Genotype × Environment Interaction: A Comparison between Joint

Regression Analysis and Weighted Biplot Models

CRISTINA DIAS1,4, CARLA SANTOS2,4, JOÃO TIAGO MEXIA3,4

1Polytechnic Institute of Portalegre,

Portalegre

PORTUGAL

2Polytechnic Institute of Beja,

Beja,

PORTUGAL

3Department of Mathematics, SST,

New University of Lisbon,

Caparica,

PORTUGAL

4NOVAMATH - Center for Mathematics and Applications, SST,

New University of Lisbon,

Caparica,

PORTUGAL

Abstract: - This work examines the obstacles presented by genotype-environment interaction (GEI) in plant

breeding and the significance of accurate analysis in selecting superior genotypes. Although current models,

such as the JRA and GGE biplot models, have their limitations, especially when dealing with multi-

environment data with a single trait and several environments, our approach addresses the challenges of GEI in

plant breeding. We introduce new models and conduct a comprehensive comparison with the existing ones. The

inclusion of mega-environment analysis and the evaluation of individual test environments within each mega-

environment adds depth to this study, aiming to provide a more nuanced understanding of the causes and

effects of GEI. We intend to validate and test the proposed models on real-world datasets to assess their

effectiveness and practical applicability in the field of plant breeding. Additionally, communicating the benefits

and potential limitations of your proposed models will contribute to the broader understanding and adoption of

improved methods for analyzing GEI in plant breeding. We conclude that joint use of the JRA and GGE Biplot

models has proven effective in exploring genotype × environment interaction, particularly for multi-

environment data (MET). JRA model provides a most robust and reliable representation of patterns in the data

related to genotypes and environments.

Key-Words: - Genotype evaluation, Mega-environment (MET), JRA Model, GGE Biplot models, Principal

component analysis, Stability analysis, Zigzag algorithm,  environmental indexes, Scheffé

multiple comparison tests, Upper contour.

Received: April 15, 2023. Revised: November 11, 2023. Accepted: December 23, 2023. Published: December 31, 2023.

1 Introduction

When different genotypes are adapted to different

groups of environments and the variation between

groups is greater than within the group, a mega-

environment (MET), is formed, [1]. The definition

of mega-environments and the relationship between

environments helps plant breeders to identify

genotypes that have broad or specific adaptations to

certain environments or groups of environments. In

the field of statistical genetics, identifying superior

genotypes is a significant challenge. Scientists,

farmers, and breeders struggle to identify genotypes

that show high potential productivity associated

with desirable agronomic characteristics in a vast

range of environmental conditions. However, one of

the most difficult problems to overcome in this type

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of research, is that when genotypes are put into

competition in various environments, the relative

ranking between them may not coincide, which

makes it very hard to identify those that are

superior. This oscillation in the behavior of

genotypes in the face of environmental variations is

called the genotype-by-environment interaction

(GEI). A genotype that thrives in well-watered

environments, for instance, could not produce as

well in arid environments. Crossbreeding, or

differences in the order of performance between two

genotypes in different contexts, or the inconsistent

responses of certain genotypes in different situations

without changes in the order of performance, have

been the subject of study by breeders and agronomic

researchers, that aim to identify superior performing

genotypes across a wide range of environmental

conditions. The existence of GEI implies changes in

the relative performance of genotypes across diverse

environments and complicates the evaluation of

genotypes. In the absence of GEI, a single cultivar

could dominate globally, requiring only a single

experiment for evaluation, [2]. The GEI manifests in

various forms, with cross-interactions representing

its most extreme manifestation. In this scenario,

genotypic responses change between environments,

challenging the concept that the best-performing

genotypes in one location will be excellent in others.

Biometricians and quantitative geneticists have

focused on GEI, aiming to develop stability indices

to quantify and mitigate GEI effects, [3]. The last

two decades have seen an increase in GEI research,

which has led to the development and application of

various statistical methods. Among these methods,

the AMMI model, [4] and the GGE biplot model

stand out, [4]. The AMMI model is a statistical

method employing analysis of variance and

principal component analysis, that dissects the

interactions between genotypes and environments.

