In priviere/PPBstats: An R package to perform analysis found within PPB programmes

GGE (M6b) {#gge}

Theory of the model

The experimental design used is fully replicated (D1). The GGE model is the same as the AMMI model except that the PCA is done on a matrix centered on the locations: germplasm and interaction effects are merged ^[Note than in PPBstats, as for AMMI, the PCA is done on germplasm and interaction effects + residuals.] [@gauch_statistical_2006][@gauch_statistical_2008][@yan_gge_2007]. The model is based on frequentist statistics (section \@ref(section-freq)).

The GGE model can be written as followed:

$Y_{ijk} = \mu + \theta_{j} + rep_{k}(\theta_{j}) + \sum_{n}^{N} \lambda_{n} \gamma_{in} \omega_{jn} + \varepsilon_{ijk}; \quad \varepsilon_{ijk} \sim \mathcal{N} (0,\sigma^2)$

with,

• $Y_{ijk}$ the phenotypic value for replication $k$, germplasm $i$ and location $j$,
• $\mu$ the general mean,
• $\theta_{j}$ the effect of location $j$,
• $rep_{k}(\theta_{j})$ the effect of replication $k$ nested in location,
• $N$ the number of dimension (PCA componant) which has as maximum value the number of location,
• $\lambda_{n}$ the eigen value for componant $n$,
• $\gamma_{in}$ the eigen vector for germplasm $i$ for componant $n$,
• $\omega_{jn}$ the eigen vector for location $j$ for componant $n$.
• $\varepsilon_{ijk}$ the residuals.

or, if there is year effect:

$Y_{ijkl} = \mu + \theta_{j} + rep_{k}(\theta_{j}\beta_{l}) +\sum_{n}^{N} \lambda_{n} \gamma_{in} \omega_{jn} + \beta_{l} + \beta_{l}\alpha_{i} + \beta_{l}\theta_{j} + \varepsilon_{ijk}; \quad \varepsilon_{ijk} \sim \mathcal{N} (0,\sigma^2)$

With,

• $Y_{ijkl}$ the phenotypic value for replication $k$, germplasm $i$, location $j$ and year $l$,
• $\beta_{l}$ the year $l$ effect,
• $\beta_{l}\alpha_{i}$ the year $\times$ germplasm interaction effect,
• $\beta_{l}\theta_{j}$ the year $\times$ location interaction effect,
• $\varepsilon_{ijk}$ the residuals,
• and all other effects are the same as in the previous model.

Steps with PPBstats

For GGE analysis, everything is exactly the same than for AMMI analysis except you should settle gxe_analysis = "GGE" in model_GxE. You can follow these steps (Figure \@ref(fig:main-workflow)):

• Format the data with format_data_PPBstats()
• Run the model with model_GxE() and gxe_analysis = "GGE"
• Check model outputs with graphs to know if you can continue the analysis with check_model()
• Get mean comparisons for each factor with mean_comparisons() and vizualise it with plot()
• Get and visualize biplot with biplot_data() and plot()
• Get groups of each parameters with parameters_groups() and visualise it with plot()

We will not details everything as it the same than AMMI in the code. Of course the calculation is different on the interaction matrix so you'll get different results.

The workflow is therefore :

Format the data

data(data_model_GxE)
data_model_GxE = format_data_PPBstats(data_model_GxE, type = "data_agro")
head(data_model_GxE)

Run the model

out_gge = model_GxE(data_model_GxE, variable = "y1", gxe_analysis = "GGE")

Check and visualize model outputs

The tests to check the model are explained in section \@ref(check-model-freq).

Check the model

out_check_gge = check_model(out_gge)

Visualize outputs

p_out_check_gge = plot(out_check_gge)

Get and visualize mean comparisons

The method to compute mean comparison are explained in section \@ref(mean-comp-check-freq).

