Compute the Weighted Average of Absolute Scores for AMMI analysis (Olivoto et al., 2019).

This function compute the weighted average of absolute scores, estimated as follows: $WAAS_i = \sum_{k = 1}^{p} |IPCA_{ik} \times EP_k|/ \sum_{k = 1}^{p}EP_k$

where $$WAAS_i$$ is the weighted average of absolute scores of the ith genotype; $$IPCA_{ik}$$ is the score of the ith genotype in the kth IPCA; and $$EP_k$$ is the explained variance of the kth IPCA for k = 1,2,..,p, considering p the number of significant PCAs, or a declared number of PCAs. For example if prob = 0.05, all axis that are significant considering this probability level are used. The number of axis can be also informed by declaring naxis = x. This will override the number of significant axes according to the argument codeprob.

## Usage

waas(
.data,
env,
gen,
rep,
resp,
block = NULL,
mresp = NULL,
wresp = NULL,
prob = 0.05,
naxis = NULL,
ind_anova = FALSE,
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 a resolvable alpha-lattice design (Patterson and Williams, 1976) is employed. All effects, except the error, are assumed to be fixed.

mresp

The new maximum value after rescaling the response variable. By default, all variables in resp are rescaled so that de maximum value is 100 and the minimum value is 0 (i.e., mresp = NULL). It must be a character vector of the same length of resp if rescaling is assumed to be different across variables, e.g., if for the first variable smaller values are better and for the second one, higher values are better, then mresp = c("l, h") must be used. Character value of length 1 will be recycled with a warning message.

wresp

The weight for the response variable(s) for computing the WAASBY index. By default, all variables in resp have equal weights for mean performance and stability (i.e., wresp = 50). It must be a numeric vector of the same length of resp to assign different weights across variables, e.g., if for the first variable equal weights for mean performance and stability are assumed and for the second one, a higher weight for mean performance (e.g. 65) is assumed, then wresp = c(50, 65) must be used. Numeric value of length 1 will be recycled with a warning message.

prob

The p-value for considering an interaction principal component axis significant.

naxis

The number of IPCAs to be used for computing the WAAS index. Default is NULL (Significant IPCAs are used). If values are informed, the number of IPCAS will be used independently on its significance. Note that if two or more variables are included in resp, then naxis must be a vector.

ind_anova

Logical argument set to FALSE. If TRUE an within-environment ANOVA is performed.

verbose

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

## Value

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

• individual A within-environments ANOVA considering a fixed-effect model.

• model A data frame with the response variable, the scores of all Principal Components, the estimates of Weighted Average of Absolute Scores, and WAASY (the index that consider the weights for stability and productivity in the genotype ranking.

• MeansGxE The means of genotypes in the environments

• PCA Principal Component Analysis.

• ANOVA Joint analysis of variance for the main effects and Principal Component analysis of the interaction effect.

• Details A list summarizing the results. The following information are showed. WgtResponse, the weight for the response variable in estimating WAASB, WgtWAAS the weight for stability, Ngen the number of genotypes, Nenv the number of environments, OVmean the overall mean, Min the minimum observed (returning the genotype and environment), Max the maximum observed, Max the maximum observed, 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.

• augment: Information about each observation in the dataset. This includes predicted values in the fitted column, residuals in the resid column, standardized residuals in the stdres column, the diagonal of the 'hat' matrix in the hat, and standard errors for the fitted values in the se.fit column.

• probint The p-value for the genotype-vs-environment interaction.

## References

Olivoto, T., A.D.C. L\'ucio, J.A.G. da silva, V.S. Marchioro, V.Q. de Souza, and E. Jost. 2019a. 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

waas_means() waasb() get_model_data()

