Description Usage Arguments Details Value Author(s) Examples

This function was developed to solve a mixel model for multi-environmental trials and/or replicated trials when genomic is available. The model includes a semi-parametric term to account for spatial variation when the field layout of the experiment is know. Covariates (fixed effects), genetics (random effect) and spatial term are fitted all in a single step.

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`y` |
Numeric vector of phenotypes of length |

`gen` |
Numeric matrix, with dimension |

`dta` |
Data frame with |

`it` |
Integer. Total numeric of MCMC iterations used to fit the model. |

`bi` |
Integer. Burn-in of MCMC iterations, i.e., number of iteration to be discarted prior to model convergence. |

`th` |
Integer. Thinning parameter: saves only 1 interation every |

`model` |
Prediction model: The options are: |

`...` |
Pass arguments to the function that builds the spatial splines |

The general model is *y=Xb+Zu+f(x)+e*, where *y* is the response variable, *Xb* refers to the fixed effects, *Zu* regards the genetic effect, *f(x)* represents the field variation, and *e* is the vector of residuals. In this model *u* is a link function that represents the genetic component, which depends on the model specified.

For whole-genome regression models (BRR or BayesA), *u = Ma*, where *M* is the matrix of genotypes. For kernel models (RKHS and GBLUP), *u=N(0,Kσ2a)*, where K is either a Gaussian kernel (RKHS) or a linear kernel (GBLUP). To avoid over-representation of genotypes, *u* is not weighted according to the number of observations of each genotype.

Unobserved genotypes not provided in `dta`

but provided in `gen`

are predicted in the output of the function. Genotypes without genotypic information are transfered to the fixed effect (eg. checks). Missing loci are imputed with the expectation. If `dta`

is not provided, the function will work as a regular genomic prediction model, so the length of `y`

must match the number of rows of `gen`

.

In whole-genome regression models, the regularization of the genetic term is either based on chosen prior (t, Gaussian), Gaussian (from ridge regression) and t (from BayesA). Kernel models (GBLUP and RKHS) are regularized as Gaussian process, which is similar to a ridge regression of Eigenvectors where the regularization of Eigenpairs also relies on the Eigenvalues.

If there is a large number of trials and users acknowledge the necessity of sparse matrices, we recommend installing the Matrix package and run the following code that enables sparsity:

`source(system.file("add","sparseGMM.R",package="NAM"))`

The function gmm returns a list containing the fitted values (`hat`

), observed values (`obs`

), intercept (`mu`

, incidence matrix of genotypes (`Z`

) and the vector of breeding values (`EBV`

). If fixed effects are provided, it also returns the design matrices and coefficients of fixed effects (`X`

,`b`

). If the model was kernel or regression, output will include
the random effect coefficients (`g`

), variance components of markers (`Vg`

) and residuals (`Ve`

). Kernel models regress the Eigenvectors of the kernel, weighted by the Eigenvalue. The coefficient (`cxx`

) used in the `BRR`

model to convert marker variance *Vb* into genetic variance *Va*. If spatial information is provided, the output includes the fitted spatial term (`sp`

).

Alencar Xavier

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