GxE Analysis Workshop

class: center, middle, inverse, title-slide

.title[
# GxE Analysis Workshop
]
.author[
### Ellena Girma
]
.date[
### 2022/05/27 (updated: 2022-10-12)
]

---

class: middle, center

### Hello, Everyone!

#### My name is Ellena Girma and I am a breeding data systems and analytics specialist at the Alliance Bioversity International - CIAT.

#### The goal of this presentation is to demonstrate how R can be used to run GxE analyses.

#### I will take you through the workflow that I use to run my GxE analysis for multi environment trials. For each part, I will cover the main functions that I use and what outputs/results you can expect to get.

---

## Introduction to GxE Studies

.pull-left[
#### Understanding Genotype-Environment (GxE) Interaction
* GXE interaction: the interaction of genotypes and environmental conditions and its joint effect on the traits of interest

* Is there a pattern of phenotypic expression of a trait across various environmental conditions? 
]

.pull-right[
#### Selecting & recommending genotypes that have wider or more specific adaptation across various environmental conditions

* Stability: Does the performance of the genotype stay the same across environments?

* Yield Production: Does the genotype perform well across environments? 
]

---

## My Workflow

#### In this presentation I will be sharing the process I use to get from raw data to results. I will be focusing on the response variable yield, but the code I use can be altered for other response variables too.

Workflow:

* Data Preparation

* Exploratory Data Analysis

* Means and High Yielding Genotypes & Locations

* Analysis of Variance (ANOVA)

* GGE Biplots

* Stability Analysis

* Additive Main-Effects Multiplicative Interaction (AMMI)

---
class: middle, center

## Let's take a quick tour of R.

---
class: middle, center

## For this workshop, we'll be focusing on the metan package.

---

## Loading Packages and Importing Data

```r
#loading metan package into workspace
#for analysis
library(metan)

#loading tidyverse package into workspace
#for data manipulation functions
library(tidyverse)

#importing a csv file using the read.csv() function
bean_data <- read.csv(here("GxE-June2022", "csvfile.csv"), 
                      header=TRUE,
                      colClasses = c(rep("factor", 3), rep("numeric", 5)))
```

---
class: middle

## Exploratory Data Analysis (EDA)

### Exploring and summarizing datasets using summary statistics and plots/data visualization methods.

Some EDA tasks:

* finding outliers and missing values

* looking at how our data is distributed

* looking at the relationship between variables

* getting summary statistics for each variable of interest

---

## Summary Statistics

.panelset[
.panel[.panel-name[R Code]

```r
#descriptive stats using metan package
ds_all <- desc_stat(bean_data, stats= "main") 
```
]

.panel[.panel-name[desc_stats output]

```r
print(ds_all)
```

```
## # A tibble: 5 × 10
##   variable      cv    max    mean median   min sd.amo      se    ci.t n.valid
##   <chr>      <dbl>  <dbl>   <dbl>  <dbl> <dbl>  <dbl>   <dbl>   <dbl>   <dbl>
## 1 DF          7.02   48     39.5    39    33     2.77  0.093   0.182      891
## 2 pods.plant 48.1    46.6   14.8    13.4   2.6   7.11  0.238   0.467      891
## 3 seeds.pod  21.3     7.6    4.82    4.8   2.6   1.02  0.0343  0.0673     891
## 4 sw         32.9    67     35.2    35    15.3  11.6   0.389   0.763      891
## 5 yield      43.0  5153.  1754.   1602.  498.  754.   25.3    49.6        891
```
]
]