Get mean comparisons

out_mean_comparisons_gge = mean_comparisons(out_check_gge, p.adj = "bonferroni")

Visualize mean comparisons

p_out_mean_comparisons_gge = plot(out_mean_comparisons_gge)

Get and visualize biplot

Get biplot

out_biplot_gge = biplot_data(out_check_gge)

Visualize biplot

p_out_biplot_gge = plot(out_biplot_gge)

Compared to AMMI analysis, in the output of p_out_biplot_gge, which_won_where, mean_vs_stability and discrimitiveness_vs_representativeness are displayed.

The description of these following graphs are greatly inspired from @ceccarelli_manual_2012.

biplot = p_out_biplot_gge$biplot

• The which won where graph

This graph allow to detect location where germplasm (and the interaction) behave better : 'which won where' [@gauch_statistical_2008,@yan_gge_2007]. The germplasms which have the largest value in a sector "win" in the location present in that sector. The information is summarized in the legend of the plot. See ?plot.PPBstats for more details on this plot.

biplot$which_won_where

• The mean vs stability graph.

  • mean A red circle define the average location. An high score mean a greater mean performance of an entry. Entries with a score above zero means entries with above-average means. Entries with a score below zero means entries with below-average means. Note that the distance from the biplot origin to the average location circle (represented with an arrow), is a measure of the relative importance of the germplasm main effect versus the entry by location interaction. The longer the arrow is, the more important is germplasm effect and the more meaningful is the selection based on mean performance. r biplot$mean_vs_stability$mean_performance

  • stability This information is related to the ecovalence graph. The score is equal to the length of the projection. A high score represents a low stability (i.e. an high entry by location interaction). r biplot$mean_vs_stability$stability_performance

• The discrimitiveness vs representativeness graph. It is interessting to assess the ability of the locations to discriminate the germplasms and their ability to represent the target locations.

The closer a given location is next to the averge location (represented by a red circle), the more desirable it is judged on both discrimination and representativeness.

• discrimitiveness The higher the value, the highest the discrimitiveness for locations.

biplot$discrimitiveness_vs_representativeness$discrimitiveness

• representativeness The highest the value, the less representative the location.

biplot$discrimitiveness_vs_representativeness$representativeness

• discrimitiveness vs representativeness The location combining better score (i. e.discrimination and representativeness) are the ones that could be used to test germplasms as they are more representative of all the locations. This has to be done severals year to get robust results. The highest the score, the more representative the location.

biplot$discrimitiveness_vs_representativeness$discrimitiveness_vs_representativeness

Get and vizualise groups of parameters

Get groups of parameters

# First run the models
gge_2 = model_GxE(data_model_GxE, variable = "y2", gxe_analysis = "GGE")
gge_3 = model_GxE(data_model_GxE, variable = "y3", gxe_analysis = "GGE")

# Then check the models
out_check_gge_2 = check_model(gge_2)
out_check_gge_3 = check_model(gge_3)

# Then run the function for germplasm. It can also be done on location or intra germplasm variance
out_parameter_groups = parameter_groups(
  list("y1" = out_check_gge, "y2" = out_check_gge_2, "y3" = out_check_gge_3), 
  "germplasm"
  )

Visualize groups of parameters

p_germplasm_group = plot(out_parameter_groups)

Apply the workflow to several variables

If you wish to apply the AMMI workflow to several variables, you can use the function workflow_gxe() presented in section \@ref{workflow-gxe} with the following code :

vec_variables = c("y1", "y2", "y3")

out = lapply(vec_variables, workflow_gxe, "GGE")
names(out) = vec_variables

out_parameter_groups = parameter_groups(
  list("y1" = out$y1$out_check_gxe, "y2" = out$y2$out_check_gxe, "y3" = out$y3$out_check_gxe), 
  "germplasm" )

p_germplasm_group = plot(out_parameter_groups)

priviere/PPBstats documentation built on May 6, 2021, 1:20 a.m.