## Author

Tiago Olivoto tiagoolivoto@gmail.com

## Examples

# \donttest{
library(metan)
#===============================================================#
# Example 1: Analyzing all numeric variables considering p-value#
# <= 0.05 to compute the WAAS.                                  #
#===============================================================#
model <- waas(data_ge,
env = ENV,
gen = GEN,
rep = REP,
resp = everything())
#> variable GY
#> ---------------------------------------------------------------------------
#> AMMI analysis table
#> ---------------------------------------------------------------------------
#>     Source  Df  Sum Sq Mean Sq F value   Pr(>F) Proportion Accumulated
#>        ENV  13 279.574 21.5057   62.33 0.00e+00         NA          NA
#>   REP(ENV)  28   9.662  0.3451    3.57 3.59e-08         NA          NA
#>        GEN   9  12.995  1.4439   14.93 2.19e-19         NA          NA
#>    GEN:ENV 117  31.220  0.2668    2.76 1.01e-11         NA          NA
#>        PC1  21  10.749  0.5119    5.29 0.00e+00       34.4        34.4
#>        PC2  19   9.924  0.5223    5.40 0.00e+00       31.8        66.2
#>        PC3  17   4.039  0.2376    2.46 1.40e-03       12.9        79.2
#>        PC4  15   3.074  0.2049    2.12 9.60e-03        9.8        89.0
#>        PC5  13   1.446  0.1113    1.15 3.18e-01        4.6        93.6
#>        PC6  11   0.932  0.0848    0.88 5.61e-01        3.0        96.6
#>        PC7   9   0.567  0.0630    0.65 7.53e-01        1.8        98.4
#>        PC8   7   0.362  0.0518    0.54 8.04e-01        1.2        99.6
#>        PC9   5   0.126  0.0252    0.26 9.34e-01        0.4       100.0
#>  Residuals 252  24.367  0.0967      NA       NA         NA          NA
#>      Total 536 389.036  0.7258      NA       NA         NA          NA
#> ---------------------------------------------------------------------------
#>
#> variable HM
#> ---------------------------------------------------------------------------
#> AMMI analysis table
#> ---------------------------------------------------------------------------
#>     Source  Df  Sum Sq Mean Sq F value   Pr(>F) Proportion Accumulated
#>        ENV  13 5710.32 439.255   57.22 1.11e-16         NA          NA
#>   REP(ENV)  28  214.93   7.676    2.70 2.20e-05         NA          NA
#>        GEN   9  269.81  29.979   10.56 7.41e-14         NA          NA
#>    GEN:ENV 117 1100.73   9.408    3.31 1.06e-15         NA          NA
#>        PC1  21  381.13  18.149    6.39 0.00e+00       34.6        34.6
#>        PC2  19  319.43  16.812    5.92 0.00e+00       29.0        63.6
#>        PC3  17  114.26   6.721    2.37 2.10e-03       10.4        74.0
#>        PC4  15   81.96   5.464    1.92 2.18e-02        7.4        81.5
#>        PC5  13   68.11   5.240    1.84 3.77e-02        6.2        87.7
#>        PC6  11   59.07   5.370    1.89 4.10e-02        5.4        93.0
#>        PC7   9   46.69   5.188    1.83 6.33e-02        4.2        97.3
#>        PC8   7   26.65   3.808    1.34 2.32e-01        2.4        99.7
#>        PC9   5    3.41   0.682    0.24 9.45e-01        0.3       100.0
#>  Residuals 252  715.69   2.840      NA       NA         NA          NA
#>      Total 536 9112.21  17.000      NA       NA         NA          NA
#> ---------------------------------------------------------------------------
#>
#> All variables with significant (p < 0.05) genotype-vs-environment interaction
#> Done!
# Residual plot (first variable)
plot(model)

# Get the WAAS index
get_model_data(model, "WAAS")
#> Class of the model: waas
#> Variable extracted: WAAS
#> # A tibble: 10 × 3
#>    GEN      GY    HM
#>    <chr> <dbl> <dbl>
#>  1 G1    0.151 0.455
#>  2 G10   0.652 1.33
#>  3 G2    0.283 0.985
#>  4 G3    0.106 0.418
#>  5 G4    0.326 0.752
#>  6 G5    0.270 1.16
#>  7 G6    0.233 0.478
#>  8 G7    0.428 0.766
#>  9 G8    0.327 0.541
#> 10 G9    0.507 0.701

# Plot WAAS and response variable
plot_scores(model, type = 3)

#===============================================================#
# Example 2: Declaring the number of axis to be used for        #
# computing WAAS and assigning a larger weight for the response #
# variable when computing the WAASBY index.                     #
#===============================================================#

