[Stable]

Genotype analysis in multi-environment trials using mixed-effect or random-effect models.

The nature of the effects in the model is chosen with the argument random. By default, the experimental design considered in each environment is a randomized complete block design. If block is informed, a resolvable alpha-lattice design (Patterson and Williams, 1976) is implemented. The following six models can be fitted depending on the values of random and block arguments.

  • Model 1: block = NULL and random = "gen" (The default option). This model considers a Randomized Complete Block Design in each environment assuming genotype and genotype-environment interaction as random effects. Environments and blocks nested within environments are assumed to fixed factors.

  • Model 2: block = NULL and random = "env". This model considers a Randomized Complete Block Design in each environment treating environment, genotype-environment interaction, and blocks nested within environments as random factors. Genotypes are assumed to be fixed factors.

  • Model 3: block = NULL and random = "all". This model considers a Randomized Complete Block Design in each environment assuming a random-effect model, i.e., all effects (genotypes, environments, genotype-vs-environment interaction and blocks nested within environments) are assumed to be random factors.

  • Model 4: block is not NULL and random = "gen". This model considers an alpha-lattice design in each environment assuming genotype, genotype-environment interaction, and incomplete blocks nested within complete replicates as random to make use of inter-block information (Mohring et al., 2015). Complete replicates nested within environments and environments are assumed to be fixed factors.

  • Model 5: block is not NULL and random = "env". This model considers an alpha-lattice design in each environment assuming genotype as fixed. All other sources of variation (environment, genotype-environment interaction, complete replicates nested within environments, and incomplete blocks nested within replicates) are assumed to be random factors.

  • Model 6: block is not NULL and random = "all". This model considers an alpha-lattice design in each environment assuming all effects, except the intercept, as random factors.

gamem_met(
  .data,
  env,
  gen,
  rep,
  resp,
  block = NULL,
  by = NULL,
  random = "gen",
  prob = 0.05,
  verbose = TRUE
)

Arguments

.data

The dataset containing the columns related to Environments, Genotypes, replication/block and response variable(s).

env

The name of the column that contains the levels of the environments.

gen

The name of the column that contains the levels of the genotypes.

rep

The name of the column that contains the levels of the replications/blocks.

resp

The response variable(s). To analyze multiple variables in a single procedure a vector of variables may be used. For example resp = c(var1, var2, var3).

block

Defaults to NULL. In this case, a randomized complete block design is considered. If block is informed, then an alpha-lattice design is employed considering block as random to make use of inter-block information, whereas the complete replicate effect is always taken as fixed, as no inter-replicate information was to be recovered (Mohring et al., 2015).

by

One variable (factor) to compute the function by. It is a shortcut to dplyr::group_by().This is especially useful, for example, when the researcher want to analyze environments within mega-environments. In this case, an object of class waasb_grouped is returned.

random

The effects of the model assumed to be random. Defaults to random = "gen". See Details to see the random effects assumed depending on the experimental design of the trials.

prob

The probability for estimating confidence interval for BLUP's prediction.

verbose

Logical argument. If verbose = FALSE the code will run silently.

Value

An object of class waasb with the following items for each variable:

  • fixed Test for fixed effects.

  • random Variance components for random effects.

  • LRT The Likelihood Ratio Test for the random effects.

  • BLUPgen The random effects and estimated BLUPS for genotypes (If random = "gen" or random = "all")

  • BLUPenv The random effects and estimated BLUPS for environments, (If random = "env" or random = "all").

  • BLUPint The random effects and estimated BLUPS of all genotypes in all environments.

  • MeansGxE The phenotypic means of genotypes in the environments.

  • modellme The mixed-effect model of class lmerMod.

  • residuals The residuals of the mixed-effect model.

  • model_lm The fixed-effect model of class lm.

  • residuals_lm The residuals of the fixed-effect model.

  • Details A list summarizing the results. The following information are shown: Nenv, the number of environments in the analysis; Ngen the number of genotypes in the analysis; Mean the grand mean; SE the standard error of the mean; SD the standard deviation. CV the coefficient of variation of the phenotypic means, estimating WAASB, Min the minimum value observed (returning the genotype and environment), Max the maximum value observed (returning the genotype and environment); MinENV the environment with the lower mean, MaxENV the environment with the larger mean observed, MinGEN the genotype with the lower mean, MaxGEN the genotype with the larger.