The GGE biplot model considers the main effect of

the genotype plus the interaction between genotype

and environment. Both analyses are based on biplot

graphs and represent a data matrix. The difference

between the AMMI and GGE models lies in the

initial stage of the analysis. The GGE model directly

analyses the effect of genotype plus GEI, while the

AMMI analysis separates the effect of genotype

from GEI. However, this separation is not superior

to AMMI analysis. According to [5], the GGE biplot

model is more suitable for identifying mega-

environments, selecting representative and

discriminating environments, and indicating

cultivars that are more adapted and stable in specific

environments. Another technique widely used in the

study and analysis of GEI is Joint Regression

Analysis (JRA), [6]. According to [7], precision in

analyzing series of randomized block experiments

increased considerably when environmental indices

were introduced for individual blocks, instead of

using just one environmental index per experiment.

Being a flexible technique, it allows one or more

genotypes to be selected for each productivity value

measured by the environmental index. The upper

contour, which is defined by the fitted regression

lines, can also be used for this selection. In this

work, we intend to validate and test the proposed

techniques on real datasets to assess their

effectiveness and practical applicability in plant

breeding. In addition, communicating the benefits

and potential limitations of our proposed models

will contribute to a wider understanding and

adoption of improved methods for analyzing GEI in

plant breeding.

2 Methods and Material

2.1 Joint Regression Analysis

The goal of plant breeding is to develop varieties

that can be used in a wide range of areas. To achieve

this, cultivar comparison trials are conducted within

trial networks that accurately represent these areas.

These trial networks aim to gather data that can be

interpreted collectively. JRA, a method based on

Regression Analysis, is used to analyze these trial

networks. In JRA, an environmental index is used as

a synthetic variable to measure the productivity of

each pair (local, year). Specific interactions occur

when certain cultivars perform exceptionally well or

poorly in a particular year and location. This leads to

unusually high or low values in the corresponding

residuals for the genotype or genotypes. When there

is a specific interaction, the residuals for the

genotype involved should show significant

differences compared to the others. Suppose the data

is organized in a two-dimensional array with I rows

and b columns. If a specific genotype is present, we

can consider  as a continuous response variable,

like yield, for genotype  in block . The joint

regression model we are discussing is based on the

model proposed by [7]. However, instead of

calculating environmental indexes for each

environment, we calculate them for each block, [8].

It's important to note that we assume the yield

vectors are independent follow a normal distribution

and have equal variances, and that block  (or

replicate) contains genotype . The joint regression

model may be written as:

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  (1)

(󰇜

where  is the intercept,  the slope for genotype

,  the block environmental index, and  the

residuals. The environmental indexes serve to

measure productivity and are based on means within

a block/superblock. [9], suggests that for a more

precise analysis of randomized block experiments, it

can be more accurate to use environmental indexes

for each block rather than just one index for the

entire environment. This method entails conducting

 experiments, each with b blocks, resulting in 

supporting points per regression. In contrast, the

traditional model only utilizes  points for

regression.

To determine the parameters of the model, it is

our objective to minimize:

󰇛󰇜  





 , (2)

It is possible to simplify the treatment by

introducing weights , which have the value, [10],

respectively when the j-th cultivar is present[absent]

in the i-th block. Since b is the number of pairs

(location, year), we will have b environmental

indices,  which will be the components of

the structure vector xb and which will also have to

be adjusted. Where  is the yield of the j-th cultivar

in the i-th block if   and any value if  .

As we've mentioned, it is necessary to start by

choosing initial values for the components of the

structure vector xb. When all the trials have

complete blocks, the mean yields of these blocks are

considered as the initial values for the

environmental indexes. If the design is of the type

, the blocks are grouped into super-blocks, [10].

2.2 Performance of Different Genotypes

Influenced by Specific Environment

GGE biplot analysis is based on graphical

representation and uses Principal Component

Analysis (PCA), to study multi-environmental trials.

The graphical representation shows the relationships

between, test environments, genotypes, and the

genotype-environment interaction. This technique

considers the main effect of the genotype plus the

GEI and graphically represents a data matrix. The

biplot analysis is carried out with the first two

principal axes of the PCA, using Local Regression

Models (SREG). The first principal component

(PC1) represents the proportion of the yield that is

due exclusively to the characteristics of the

genotype, whenever it is observed to be strongly

related to the main effect of the genotype. The

second component (PC2) represents the part of the

yield resulting from GEI, [11]. The use of the GGE

biplot in MET research makes it possible to assess

genotypic adaptability and stability in multiple

environments. This way, the definition of MET and

the relationship between environments can help

breeders and agronomic researchers to identify

genotypes that have greater or specific adaptation in

certain environments or groups of environments.

