# Parametric and non-parametric stability statistics

#### Tiago Olivoto

#### 2021-12-29

Source:`vignettes/vignettes_stability.Rmd`

`vignettes_stability.Rmd`

# Getting started

In this section, we will use the data in `data_ge`

and `data_ge2`

. For more information see `?data_ge`

and `?data_ge2`

, respectively. Other data sets can be used provided that the following columns are in the data_ge: environment, genotype, block/replicate and response variable(s).

See the section Rendering engine to know how HTML tables were generated.

# Genotype-environment performance

The function `ge_plot()`

may be used to visualize the genotype’s performance across the environments.

```
library(metan)
a <- ge_plot(data_ge, ENV, GEN, GY)
b <- ge_plot(data_ge, ENV, GEN, GY) + ggplot2::coord_flip()
arrange_ggplot(a, b, tag_levels = "a")
```

To identify the winner genotype within each environment, we can use the function `ge_winners()`

.

`ge_winners(data_ge2, ENV, GEN, resp = everything()) %>% print_table()`

Or get the genotype ranking within each environment.

```
ge_winners(data_ge2, ENV, GEN, resp = everything(), type = "ranks") %>%
print_table()
```

For more details about the trials, we can use `ge_details()`

```
ge_details(data_ge2, ENV, GEN, resp = everything()) %>%
print_table()
```

# Within-environment analysis of variance

The function `anova_ind()`

can be used to compute an within-environment analysis of variance. Environment with values in blue had a significant (p < 0.05) genotype effect.

```
ind <- anova_ind(data_ge, ENV, GEN, REP, GY)
# Evaluating trait GY |============================================| 100% 00:00:00
print_table(ind$GY$individual)
```

# Genotype-by-environment means

The function `make_mat()`

can be used to produce a two-way table for the genotype-environment means.

# Genotype-by-environment interaction effects

The function `ge_effects()`

is used to compute the genotype-environment effects.

```
ge_ef <- ge_effects(data_ge, ENV, GEN, GY)
print_table(ge_ef$GY)
```

# Genotype plus Genotype-by-environment effects

To obtain the genotype plus genotype-environment effects, we can use the argument `type = "gge"`

in the function `ge_effects()`

.

```
gge_ef <- ge_effects(data_ge, ENV, GEN, GY, type = "gge")
print_table(gge_ef$GY)
```

# Clustering tester locations

The function `ge_cluster()`

computes a cluster analysis for grouping environments based on its similarities using an Euclidean distance based on standardized data. Line means are divided by the phenotypic standard error of the relevant environment after its mean has been subtracted. By standardizing the data each environment will have a mean of zero and unit variance, and the effect of variability in phenotypic variance (as well as the mean) should be reduced (Fox and Rosielle 1982).

```
d1 <- ge_cluster(data_ge, ENV, GEN, GY, nclust = 4)
plot(d1, nclust = 4)
```

The function `env_dissimilarity()`

computes the dissimilarity between test environment using:

The partition of the partition of the mean square of the genotype-environment interaction (MS_GE) into single (S) and complex (C) parts, according to Robertson (1959), where \(S = \frac{1}{2}(\sqrt{Q_1}-\sqrt{Q_2})^2)\) and \(C = (1-r)\sqrt{Q1-Q2}\), being \(r\) the correlation between the genotype’s average in the two environments; and \(Q_1\) and \(Q_2\) the genotype mean square in the environments 1 and 2, respectively

The decomposition of the MS_GE, in which the complex part is given by \(C = \sqrt{(1-r)^3\times Q1\times Q2}\) (Cruz and Castoldi 1991).

The interaction mean square between genotypes and pairs of environments.

The correlation coefficients between genotypes’s average in each pair of environment.

```
mod <- env_dissimilarity(data_ge, ENV, GEN, REP, GY)
# Evaluating trait Y |=============================================| 100% 00:00:00
# Pearson's correlation coefficient
print_table(mod$GY$correlation, rownames = TRUE)
```

To obtain dendrograms based on the above matrix we can use plot(). The dendrograms are based on the hierarchical clustering algorithm UPGMA (Unweighted Pair Group Method using Arithmetic averages).