model2 <- waas(data_ge,
env = ENV,
gen = GEN,
rep = REP,
resp = everything(),
naxis = 1, # Only to compare with PC1
wresp = 60)
#> Warning: Invalid length in 'naxis'. Setting naxis = 1 to all the 2 variables.
#> Warning: Invalid length in 'wresp'. Setting wresp = 60 to all the 2 variables.
#> variable GY
#> ---------------------------------------------------------------------------
#> AMMI analysis table
#> ---------------------------------------------------------------------------
#>     Source  Df  Sum Sq Mean Sq F value   Pr(>F) Proportion Accumulated
#>        ENV  13 279.574 21.5057   62.33 0.00e+00         NA          NA
#>   REP(ENV)  28   9.662  0.3451    3.57 3.59e-08         NA          NA
#>        GEN   9  12.995  1.4439   14.93 2.19e-19         NA          NA
#>    GEN:ENV 117  31.220  0.2668    2.76 1.01e-11         NA          NA
#>        PC1  21  10.749  0.5119    5.29 0.00e+00       34.4        34.4
#>        PC2  19   9.924  0.5223    5.40 0.00e+00       31.8        66.2
#>        PC3  17   4.039  0.2376    2.46 1.40e-03       12.9        79.2
#>        PC4  15   3.074  0.2049    2.12 9.60e-03        9.8        89.0
#>        PC5  13   1.446  0.1113    1.15 3.18e-01        4.6        93.6
#>        PC6  11   0.932  0.0848    0.88 5.61e-01        3.0        96.6
#>        PC7   9   0.567  0.0630    0.65 7.53e-01        1.8        98.4
#>        PC8   7   0.362  0.0518    0.54 8.04e-01        1.2        99.6
#>        PC9   5   0.126  0.0252    0.26 9.34e-01        0.4       100.0
#>  Residuals 252  24.367  0.0967      NA       NA         NA          NA
#>      Total 536 389.036  0.7258      NA       NA         NA          NA
#> ---------------------------------------------------------------------------
#>
#> variable HM
#> ---------------------------------------------------------------------------
#> AMMI analysis table
#> ---------------------------------------------------------------------------
#>     Source  Df  Sum Sq Mean Sq F value   Pr(>F) Proportion Accumulated
#>        ENV  13 5710.32 439.255   57.22 1.11e-16         NA          NA
#>   REP(ENV)  28  214.93   7.676    2.70 2.20e-05         NA          NA
#>        GEN   9  269.81  29.979   10.56 7.41e-14         NA          NA
#>    GEN:ENV 117 1100.73   9.408    3.31 1.06e-15         NA          NA
#>        PC1  21  381.13  18.149    6.39 0.00e+00       34.6        34.6
#>        PC2  19  319.43  16.812    5.92 0.00e+00       29.0        63.6
#>        PC3  17  114.26   6.721    2.37 2.10e-03       10.4        74.0
#>        PC4  15   81.96   5.464    1.92 2.18e-02        7.4        81.5
#>        PC5  13   68.11   5.240    1.84 3.77e-02        6.2        87.7
#>        PC6  11   59.07   5.370    1.89 4.10e-02        5.4        93.0
#>        PC7   9   46.69   5.188    1.83 6.33e-02        4.2        97.3
#>        PC8   7   26.65   3.808    1.34 2.32e-01        2.4        99.7
#>        PC9   5    3.41   0.682    0.24 9.45e-01        0.3       100.0
#>  Residuals 252  715.69   2.840      NA       NA         NA          NA
#>      Total 536 9112.21  17.000      NA       NA         NA          NA
#> ---------------------------------------------------------------------------
#>
#> All variables with significant (p < 0.05) genotype-vs-environment interaction
#> Done!
# Get the WAAS index (it will be |PC1|)
get_model_data(model2)
#> Class of the model: waas
#> Variable extracted: WAAS
#> # A tibble: 10 × 3
#>    GEN       GY     HM
#>    <chr>  <dbl>  <dbl>
#>  1 G1    0.317  0.280
#>  2 G10   1.00   1.78
#>  3 G2    0.139  1.56
#>  4 G3    0.0434 0.342
#>  5 G4    0.325  0.202
#>  6 G5    0.326  1.58
#>  7 G6    0.0984 0.547
#>  8 G7    0.285  1.22
#>  9 G8    0.499  0.0418
#> 10 G9    0.467  1.07

# Get values for IPCA1
get_model_data(model2, "PC1")
#> Class of the model: waas
#> Variable extracted: PC1
#> # A tibble: 10 × 3
#>    GEN        GY      HM
#>    <chr>   <dbl>   <dbl>
#>  1 G1     0.317   0.280
#>  2 G10   -1.00   -1.78
#>  3 G2     0.139   1.56
#>  4 G3     0.0434  0.342
#>  5 G4    -0.325  -0.202
#>  6 G5    -0.326   1.58
#>  7 G6    -0.0984  0.547
#>  8 G7     0.285  -1.22
#>  9 G8     0.499  -0.0418
#> 10 G9     0.467  -1.07

