BGLR (version 1.1.3)

Multitrait: Multi trait models

Description

The Multitrait function fits Bayesian multitrait models with arbitrary number of random effects ussing a Gibbs sampler. The data equation is as follows:

y_j = 1μ_j + X_Fj β_Fj+X₁β_j1 + ... +X_kβ_jk + u_j1+ ... + u_jq + e_j,

where:

• y_j is a n-dimensional response vector of phenotypes (NAs allowed) with y_ij representing the phenotypic record of the i-th subject for the j-th trait.

• μ_j is an intercept.

• X_Fj is a matrix of fixed effects.

• β_Fj is a vector of fixed effects.

• X_s is an incidence matrix for predictors that are common for all individuals (e.g. markers), s=1,...,k.

• β_js is a vector of regression coefficients, s=1,..,k. Different priors can be assigned to these regression coefficients (spike-slab and Gaussian) and regression coefficients are correlated across traits.

• u_jr is a n-dimensional vector of random effects.

• e_j is a n-dimensional vector of residuals.

Usage

Multitrait(y, ETA, intercept=TRUE,resCov = list(df0=5,S0=NULL,type="UN"), 
   R2=0.5, nIter=1000,burnIn=500,thin=10, saveAt="",verbose=TRUE)

Value

List containing estimated posterior means and estimated posterior standard deviations.

Arguments

y

a matrix of dimension \(n \times t\), where \(n\) is the number of individials in each trait, \(t\) is the number of traits, NAs are allowed.

ETA

(list) This is a two-level list used to specify the regression function (or linear predictor). Regression on covariates and other types of random effects are specified in this two-level list. For instance:

ETA=list(list(X=W, model="FIXED"), 
      list(X=Z,model="BRR"), 
      list(K=G,model="RKHS"))

specifies that the linear predictor should include: a linear regression on W with regression coefficients treated as fixed effects (i.e., flat prior), plus regression on Z, with regression coefficients modeled as in the Ridge Regression plus and a random effect with co-variance structure G.

intercept

logical, if TRUE (default) an intercept is included.

nIter,burnIn, thin

(integer) the number of iterations, burn-in and thinning.

resCov

A list used to define the co-variance matrix for model residuals (R). Four covariance strucures are supported: i) Unstructured ("UN"), ii)Diagonal ("DIAG"), iii)Factor Analytic ("FA") and iv)Recursive ("REC"), for example:

resCov=list(type="UN", df0=4, S0=V)

specifies an UN-structured covariance matrix, with an Inverse Whishart prior with degree of freedom df0 (scalar) and scale matrix (t x t) V.

saveAt

(string) this may include a path and a pre-fix that will be added to the name of the files that are saved as the program runs.

R2

(numeric, 0<R2<1) The proportion of co-variance that one expects, a priori, to be explained by the regression. Only used if the hyper-parameters are not specified; if that is the case, internaly, hyper-paramters are set so that the prior modes are consistent with the co-variance partition specified by R2 and the prior distribution is relatively flat at the mode.

verbose

(logical) if TRUE the iteration history is printed, default TRUE.

Author

Gustavo de los Campos, Paulino Perez-Rodriguez

Details

Conditional distribution of the data

Model residuals are assumed to follow a multivariate normal distribution, with null mean and covariance matrix Cov((e'₁,...,e'_n)')=R₀ ⊗ I where R₀ is a t x t (within-subject) covariance matrix of model residuals and n-dimensional identity matrix. Therefore:

p(y_i1,...,y_it | θ )=MN(η_i, R₀)

where MN(.,.), denotes the multivariate normal distribution with mean η_i and covariance matrix R₀; here η_i is a t-dimensional vector whose entries are the expected values of the response variable for the i-th individual.

Prior distribution

The prior distribution is structured hierarchically. The first level of the prior specifies the distribution of the fixed and random effects given the codispersion parameters (the covariance matrices of the random effects, see below). The priors for the codispersion parameters are specified in a deeper level of the hierarchy.