When studying METs, using the GGE biplot

analysis, the mean on the graph is not related to the

overall mean, but to the mean of the mega

environment. The GGE biplot model does not

separate the effects of GEI and genotype but rather

combines them into two multiplicative terms. For

genotype evaluation, test environment evaluation,

and mega-environment design, we only need to

focus on genotype and GEI, this way, the main

effect of the environment and the general mean

should be removed from each element to only keep

the effect of the genotype and GEI in the double

entry table, [12]. The biplot approximation of a

double-entry table allows the geometric

visualization of the data structure, making its

variability known, and showing that biplots are very

intuitive when it comes to describing data sets. Once

the MET data has been appropriately dimensioned,

it should be subjected to Singular Value

Decomposition (SVD) and biplot analysis.

2.3 Biplot for Double Entry Table

Gabriel's initial biplot, is an extension of PCA based

on SVD, to Canonical Analysis for group

separation, [13]. [14], wrote the first monograph on

biplots in which they included several types

associated with various multivariate techniques.

[15], produced a symmetric version like

Correspondence Analysis that represents rows and

columns with the same quality, and described

several methods that can be used to visualize and

interpret a biplot, especially in the context of the

study of genotype-environment interaction in

agricultural studies. The biplot method can be used

to graphically represent the results of principal

component analysis or SVD, where the value of

each element in a double-entry table can be

visualized by the product of vectors and the cosine

of the angle between two vectors. So, considering

two matrices,  and , that have the same number

of rows and columns, they can be multiplied. The

new matrix generated by multiplying the two

previous ones has the same number of rows and

columns as both matrices. The SVD decomposition

expresses the matrix  as the product of three

matrices:

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, (3)

where  is a matrix  and the columns of  are

the left singular vectors representing the

relationships among genotypes. The matrix 

captures the genotype structure in the data. The

covariance diagonal matrix  as size  with

diagonal elements  that are singular

values, representing the importance of each

principal component. These singular values are

arranged in descending order, indicating the

contribution of each PC to the variability in the data.

 is a matrix of size  that captures the

environmental structure in the data. The rows of 

are the right singular vectors representing the

relationship among environments. The SVD

decomposition allows for a reduction in

dimensionality by retaining only the first  singular

values, corresponding left singular vectors from

󰇛󰇜, and corresponding right singular vectors

from 󰇛

󰇜. This provides an approximation of

the original data matrix:



, (4)

where  depends on how much variability in the

data needs to be captured. If  is a two-entry data

matrix with elements , where  rows

(genotypes) and  columns

(environments), which can be decomposed by SVD

in p principal components (PC):

 



 . (5)

The maximum number of Principal Components

(PCs) required to fully represent this matrix is 

󰇛 󰇜. Every PC is composed of a matrix

of scores , the environmental matrix , the

singular values , and  is the residual for

genotype  in environment  that is not explained by

the model. If  indicates that there is an

association between genotypes and, if ,

then there is an association between environments.

If there is no correlation between the environments,

all  PCs must be completely independent, and the

proportion of the total variation explained by each

PC must be exactly 1/p. When there is some

correlation between the environments, the

proportion of variation explained by the first CPs

must be greater than 1/p, and the variation explained

by the other CPs must be less than or equal to 1/p.

The model with some restrictions (

) and with orthonormality in scores , i.e,





 if and 



 if

, with the same restrictions for . When the

matrix  of rank p can be sufficiently approximated

by a rank 2 matrix (the first two components are the

most important since they explain the most variation

in the data) that is:

 +, (6)

It can be presented graphically in a biplot of

dimension 2 after partitioning into an appropriate

singular value:

 󰇛

󰇜󰇛

󰇜󰇛

󰇜󰇛

󰇜(7)

where 󰇟󰇠 is the singular value partition

factor (SVP). The biplot is constructed by graphing



as the abscissa, 

 as an ordinate for each

genotype, and at the same time plotting 

 as

abscissa and 

 as an ordinate for each

genotype, and at the same time plotting 

 as

abscissa and 

 as ordered for each

environment. The exponent  is used to resize the

row and column scores to improve the visual

interpretation of the biplot for a particular purpose.