`plot(mod)`

# Joint regression analysis

Eberhart and Russell (1966) popularized the regression-based stability analysis. In these procedures, the adaptability and stability analysis is performed by means of adjustments of regression equations where the dependent variable is predicted as a function of an environmental index, according to the following model:

\[
\mathop Y\nolimits_{ij} = {\beta _{0i}} + {\beta _{1i}}{I_j} + {\delta _{ij}} + {\bar \varepsilon _{ij}}
\] where \({\beta _{0i}}\) is the grand mean of the genotype *i* (*i* = 1, 2, …, I); \({\beta _{1i}}\) is the linear response (slope) of the genotype *i* to the environmental index; *Ij* is the environmental index (*j* = 1, 2, …, *e*), where \({I_j} = [(y_{.j}/g)- (y_{..}/ge)]\), \({\delta _{ij}}\) is the deviation from the regression, and \({\bar \varepsilon _{ij}}\) is the experimental error. The model is fitted with the function `ge_reg()`

. The S3 methods `plot()`

and `summary()`

may be used to explore the fitted model.

```
reg_model <- ge_reg(data_ge, ENV, GEN, REP, GY)
# Evaluating trait GY |============================================| 100% 00:00:00
print_table(reg_model$GY$anova)
```

# Genotypic confidence index

Annicchiarico (1992) proposed a stability method in which the stability parameter is measured by the superiority of the genotype in relation to the average of each environment, according to the following model:

\[
{Z_{ij}} = \frac{{{Y_{ij}}}}{{{{\bar Y}_{.j}}}} \times 100
\] The genotypic confidence index of the genotype *i* (\(W_i\)) is then estimated as follows:

\[
W_i = Z_{i.}/e - \alpha \times sd(Z_{i.})
\] Where \(\alpha\) is the quantile of the standard normal distribution at a given probability error (\(\alpha \approx 1.64\) at 0.05). The method is implemented using the function `Annicchiarico()`

. The confidence index is estimated considering all environment, favorable environments (positive index) and unfavorable environments (negative index), as follows:

## Computing the index

```
ann5 <- Annicchiarico(data_ge, ENV, GEN, REP, GY)
# Evaluating trait GY |============================================| 100% 00:00:00
ann1 <- Annicchiarico(data_ge, ENV, GEN, REP, GY, prob = 0.01)
# Evaluating trait GY |============================================| 100% 00:00:00
```

# Superiority index

The function `superiority()`

implements the nonparametric method proposed by Lin and Binns (1988), which considers that a measure of cultivar general superiority for cultivar x location data is defined as the distance mean square between the cultivar’s response and the maximum response averaged over all locations, according to the following model.

\[
P_i = \sum\limits_{j = 1}^n{(y_{ij} - y_{.j})^2/(2n)}
\] where *n* is the number of environments

Similar then the genotypic confidence index, the superiority index is calculated by all environments, favorable, and unfavorable environments.

```
super <- superiority(data_ge, ENV, GEN, GY)
# Evaluating trait GY |============================================| 100% 00:00:00
print_table(super$GY$index)
```

# Environmental stratification

A method that combines stability analysis and environmental stratification using factor analysis was proposed by Murakami and Cruz (2004). This method is implemented with the function `ge_factanal()`

, as follows:

```
fact <- ge_factanal(data_ge, ENV, GEN, REP, GY)
plot(fact)
```

`print_table(fact$GY$PCA)`

# Wrapper function

The easiest way to compute the above-mentioned stability indexes is by using the function `ge_stats()`

. It is a wrapper function that computes all the stability indexes at once. To get the results into a *“ready-to-read”* file, use `get_model_data()`

.

```
<- data_ge2 %>% ge_stats(ENV, GEN, REP, resp = c(EH, PH))
stat_ge # Evaluating trait EH |====================== | 50% 00:00:02
|============================================| 100% 00:00:04
Evaluating trait PH get_model_data(stat_ge, "stats") %>% print_table()
# Class of the model: ge_stats
# Variable extracted: stats
```

It is also possible to obtain the Spearman’s rank correlation between the stability indexes by using `corr_stab_ind()`

.

`corr_stab_ind(stat_ge)`

# Rendering engine

This vignette was built with pkgdown. All tables were produced with the package `DT`

using the following function.

```
library(DT) # Used to make the tables
# Function to make HTML tables
print_table <- function(table, rownames = FALSE, digits = 3, ...){
df <- datatable(table, rownames = rownames, extensions = 'Buttons',
options = list(scrollX = TRUE,
dom = '<<t>Bp>',
buttons = c('copy', 'excel', 'pdf', 'print')), ...)
num_cols <- c(as.numeric(which(sapply(table, class) == "numeric")))
if(length(num_cols) > 0){
formatSignif(df, columns = num_cols, digits = digits)
} else{
df
}
}
```

# References

*Journal of Genetics and Breeding*46: 269–78.

*Ceres.ufv.br*38: 422–30. http://www.ceres.ufv.br/ojs/index.php/ceres/article/view/2165.

*Crop Science*6 (1): 36–40. https://doi.org/10.2135/cropsci1966.0011183X000600010011x.

*Euphytica*31 (3): 645–56. https://doi.org/10.1007/BF00039203.

*Canadian Journal of Plant Science*68 (1): 193–98. https://doi.org/10.4141/cjps88-018.

*Crop Breeding and Applied Biotechnology*4 (1): 7–11.

*Experimental design on the measurement of heritabilities and genetic correlations: biometrical genetics*. New York, US: Pergamon.