The intercepts and vectors of fixed effects are assigned flat prior (each unknown is assigned a Gaussian prior with null mean and very large variance).

The vectors of random effects u_r are assigned independent multivariate normal priors with null mean and covariance matrices Cov(u_r)=G_r⊗K_r, u_r represent the vector of effects for the r-th random effects (sorted by subject first and trait within subject), G_r is an t x t (within-subject) covariance matrix of the r-th random effect and K_r is a user defined (between subjects) covariance matrix for the r-th random effect, for instance, may be a pedigree or marker-based relationship matrix. The covariance matrix of random effects are assigned Inverse Wishart priors (for the case of unstructured options) or priors that are structured according to some model (diagonal, factor analytic or recursive).

The vector or regression coefficients β_s are assigned Gaussian and Spike Slab priors whose covariance matrixes depend on a Ω_s covariance matrix of dimmensions t x t (within subject). The covariance matrix of random effects are assigned Inverse Wishart priors (for the case of unstructured options) or priors that are structured according to some model (diagonal, factor analytic or recursive).

Algorithm

Internally, samples from all the model unknowns are drawn using a Gibbs sampler (i.e., based on fully conditional distributions).

References

Burgueno, J., G. de los Campos, K. Weigel and J. Crossa. 2012. Genomic Prediction of Breeding Values when Modelling Genotype x Environment Interaction using Pedigree and Dense Molecular Markers. Crop Sci., 52(2):707-719.

de los Campos, G. and D. Gianola. 2007. Factor analysis models for structuring covariance matrices of additive genetic effects: a Bayesian implementation. Genet. Sel. Evol., 39:481-494.

Cheng, H., K. Kadir, J., Zeng, D. Garrick and R. Fernando. 2018. Genomic Prediction from Multiple-Trait Bayesian Regression Methods Using Mixture Priors. Genetics, 209(1): 89-103.

Sorensen, D. and D. Gianola. 2002. Likelihood, Bayesian, and MCMC methods in quantitative genetics. Springer-Verlag, New York.

Examples

if (FALSE) {
	
library(BGLR)
data(wheat)
X<-wheat.X
K<-wheat.A
y<-wheat.Y

#Example 1, Spike Slab regression
ETA1<-list(list(X=X,model="SpikeSlab"))

fm1<-Multitrait(y=y,ETA=ETA1,nIter=1000,burnIn=500)

#Example 2, Ridge Regression
ETA2<-list(list(X=X,model="BRR"))
fm2<-Multitrait(y=y,ETA=ETA2,nIter=1000,burnIn=500)

#Example 3, Random effects with user defined covariance structure
#for individuals derived from pedigree
ETA3<-list(list(K=K,model="RKHS"))
fm3<-Multitrait(y=y,ETA=ETA3,nIter=1000,burnIn=500)

#Example 4, Markers and pedigree
ETA4<-list(list(X=X,model="BRR"), list(K=K,model="RKHS"))

fm4<-Multitrait(y=y,ETA=ETA4,nIter=1000,burnIn=500)

#Example 5, recursive structures for within subject covariance matrix
M1 <- matrix(nrow = 4, ncol = 4, FALSE)
M1[3, 2] <- M1[4, 2] <- TRUE # Adding recursion from trait 2 onto traits 3 and 4
M1[4, 3] <- TRUE # Adding recursion from trait 3 on trait 4

ETA5<-list(list(K=K,model="RKHS",Cov=list(type="REC",M=M1)))
fm5<-Multitrait(y=y,ETA=ETA5,nIter=1000,burnIn=500)

#Example 6, diagonal residual covariance matrix with the predictor 
#used in example 5
residual1<-list(type="DIAG")
fm6<-Multitrait(y=y,ETA=ETA5,resCov=residual1,nIter=1000,burnIn=500)

}