In the context of MET data, singular values are

allocated entirely to genotype scores (row) if ,

i.e. genotype-centered singular value partitioning or

, or entirely to environment scores

(column) if  (environment-centered singular

value partitioning or ); and  will

allocate the square roots of the singular values to

both genotype and environment scores (symmetric

singular value partitioning or ). The

environment-focused model is important because

the correlation can be obtained through the cosine of

the angles, indicating the ability of environments to

discriminate. When the interest is in visualizing the

similarity between genotypes, then the genotype-

focused model is the appropriate way to visualize

the data. In GGE biplot analysis, the genotype-

centered and environment-centered singular value

partitions are used to evaluate genotypes and test

environments, respectively. An important property

of the biplot is that the approximation of any

element of the original matrix  of rank 2 can be

estimated visually by the inner product of genotype

and environment vectors, respectively, and the

cosine of the angle between them. This is known as

the inner product property of the biplot. By

examining the angles and lengths of vectors in the

biplot, we can interpret the nature and magnitude of

the genotype-environment interaction patterns.

Smaller angles indicate similar environments, while

larger angles indicate distinct environments.

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2.4 GGE Biplot Model

The GGE biplot graph was constructed according to

the methodology of [16], which considers the main

effect of genotype plus the GEI, is based on biplot

graphs, graphically representing a matrix of data.

The graph is constructed based on the first two

principal components of a PCA using local

regression models (SREG). The first principal

component represents the yield attributed to the

genotype, and the second principal component

represents the part of the yield due to the genotype-

by-environment interaction. Many statistical tools

and models have been put in place to analyze GEI

interaction effects under mega-environment

experiments. The idea of a mega-environment

refers to a large-scale environmental factor that has

a significant impact on the performance and

adaptation of different genotypes. In the analysis of

GGE biplots, when examining mega-environments,

the mean shown in the graph is not the global mean,

but rather the mean of the mega-environment. This

approach assists in identifying genotypes that have

wide or specific adaptation to environments or

groups of environments, [17]. The GGE biplot

model does not distinguish between the genotype

and GEI effects, as they are combined into two

multiplicative terms. This is because only G and

GEI are relevant for genotype evaluation. Therefore,

during this evaluation, the test environment, mega-

environment, and overall mean must be subtracted

from each element, leaving only G and GEI in the

double-entry table. Once the MET data is centered

around the environment and appropriately scaled, it

suffers a singular value decomposition and biplot

analysis. This specific mean provides more detailed

information compared to the general mean, enabling

the identification of genotypes that perform better in

particular clusters of environments or mega-

environments. In a MET dataset, each value in the

table typically represents the mean performance or

productivity of a specific genotype in a particular

environment. The relationship between the typical

production of a specific genetic composition in each

environment () and its constituent elements

(  󰇜, without considering any random

errors. The model can be described as follows:

 , (8)

where:

  is the mean yield of a specific genotype i

in environment j.

  is the overall mean yield across all

genotypes and environments.

  is the main effect of genotype i.

  is the main effect of environment j.

 represents the specific interaction

between genotype i and environment j (the

deviation from the main effects due to the

interaction).

This model illustrates how the mean yield in a

specific environment for a particular genotype is

composed of the overall mean yield (), the

genotype effect (), the environment effect (),

and the interaction effect (). This representation

simplifies the relationship between the components

to show their contributions to mean performance in

a given environment. In reality, this model doesn't

account for random errors or other factors that might

influence yield variations in a MET dataset.

However, it provides a fundamental understanding

of how the mean yield is formed from various

components in a genotype-by-environment study.

The equation in the SREG methodology referenced

in [18], likely adjusts the data by removing the main

effect of the environment (E) and the overall mean

from each entry in the dataset. This adjustment is

made to isolate and emphasize the genotype (G) and

GEI interaction effects in the analysis.