---

#### Checking for outliers: Plots

.panelset[
.panel[.panel-name[Code]
This is for the variable yield. To get potential outliers for another variable, set the var argument to the variable name.

```r
#Checking for potential outliers in yield variable
find_outliers(bean_data, var = yield, plots =T)
```

```
## Trait: yield 
## Number of possible outliers: 37 
## Line(s): 10 28 32 35 58 66 68 69 74 75 84 93 109 127 131 134 157 165 167 168 173 174 183 192 208 209 226 230 233 256 264 266 267 272 273 282 291 
## Proportion: 4.3%
## Mean of the outliers: 4314.67 
## Maximum of the outliers: 5152.6  | Line 208 
## Minimum of the outliers: 3241.9  | Line 28 
## With outliers:    mean = 1754.348 | CV = 42.991%
## Without outliers: mean = 1643.421 | CV = 32.131%
```
]
.panel[.panel-name[Plot]

```r
find_outliers(bean_data, var = yield, plots =T)
```

<img src="index_files/figure-html/yield-outliers-plots-1.png" width="504" />
]
]

---

### Checking for Outliers: Looking at Data Points

Let's take a look at which data point are the outliers. Are they all the same location or same genotypes?

.panelset[
.panel[.panel-name[Code]

```r
#potential outliers
yield_outliers <- c(10, 28, 32, 35, 58, 66, 68, 69, 74 ,75, 84, 93, 109, 127, 131, 134, 157, 165, 167, 168, 173, 174, 183, 192, 208, 209, 226, 230, 233, 256, 264, 266, 267, 272, 273, 282, 291)

yo <- bean_data %>%
  slice(yield_outliers) %>%
  arrange(Genotype) %>%
  slice(1:15)

yo
```

]

.panel[.panel-name[Output]

```
##      Site Rep      Genotype DF pods.plant seeds.pod sw  yield
## 1  Selian   1        Cheupe 44       44.8       7.6 33 5085.3
## 2  Selian   2        Cheupe 44       46.2       7.4 33 5127.8
## 3  Selian   3        Cheupe 43       46.6       6.8 33 5152.6
## 4  Selian   3 Chumba Neroza 38       15.4       5.8 31 3478.7
## 5  Selian   1        Kanade 38       24.2       5.4 42 3241.9
## 6  Selian   2        Kanade 38       23.8       5.4 42 3245.4
## 7  Selian   3        Kanade 39       24.8       5.4 42 3291.5
## 8  Selian   1       Katuku2 39       36.2       5.2 40 4309.2
## 9  Selian   2       Katuku2 39       35.2       5.0 40 4256.0
## 10 Selian   3       Katuku2 39       35.0       5.4 40 4245.4
## 11 Selian   1        Kikobe 39       35.2       6.2 27 3306.3
## 12 Selian   2        Kikobe 39       36.8       6.4 27 3285.1
## 13 Selian   3        Kikobe 39       34.0       5.8 27 3355.9
## 14 Selian   1  Nyeupe Kubwa 39       27.6       5.8 35 4419.1
## 15 Selian   2  Nyeupe Kubwa 39       27.0       6.2 35 4298.6
```

]
]

---

## Grand Mean Yield of Genotypes Across Locations

We can see the grand mean yield for each genotype across all three locations using the means_by() function.

.panelset[
.panel[.panel-name[R Code]

```r
#getting means by Genotype
mg <- means_by(bean_data, Genotype)

#Creating a new object with Ordered Means by Genotype
mg_y <- mg %>%
  select(Genotype, yield) %>%
  arrange(desc(yield))
```
]

.panel[.panel-name[Table]

```r
slice(mg_y, 1:10)
```

```
## # A tibble: 10 × 2
##    Genotype     yield
##    <fct>        <dbl>
##  1 Cheupe       3144.
##  2 Selian 15    3086.
##  3 Ruondera     2906.
##  4 Nyeupe Kubwa 2894.
##  5 Selian 14    2778.
##  6 Uyole 84     2722.
##  7 Selian 05    2711.
##  8 Soya Mbeya   2680.
##  9 Kikobe       2546.
## 10 Jabeyila     2476.
```
]
]

---

## Yield Means Across Genotypes at Each Location

Now, let's take a look at the grand mean at each location using the means_by() function.

