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.

## Usage

```
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.

## 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% 00:00:01
Evaluating trait HM |============================================| 100% 00:00:03
#> 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 × 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% 00:00:01
#> 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 × 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
# }
```