve(S), "sparseMatrix") )
#system.time(mox <- MCMCglmm(Yield ~ Location, random = ~ GCA1 + GCA2 + SCA,
#      + data = hybrid2, verbose = T, ginverse=pro))
## Takes 7:13 minutes in MCMCglmm, in sommer only takes 7 seconds

####=========================================####
#### it is also possible to do GWAS for hybrids, separatting 
#### and accounting for effects of GCA1, GCA2, SCA
####=========================================####

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#### EXAMPLE 4
#### COMPARE WITH cpgen
####=========================================####
####=========================================####

#Z_list = list(Z1,Z2,Z3)
#G_list = list(solve(K1), solve(K2), solve(S))
#fit <- clmm(y = y, Z = Z_list, ginverse=G_list, niter=15000, burnin=5000)
####=========================================####
#### inspect results and notice that variance 
#### components were NOT estimated correctly!!
#### also takes longer and no user-friendly 
####=========================================####
#str(fit)

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####=========================================####
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#### EXAMPLE 5
#### COMPARE WITH pedigreemm example
####=========================================####
####=========================================####

#library(pedigreemm)
#A <- as.matrix(getA(pedCowsR))
#y <- milk$milk
#Z1 <- model.matrix(~id-1, data=milk); dim(Z1)
#vv <- match(unique(milk$id), gsub("id","",colnames(Z1)))
#K1<- A[vv,vv]; dim(K1) 
#Z2 <- model.matrix(~as.factor(herd)-1, data=milk); dim(Z2)
#ETA<- list(list(Z=Z1, K=K1),list(Z=Z2))
#fm3 <- mmer(y=y, Z=ETA) 
####=========================================####
##### or using mmer2 would look:
####=========================================####
#fm3 <- mmer2(fixed=milk ~ 1, random = ~ id + herd, 
#             G=list(id=K1), data=milk)
#summary(fm3)
####=========================================####
#### Try pedigreemm but takes longer, 
#### is an extension of lme4
####=========================================####
#fm2 <- pedigreemm(milk ~ (1 | id) + (1 | herd),data = milk, pedigree = list(id= pedCowsR))
#plot(fm3$u.hat[[1]], ranef(fm2)$id[,1])
#plot(fm3$u.hat[[2]], ranef(fm2)$herd[,1])
####=========================================####
#### a big data frame with 3397 rows and 1359 animals analyzed
#### pedigreemm takes 4 min, sommer takes 1 minute
####=========================================####

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####=========================================####
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#### EXAMPLE 6
#### PREDICTING SPECIFIC PERFORMANCE 
#### within biparental population    
####=========================================####
####=========================================####

#data(CPdata)
#CPpheno <- CPdata$pheno
#CPgeno <- CPdata$geno
## look at the data
#head(CPpheno)
#CPgeno[1:5,1:5]
####=========================================####
#### fit a model including additive and dominance effects
####=========================================####
#y <- CPpheno$color
#Za <- diag(length(y))
#Zd <- diag(length(y))
#A <- A.mat(CPgeno)
#D <- D.mat(CPgeno)

#y.trn <- y # for prediction accuracy
#ww <- sample(c(1:dim(Za)[1]),72) # delete data for 1/5 of the population
#y.trn[ww] <- NA

####================####
#### ADDITIVE MODEL ####
####================####
#ETA.A <- list(list(Z=Za,K=A))
#ans.A <- mmer(y=y.trn, Z=ETA.A)
#cor(ans.A$fitted.y[ww], y[ww], use="pairwise.complete.obs")
####=========================####
#### ADDITIVE-DOMINANT MODEL ####
####=========================####
#ETA.AD <- list(list(Z=Za,K=A),list(Z=Zd,K=D))
#ans.AD <- mmer(y=y.trn, Z=ETA.AD)
#cor(ans.AD$fitted.y[ww], y[ww], use="pairwise.complete.obs")
### greater accuracy !!!! 4 percent increment!!
### we run 100 iterations, 4 percent increment in general
####===================================####
#### ADDITIVE-DOMINANT-EPISTATIC MODEL ####
####===================================####
#ETA.ADE <- list(list(Z=Za,K=A),list(Z=Zd,K=D),list(Z=Ze,K=E))
#ans.ADE <- mmer(y=y.trn, Z=ETA.ADE)
#cor(ans.ADE$fitted.y[ww], y[ww], use="pairwise.complete.obs")
#### adding more effects doesn't necessarily increase prediction accuracy!