.panelset[
.panel[.panel-name[Code]

```r
#getting means for all variables by Location
me <- means_by(bean_data, Site)

#Creating a new object with Ordered Means by Location
me_y <- me %>%
  select(Site, yield) %>%
  arrange(desc(yield))
```
]

.panel[.panel-name[Output]

```r
me_y
```

```
## # A tibble: 3 × 2
##   Site   yield
##   <fct>  <dbl>
## 1 Selian 2336.
## 2 Uyole  1579.
## 3 SUA    1347.
```
]
]

---

## Top Performers at Each Location

We can use the ge_winners() function to get the top performing genotype at each location across all response variables.

```r
#top performing genotype in each env for yield
win <- ge_winners(.data = bean_data, 
                  env = Site, 
                  gen = Genotype, 
                  resp = c(DF, pods.plant, seeds.pod, sw, yield), type = "winners")
```

```r
print(win)
```

```
## # A tibble: 3 × 6
##   ENV    DF              pods.plant   seeds.pod sw          yield    
##   <fct>  <chr>           <chr>        <chr>     <chr>       <chr>    
## 1 Selian Kamosi          Cheupe       Cheupe    Lyamungo 90 Cheupe   
## 2 SUA    Kaisho kamugole Cheupe       Kaempu    Lyamungo 90 Jabeyila 
## 3 Uyole  Selian 15       Wifi Nyegela Cheupe    Selian 15   Selian 14
```

---
class: middle

## Analysis of Variance (ANOVA): Introduction

#### Running an ANOVA allows us to

* Evaluate differences in means among groups

* see if genotype main effect, environment main effect and GxE interaction effect are significant

* see which effect contributes most to the variation in performance of the different traits
* Divide the variance into parts to estimate heritability

#### The metan package provides <a href = "https://tiagoolivoto.github.io/metan/reference/index.html">many functions</a> that can be used to run ANOVA. Different functions are available based on the ANOVA type (one-way, two-way), type of model (fixed or mixed effect), etc.

---

## ANOVA for RCBD

#### anova_joint() is the function used for combined site analysis.

#### get the anova table for the response variable you want through anova_object_name$var_name$anova to

.panelset[
.panel[.panel-name[R Code]

```r
#Running joint anova for rcbd layout/plot design for all repsonse variables

anova_joint(bean_data, env = "Site", gen = "Genotype", rep = "Rep", resp = everything())
```
]
.panel[.panel-name[Yield ANOVA Table]

```r
#getting the ANOVA table for yield 
rcbd_anova_yield <- rcbd_anova$yield$anova
rcbd_anova_yield
```

```
##      Source          Df      Sum Sq     Mean Sq     F value        Pr(>F)
## 1       ENV    2.000000 158874976.4 79437488.20 3943.977619  0.000000e+00
## 2  REP(ENV)    6.000000    759885.9   126647.66    6.287907  2.023559e-06
## 3       GEN   98.000000 199047400.1  2031095.92  100.841517 1.451699e-310
## 4   GEN:ENV  196.000000 135738882.9   692545.32   34.384059 1.554893e-232
## 5 Residuals  588.000000  11843181.6    20141.47          NA            NA
## 6     CV(%)    8.089650          NA          NA          NA            NA
## 7 MSR+/MSR-    6.288451          NA          NA          NA            NA
## 8    OVmean 1754.348260          NA          NA          NA            NA
```
]
.panel[.panel-name[SW ANOVA Table]

```r
#getting the ANOVA table for 100g seed weight
rcbd_anova$sw$anova
```

```
##      Source         Df      Sum Sq      Mean Sq      F value    Pr(>F)
## 1       ENV   2.000000 27392.54795 1.369627e+04 51909.689778 0.0000000
## 2  REP(ENV)   6.000000     2.41064 4.017733e-01     1.522745 0.1681973
## 3       GEN  98.000000 85390.26909 8.713293e+02  3302.389579 0.0000000
## 4   GEN:ENV 196.000000  6922.37872 3.531826e+01   133.858293 0.0000000
## 5 Residuals 588.000000   155.14269 2.638481e-01           NA        NA
## 6     CV(%)   1.458387          NA           NA           NA        NA
## 7 MSR+/MSR-   2.315797          NA           NA           NA        NA
## 8    OVmean  35.221212          NA           NA           NA        NA
```
]
.panel[.panel-name[DF ANOVA Table]

```r
#getting the ANOVA table for days to 75% flowering
rcbd_anova$DF$anova
```

```
##      Source         Df      Sum Sq    Mean Sq    F value        Pr(>F)
## 1       ENV   2.000000  376.141414 188.070707 744.253705 7.959563e-162
## 2  REP(ENV)   6.000000    6.747475   1.124579   4.450306  2.041800e-04
## 3       GEN  98.000000 3804.080808  38.817151 153.611421  0.000000e+00
## 4   GEN:ENV 196.000000 2515.858586  12.836013  50.796057 6.659219e-278
## 5 Residuals 588.000000  148.585859   0.252697         NA            NA
## 6     CV(%)   1.271386          NA         NA         NA            NA
## 7 MSR+/MSR-   2.591123          NA         NA         NA            NA
## 8    OVmean  39.538721          NA         NA         NA            NA
```
]
]

