# Indexes for simultaneous selection

#### Tiago Olivoto

#### 2021-12-29

Source:`vignettes/vignettes_indexes.Rmd`

`vignettes_indexes.Rmd`

# Getting started

In this section we will use the data examples `data_ge`

and `data_ge2`

provided in the **metan** package. For more information, please, see `?data_ge`

and `?data_ge2`

. Other data sets can be used provided that the following columns are in the dataset: environment, genotype, block/replicate and response variable(s). See the section Rendering engine to know how HTML tables were generated.

# Multi-trait stability index

The function `mtsi()`

is used to compute the multi-trait stability index (*MTSI*) proposed by Olivoto et al. (2019). The first argument is a model of the class `waasb`

or `waas`

. It is possible to compute the *MTSI* for both *WAASB* -stability only- and *WAASBY* -simultaneous selection for mean performance and stability.

## Based on stability only

In the following example, the selection of stable genotypes will consider five traits, namely, KW, NKE, PH, EH, and TKW. Note that the output of the function `waasb()`

is passed to the function `mtsi()`

by the forward-pipe operator `%>%`

. Finally, the MTSI index is plotted using the function `plot()`

.

## Based on mean performance and stability

The following code considers that higher values for KW, NKE, TKW are better, and lower values for PH and EH are better. By using `wresp = 65`

, the simultaneous selection for mean performance and stability will prioritize the mean performance (mean of the variables) rather than their stability across environments.

# MGIDI index

The MGIDI index can be seen as the MTSI index with a weigth of 100 for mean performance. This index is computed with the function `mgidi()`

. Here, we will use the example data `data_g()`

. By default, all traits are assumed to be increased. To change this default, use the argument `ideotype`

. For example, if for three traits, the first one is assumed to be decreased and the last two are assumed to be increased, use `ideotype = c("l, h, h")`

.

```
mod <- gamem(data_g,
gen = GEN,
rep = REP,
resp = everything())
mgidi_index <- mgidi(mod,
SI = 20) # Selection intensity
```

```
p1 <- plot(mgidi_index, SI = 20)
p2 <- plot(mgidi_index, type = "contribution")
arrange_ggplot(p1, p2)
```

# Smith-Hazel index

The Smith-Hazel index ((Smith 1936; Hazel 1943)) is computed with the function `Smith_Hazel()`

. Users can compute the index either by declaring known genetic and phenotypic variance-covariance matrices or by using as inpute data, a model fitted with the function `gamem()`

. In this case, the variance-covariance are extracted internally. The economic weights in the argument `weights`

are set by default to 1 for all traits.

`smith <- Smith_Hazel(mod, SI = 20)`

# FAI-BLUP index

The FAI-BLUP is a multi-trait index based on factor analysis and ideotype-design recently proposed by Rocha, Machado, and Carneiro (2018). It is based on factor analysis, when the factorial scores of each ideotype are designed according to the desirable and undesirable factors. Then, a spatial probability is estimated based on genotype-ideotype distance, enabling genotype ranking (Rocha, Machado, and Carneiro 2018). Here we will use the mixed-model `mod`

as inpute data. By default, the selection is made to increase the value of all traits. Change this default with the arguments `DI`

and `UI`

.

```
fai <- fai_blup(mod, SI = 20)
```

`plot(fai)`

# Comparing the indexes

The function `coincidence_indexes()`

can be used to compute the coincidence index described by Hamblin and Zimmermann (1986), among the multi-trait indexes implemented in `metan`

. To do that, just use a comma-separated list of indexes as inpute data and inform the total number of genotypes.

```
coincidence <- coincidence_index(mgidi_index, fai, smith, total = 13)
coincidence
# ---------------------------------------------------------------------------
# Coincidence index and common genotypes
# ---------------------------------------------------------------------------
# # A tibble: 3 x 5
# V1 V2 index common genotypes
# <chr> <chr> <dbl> <int> <chr>
# 1 mgidi_index fai 100 3 H13,H5,H2
# 2 mgidi_index smith 56.71 2 H13,H5
# 3 fai smith 56.71 2 H5,H13
```

We can also produce a Venn plot to show the relationships between the indexes

```
MGIDI <- gmd(mgidi_index, "sel_gen")
# Class of the model: mgidi
# Variable extracted: sel_gen
FAI <- gmd(fai, "sel_gen")
# Class of the model: fai_blup
# Variable extracted: sel_gen
SH <- gmd(smith, "sel_gen")
# Class of the model: sh
# Variable extracted: sel_gen
# Create the plot
venn_plot(MGIDI, FAI, SH, show_elements = TRUE)
```

# 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

*Plant Breeding Reviews*, 245–72. Hoboken, NJ, USA: John Wiley & Sons, Inc. https://doi.org/10.1002/9781118061015.ch8.

*Genetics*28 (6): 476–90. http://www.ncbi.nlm.nih.gov/pubmed/17247099 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC1209225.

*Agronomy Journal*111 (6): 2961–69. https://doi.org/10.2134/agronj2019.03.0221.

*GCB Bioenergy*10 (1): 52–60. https://doi.org/10.1111/gcbb.12443.

*Annals of Eugenics*7 (3): 240–50. https://doi.org/10.1111/j.1469-1809.1936.tb02143.x.