  • ESTIMATES A tibble with the genetic parameters (if random = "gen" or random = "all") with the following columns: Phenotypic variance the phenotypic variance; Heritability the broad-sense heritability; GEr2 the coefficient of determination of the interaction effects; h2mg the heritability on the mean basis; Accuracy the selective accuracy; rge the genotype-environment correlation; CVg the genotypic coefficient of variation; CVr the residual coefficient of variation; CV ratio the ratio between genotypic and residual coefficient of variation.

  • formula The formula used to fit the mixed-model.

References

Olivoto, T., A.D.C. L\'ucio, J.A.G. da silva, V.S. Marchioro, V.Q. de Souza, and E. Jost. 2019. Mean performance and stability in multi-environment trials I: Combining features of AMMI and BLUP techniques. Agron. J. 111:2949-2960. doi: 10.2134/agronj2019.03.0220

Mohring, J., E. Williams, and H.-P. Piepho. 2015. Inter-block information: to recover or not to recover it? TAG. Theor. Appl. Genet. 128:1541-54. doi: 10.1007/s00122-015-2530-0

Patterson, H.D., and E.R. Williams. 1976. A new class of resolvable incomplete block designs. Biometrika 63:83-92.

See also

Author

Tiago Olivoto tiagoolivoto@gmail.com

Examples

# \donttest{
library(metan)
#===============================================================#
# Example 1: Analyzing all numeric variables assuming genotypes #
# as random effects                                             #
#===============================================================#
model <- gamem_met(data_ge,
                  env = ENV,
                  gen = GEN,
                  rep = REP,
                  resp = everything())
#> Evaluating trait GY |======================                      | 50% 00:00:00 
Evaluating trait HM |============================================| 100% 00:00:01 

#> Method: REML/BLUP
#> Random effects: GEN, GEN:ENV
#> Fixed effects: ENV, REP(ENV)
#> Denominador DF: Satterthwaite's method
#> ---------------------------------------------------------------------------
#> P-values for Likelihood Ratio Test of the analyzed traits
#> ---------------------------------------------------------------------------
#>     model       GY       HM
#>  COMPLETE       NA       NA
#>       GEN 1.11e-05 5.07e-03
#>   GEN:ENV 2.15e-11 2.27e-15
#> ---------------------------------------------------------------------------
#> All variables with significant (p < 0.05) genotype-vs-environment interaction
# Distribution of random effects (first variable)
plot(model, type = "re")


# Genetic parameters
get_model_data(model, "genpar")
#> Class of the model: waasb
#> Variable extracted: genpar
#> # A tibble: 9 x 3
#>   Parameters              GY     HM
#>   <chr>                <dbl>  <dbl>
#> 1 Phenotypic variance  0.181 5.52  
#> 2 Heritability         0.154 0.0887
#> 3 GEIr2                0.313 0.397 
#> 4 h2mg                 0.815 0.686 
#> 5 Accuracy             0.903 0.828 
#> 6 rge                  0.370 0.435 
#> 7 CVg                  6.26  1.46  
#> 8 CVr                 11.6   3.50  
#> 9 CV ratio             0.538 0.415 



#===============================================================#
# Example 2: Unbalanced trials                                  #
# assuming all factors as random effects                        #
#===============================================================#
un_data <- data_ge %>%
             remove_rows(1:3) %>%
             droplevels()

model2 <- gamem_met(un_data,
                   env = ENV,
                   gen = GEN,
                   rep = REP,
                   random = "all",
                   resp = GY)
#> Evaluating trait GY |============================================| 100% 00:00:00 

#> Method: REML/BLUP
#> Random effects: GEN, REP(ENV), ENV, GEN:ENV
#> Fixed effects: -
#> Denominador DF: Satterthwaite's method
#> ---------------------------------------------------------------------------
#> P-values for Likelihood Ratio Test of the analyzed traits
#> ---------------------------------------------------------------------------
#>     model       GY
#>  COMPLETE       NA
#>       GEN 1.31e-05
#>  REP(ENV) 9.23e-08
#>       ENV 9.33e-17
#>   GEN:ENV 2.11e-11
#> ---------------------------------------------------------------------------
#> All variables with significant (p < 0.05) genotype-vs-environment interaction
get_model_data(model2)
#> Class of the model: waasb
#> Variable extracted: genpar
#> # A tibble: 9 x 2
#>   Parameters               GY
#>   <chr>                 <dbl>
#> 1 Phenotypic variance  0.907 
#> 2 Heritability         0.0308
#> 3 GEIr2                0.314 
#> 4 h2mg                 0.813 
#> 5 Accuracy             0.902 
#> 6 rge                  0.371 
#> 7 CVg                  6.24  
#> 8 CVr                 11.6   
#> 9 CV ratio             0.536 
# }