---

## Estimating Heritability

#### Variance components can be calculated using the results from the ANOVA table. Using these variance components, we can estimate heritability.
#### Use indexing to get the correct value from the anova table: anova_table[row_number, col_number]

.panelset[
.panel[.panel-name[Calc Values 1]

V of line = (Mean sq of treat - Mean sq of site:treat) / (site*rep)

V of line*site = (Mean sq of site:treat - Mean sq of residual) / (rep)

V of error = mean sq of residuals

H2= Vline / Vline + Vline*site + Verror

H2L= Vline/ (line + (Vline*site/nsites) + (Verror/nsite*nrep)

]
.panel[.panel-name[Calc Values 2]

```r
#variance of line
Vl.yd <- (rcbd_anova_yield[3,4] - rcbd_anova_yield[4,4])/(3*3)

#variance of line by site
Vls.yd <- (rcbd_anova_yield[4,4] - rcbd_anova_yield[5,4])/(3)

#variance of error
Ve.yd <- rcbd_anova_yield[5,4]

#Broad sense hertitability 
H2.yd <- Vl.yd/(Vl.yd + Vls.yd + Ve.yd)

#Hertitability of Line Means
H2L.yd <- Vl.yd/(Vl.yd + Vls.yd/3 + Ve.yd/(3*3))
```
]

.panel[.panel-name[Combing Values into Table]

```r
comb.site.vc <- cbind(rbind(Vl.yd, Vls.yd, Ve.yd, H2.yd, H2L.yd))

colnames(comb.site.vc) <- c("Yield")
rownames(comb.site.vc) = c("V of Line", "V of Line x Site", "V of Error", "Broad Sense Heritability", "Heritability of Line Means")
```
]

.panel[.panel-name[Final Output]

```r
print(comb.site.vc)
```

```
##                                   Yield
## V of Line                  1.487278e+05
## V of Line x Site           2.241346e+05
## V of Error                 2.014147e+04
## Broad Sense Heritability   3.784386e-01
## Heritability of Line Means 6.590287e-01
```
]
]

---
class: middle

## Genotype plus genotype-by-environment (GGE) Models

#### Use the gge() function to create a gge model.

#### First, the centering argument needs to be set to "environment" for us to get a GGE model.

#### Then the svp argument can be used to create three types of gge models. Below are the three options:

* svp = "environment"

* svp = "genotype"

* svp = "symmetrical"

---
class: middle, center

## Genotype plus genotype-by-environment (GGE) Biplots

#### Use the plot() function to create plots from your gge models.

#### Use the type argument to choose the kind of plot you want. There are 10 types to choose from.

---

## Type argument in plot

.pull-left[

1 = Basic biplot.