#===============================================================#
# Example 3: Analyzing GY and HM assuming a random-effect model.#
# Smaller values for HM and higher values for GY are better.    #
# To estimate WAASBY, higher weight for the GY (60%) and lower  #
# weight for HM (40%) are considered for mean performance.      #
#===============================================================#

model3 <- waas(data_ge,
env = ENV,
gen = GEN,
rep = REP,
resp = c(GY, HM),
mresp = c("h, l"),
wresp = c(60, 40))
#> variable GY
#> ---------------------------------------------------------------------------
#> AMMI analysis table
#> ---------------------------------------------------------------------------
#>     Source  Df  Sum Sq Mean Sq F value   Pr(>F) Proportion Accumulated
#>        ENV  13 279.574 21.5057   62.33 0.00e+00         NA          NA
#>   REP(ENV)  28   9.662  0.3451    3.57 3.59e-08         NA          NA
#>        GEN   9  12.995  1.4439   14.93 2.19e-19         NA          NA
#>    GEN:ENV 117  31.220  0.2668    2.76 1.01e-11         NA          NA
#>        PC1  21  10.749  0.5119    5.29 0.00e+00       34.4        34.4
#>        PC2  19   9.924  0.5223    5.40 0.00e+00       31.8        66.2
#>        PC3  17   4.039  0.2376    2.46 1.40e-03       12.9        79.2
#>        PC4  15   3.074  0.2049    2.12 9.60e-03        9.8        89.0
#>        PC5  13   1.446  0.1113    1.15 3.18e-01        4.6        93.6
#>        PC6  11   0.932  0.0848    0.88 5.61e-01        3.0        96.6
#>        PC7   9   0.567  0.0630    0.65 7.53e-01        1.8        98.4
#>        PC8   7   0.362  0.0518    0.54 8.04e-01        1.2        99.6
#>        PC9   5   0.126  0.0252    0.26 9.34e-01        0.4       100.0
#>  Residuals 252  24.367  0.0967      NA       NA         NA          NA
#>      Total 536 389.036  0.7258      NA       NA         NA          NA
#> ---------------------------------------------------------------------------
#>
#> variable HM
#> ---------------------------------------------------------------------------
#> AMMI analysis table
#> ---------------------------------------------------------------------------
#>     Source  Df  Sum Sq Mean Sq F value   Pr(>F) Proportion Accumulated
#>        ENV  13 5710.32 439.255   57.22 1.11e-16         NA          NA
#>   REP(ENV)  28  214.93   7.676    2.70 2.20e-05         NA          NA
#>        GEN   9  269.81  29.979   10.56 7.41e-14         NA          NA
#>    GEN:ENV 117 1100.73   9.408    3.31 1.06e-15         NA          NA
#>        PC1  21  381.13  18.149    6.39 0.00e+00       34.6        34.6
#>        PC2  19  319.43  16.812    5.92 0.00e+00       29.0        63.6
#>        PC3  17  114.26   6.721    2.37 2.10e-03       10.4        74.0
#>        PC4  15   81.96   5.464    1.92 2.18e-02        7.4        81.5
#>        PC5  13   68.11   5.240    1.84 3.77e-02        6.2        87.7
#>        PC6  11   59.07   5.370    1.89 4.10e-02        5.4        93.0
#>        PC7   9   46.69   5.188    1.83 6.33e-02        4.2        97.3
#>        PC8   7   26.65   3.808    1.34 2.32e-01        2.4        99.7
#>        PC9   5    3.41   0.682    0.24 9.45e-01        0.3       100.0
#>  Residuals 252  715.69   2.840      NA       NA         NA          NA
#>      Total 536 9112.21  17.000      NA       NA         NA          NA
#> ---------------------------------------------------------------------------
#>
#> All variables with significant (p < 0.05) genotype-vs-environment interaction
#> Done!

# Get the ranks for the WAASY index
get_model_data(model3, what = "OrWAASY")
#> Class of the model: waas
#> Variable extracted: OrWAASY
#> # A tibble: 10 × 3
#>    GEN      GY    HM
#>    <chr> <dbl> <dbl>
#>  1 G1        4     1
#>  2 G10      10     9
#>  3 G2        3     4
#>  4 G3        1     2
#>  5 G4        6     7
#>  6 G5        8    10
#>  7 G6        7     3
#>  8 G7        5     6
#>  9 G8        2     8
#> 10 G9        9     5
# }