2 =  Mean performance vs. stability (gge biplots) or the The Average Tester Coordination view for genotype-trait and genotype-yield*trait biplots.

3 = Which-won-where.

4 = Discriminativeness vs. representativeness.

5 = Examine an environment (or trait/yield*trait combination).
]
.pull-right[
6 = Ranking environments (or trait/yield*trait combination).

7 = Examine a genotype.

8 = Ranking genotypes.

9 = Compare two genotypes.

10 = Relationship among environments (or trait/yield*trait combination).
]

#### Take a look at the <a href = "https://tiagoolivoto.github.io/metan/reference/plot.gge.html">plot documentation</a> page for more information.

---
class: middle, center

# GGE Model 1
#Let's start with svp = "environment"

---

.panelset[
.panel[.panel-name[Create Model]

```r
gge_model <- gge(.data = bean_data,
                 env = Site,
                 gen = Gen_id,
                 resp = yield,
                 centering = "environment",
                 svp = "environment")
```
]
.panel[.panel-name[Create Plot]

Basic Biplot is a PC2 vs PC1 plot. Each environment will have a vector starting from the origin. The angle between environments gives you information about the correlation between the environments.

```r
#creating the basic biplot
plot(gge_model, type = 1)
```
]
.panel[.panel-name[Basic Biplot]

```r
#call the name of the plot to view it
gge_biplot
```

<img src="index_files/figure-html/plot-basic-biplot-1.png" width="504" />
]
]

---

.panelset[
.panel[.panel-name[R Code]
The discriminativeness vs. representativeness plot is created by setting the type argument to 4.

```r
plot(gge_model, type = 4)
```
]
.panel[.panel-name[Plot]

Selian has the longest vector. We can look at yield performance at Selian and get more information about the yield of the genotypes than at SUA or Uyole. Uyole has the smaller angle to the Average Environment Axis (AEA). This makes it the most representative environment.

```r
dvr
```

<img src="index_files/figure-html/plot-dvr-1.png" width="504" />
]
]

---

.panelset[
.panel[.panel-name[R Code]

Let's compare the environments using another biplot. 
Set the type to 6 to rank environments and set type to 10 to visualize a clear relationship of the environments.

```r
#creating plot 1: ranking environments
plot(gge_model, type=6)

#creating plot 2:relationship among environment
plot(gge_model, type = 10)
```

]
.panel[.panel-name[Ranking Environment]

```r
ranking_envrionment
```

<img src="index_files/figure-html/ranking-plot-1.png" width="504" />
]
.panel[.panel-name[Relationship Between Environments]

```r
relationship
```

<img src="index_files/figure-html/comparing-env-biplots-1.png" width="504" />
]
]

---
class: middle, center

# Now, for the second model, let's try svp = "genotype"

---

```r
gge_model_gen <- gge(.data = bean_data,
                 env = Site,
                 gen = Gen_id,
                 resp = yield,
                 centering = "environment",
                 svp = "genotype")
```

---

##### Svp = "genotype" Biplots

.panelset[
.panel[.panel-name[R Code]
explain plot types

```r
# mean performance vs stability 
plot(gge_model_gen, type =2)

# Ranking genotype 
plot(gge_model_gen, type=8)
```

]
.panel[.panel-name[Mean vs Stability]

```r
mean_vs_stability
```

<img src="index_files/figure-html/mean-perf-vs-stability-plot-1.png" width="504" />
]
.panel[.panel-name[Ranking Genotypes]

```r
rank_genotype
```

<img src="index_files/figure-html/type-8-plot-1.png" width="504" />
]
]

---
class: middle, center

# Finally, for the third model, let's look at svp = "symmetrical".

---

.panelset[
.panel[.panel-name[Create Biplot]

```r
gge_symmetrical <- gge(.data = bean_data,
                 env = Site,
                 gen = Gen_id,
                 resp = yield,
                 centering = "environment",
                 svp = "symmetrical")
```

]
.panel[.panel-name[R Code]

```r
# which won where 
plot(gge_symmetrical, type=3)

#Comparing 2 genotypes

compare_two_gen <- plot(gge_symmetrical, type=9, sel_gen1 = "15", sel_gen2 = "69")
```
]
.panel[.panel-name[Which Won Where]

```r
which_won_where
```

<img src="index_files/figure-html/type-3-plot-1.png" width="504" />
]
.panel[.panel-name[Compare Genotypes]

```r
compare_two_gen
```

<img src="index_files/figure-html/type-9-plot-1.png" width="504" />
]
]

---
class: middle

## Additive Main-Effects Multiplicative Interaction (AMMI)

AMMI deals with average performances and specific performances/GxE interaction.

* Addive Main-Effects:
  * Genotype Main Effect
  * Environment Main Effect
  
* Multiplicative Interaction
  * Genotype * Environment Interaction Effect

---

##### Creating an AMMI Model

.panelset[
.panel[.panel-name[AMMI Model]
Use performs_ammi() to create an ammi object. From the ammi object, you can get:

* ANOVA table
* PCA 
* AMMI model scores
* means of genotypes in the environments

```r
ammi <- performs_ammi(.data = bean_data, 
                      env = Site, 
                      gen = Gen_id, 
                      rep = Rep,
                      resp = yield)
```
]
.panel[.panel-name[AMMI ANOVA]

```r
ammi$yield$ANOVA
```

```
*##      Source   Df      Sum Sq     Mean Sq    F value        Pr(>F) Proportion
*## 1       ENV    2 158874976.4 79437488.20 627.232194  1.078604e-07         NA
*## 2  REP(ENV)    6    759885.9   126647.66   6.287907  2.023559e-06         NA
*## 3       GEN   98 199047400.1  2031095.92 100.841517 1.451699e-310         NA
*## 4   GEN:ENV  196 135738882.9   692545.32  34.384059 1.554893e-232         NA
*## 5       PC1   99 112959941.7  1141009.51  56.650000  0.000000e+00       83.2
*## 6       PC2   97  22778941.2   234834.45  11.660000  0.000000e+00       16.8
*## 7 Residuals  588  11843181.6    20141.47         NA            NA         NA
*## 8     Total 1086 642003209.7   591163.18         NA            NA         NA
##   Accumulated
## 1          NA
## 2          NA
## 3          NA
## 4          NA
## 5        83.2
## 6       100.0
## 7          NA
## 8          NA
```
]
.panel[.panel-name[AMMI Plot]

```r
residual_plots(ammi)
```

<img src="index_files/figure-html/ammi-plot-1.png" width="504" />
]
]

---

###### AMMI Biplots

.panelset[
.panel[.panel-name[Creating the plots]

```r
#ab2, basic ammi biplot
plot_scores(ammi)

#b2, Produces an AMMI2 biplot (PC1 x PC2) to make inferences related to the interaction effects.
plot_scores(ammi, type=2)

#b2 polygon
plot_scores(ammi, type=2, polygon=T)
```
]
.panel[.panel-name[ammi_biplot1]

```r
ammi_biplot1
```

<img src="index_files/figure-html/ab2-1.png" width="504" />
]

.panel[.panel-name[b2]

```r
b2
```

<img src="index_files/figure-html/b2-1.png" width="504" />
]

.panel[.panel-name[b2_polygon]

```r
b2_polygon
```

<img src="index_files/figure-html/b2-polygon-1.png" width="504" />
]
]

---
class: middle, center

# Ranks: Yield Output and Stability

#### Ammi Stability Value (ASV) and Yield Stability Index (YSI) are the metrics I use/come across most frequently. Ranking genotypes using these metrics is useful because it takes into consideration both yield output and stability.

.pull-left[
#### ASV: genotypes with low ASV values are more stable and vice versa. 
]

.pull-right[
#### YSI takes into account both stability and yield output making it a better method to identify high yielding and stable genotypes. Lower YSI have both high mean yield and stability traits. 
]

---

#### Use the <a href= "https://tiagoolivoto.github.io/metan/reference/ammi_indexes.html">ammi_indexes()</a> function to calculate many AMMI-based statistics.
.panelset[
.panel[.panel-name[ASV]

```r
ammi_index <- ammi_indexes(ammi, level = 0.95)
```
]
.panel[.panel-name[ASV Results]

```r
ammi_index$yield$ASV
```

```
##  [1]   0.3743284 105.7142141  45.3017281   5.5209325  24.1452122   8.3997479
##  [7]  41.8516322  36.1954736  13.1441321   9.6855204  10.0884941  15.5603091
## [13]  21.9185126  11.5032871   2.6378536  24.4509217  39.6016949  18.0710260
## [19]   3.2796301  13.9254193  22.7739710  14.3698118   1.7219150   6.1451389
## [25]   7.1595452 105.4762409  17.5841386  33.7633071  14.7950173  24.0321075
## [31]  22.0682420  31.6411401  12.4973625  16.1684393   7.8463305  45.7226331
## [37]  23.1230903  18.4354299  40.2764680  30.1568250  21.7140351  29.5019066
## [43]   8.2826783  15.2563748  21.2334733  25.5156696  37.4984548   8.6861028
## [49]  43.0402788   7.0901911  13.8969691   6.8373741  35.8930326  69.3553828
## [55]   2.9744217  29.8567655   2.9008277  32.5698178   8.8062425  52.7442305
## [61]   8.5565234  18.1772123  82.1457799  15.3907617 115.9391135 141.8757472
## [67]  22.2635780  23.0671911  11.6343117  33.3673054  15.0946263   8.8537557
## [73]  79.4979963   5.2581231  37.8284771  12.3913684  28.5883873  15.7815954
## [79]  28.5323632  32.5604059   2.9355086   6.7212823  84.5058132  37.2137729
## [85]  14.4411885  20.4230591  24.3445197  11.7288386  35.5159709   6.6535237
## [91]  15.2624635  55.0649012 140.7028315  13.8009329   8.8636252   7.4107872
## [97]  24.8599914  26.9057554  58.4753660
```
]
.panel[.panel-name[ASVR Results]

```r
ammi_index$yield$ASV_R
```

```
##  [1]  1 96 86  9 60 19 84 78 32 25 26 43 53 27  3 62 82 47  7 35 56 36  2 10 15
## [26] 95 46 75 38 59 54 71 31 45 17 87 58 49 83 70 52 68 18 40 51 64 80 21 85 14
## [51] 34 13 77 91  6 69  4 73 22 88 20 48 93 42 97 99 55 57 28 74 39 23 92  8 81
## [76] 30 67 44 66 72  5 12 94 79 37 50 61 29 76 11 41 89 98 33 24 16 63 65 90
```
]

.panel[.panel-name[YSI]

### YSI = RASV + RY

]
]

---
class: middle

# Follow Up Resources

#### The link to this presentation will be emailed to you.

#### Check out my <a href = "https://rforplantbreeders.netlify.app/"> blog </a> for more detailed written tutorials for plant breeders who want to learn how to use R.

#### Current Posts:

*  <a href = "https://rforplantbreeders.netlify.app/posts/introduction-to-r-for-plant-breeding/"> Introduction to R </a>
*  <a href = "https://rforplantbreeders.netlify.app/posts/importing-data-into-r/"> Importing in R </a>

---
class: middle, center

# Any Questions?
### Thank you for attending this workshop. You can ask any questions you may have now.