The equation mentioned involves a

transformation like the following

Adjusted_Value=Original_Value−Overall_Mean−E

nvironment_Effect

That can be rewritten as:

, (9)

This adjustment effectively removes the

contributions of the overall mean and the main

effect of each environment, leaving behind the

contributions solely related to the genotype and its

interaction with specific environments. In the

context of MET data analysis, an environment-

centered approach involves scaling the data

appropriately and then subjecting it to DVS

(Discriminant Variable Scaling). After scaling the

data using DVS, a biplot analysis is conducted (a

graphical representation that displays both

genotypes and environments simultaneously in a

single plot). This helps in visualizing complex

multivariate data and revealing patterns,

relationships, and interactions between genotypes

and environments. As mentioned before, in the

biplot, genotypes are represented as vectors,

indicating their performance across different

environments. The angles and lengths of these

vectors show how well a genotype performs in

various conditions. Environments are represented as

points, and their positions about the genotype

vectors indicate which genotypes perform well in

those specific environments. Patterns in the biplot

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can reveal clusters or mega-environments where

certain genotypes excel, assisting in the design of

strategies for specific environmental conditions. The

information ratio (IR) is a metric used to evaluate

the effectiveness of a biplot in representing the

patterns present in a double-entry table that contains

genotype and environment data. It assesses how

well the biplot captures the relationships between

genotypes and environments. To represent the major

patterns and relationships between genotypes and

environments in the dataset, the maximum number

of PCs to fully represent the double-entry table can

be calculated considering 󰇛 󰇜. This

ensures that the number of PCs used for analysis

doesn't exceed the smaller of either the number of

environments or one less than the number of

genotypes. To ensure that there’s no correlation

between the environments, each PC must be

completely independent, i.e., (1) all p principal

components must be entirely independent of each

other, in (2) the proportion of total variation

explained by each PC must be precisely 1/p. This

means that each PC accounts for an equal share of

the total variability present in the dataset. In this

scenario of correlation between environments, the

proportion of variation explained by the first PCs

may exceed 1/p, indicating that the first principal

components are capturing a greater amount of

variation than would be expected if they were

independent. However, the variation explained by

the subsequent PCs must be equal to or less than

1/p. This means that although the first PCs capture

more variation than expected, the remaining PCs

have a smaller or equal share of variation, keeping

the proportion balanced. This situation can occur

when environments are correlated, meaning that

there are similar patterns or consistent relationships

between the different environments. This leads to a

higher concentration of variation in the first PCs,

since these components capture the most significant

and dominant patterns in the data set, that is, when

there is a correlation between environments, the first

PCs may explain more variation than expected,

while subsequent PCs maintain a lower or equal

proportion of variation than expected, helping to

highlight the most relevant patterns in the data. The

IR can be calculated for each PC, which is the

proportion of the total variation explained by each

PC multiplied by p. The interpretation is as follows:

a PC with  signifies that it contains

substantial patterns or associations between

environments. This indicates that the PC captures

meaningful information and structures within the

data, and a PC with  suggests that it doesn’t

contain significant patterns or substantial

information regarding the relationships between

environments. Such a PC might not be as relevant

for representing the data's patterns. The dimension 2

biplot adequately represents the patterns in the data

if only the first two PCs have an .

3 Results and Discussion

3.1 Durum Wheat Yield Data

All comparisons presented in this paper are

illustrated with a data set resulting from a breeding

program in Portugal, carried out by the Portuguese

National Plant Breeding Station (ENMP, Elvas) in

the years 2015/2016 and 2016/2017. It contains the

yield from 22 genotypes of durum wheat (Triticum

turgidum L., Durum Group), measured in 6

environments (Benavila; Revilheira; Évora; Elvas;

Beja; Tavira) and performed in complete

randomized blocks with four replicates. Also, they

need to look over if the in-text citations of the

Tables, Equations, and Figures are properly

connected with the Tables, Equations, and Figures.

These environments were obtained in two years and

the locations were the same in both years. All the

locations in this data set are in south Portugal,

Tavira being at the seaside (Algarve) while the

remaining in the inland (Alentejo). The 22

genotypes are labeled G1 to G22. All the analyses in

this work were carried out using computer routines

implemented in the R software (R Development

Core Team, 2019). Interpreting the GGE biplot

graphs allows us to assess the stability of genotypes

in environments by observing the magnitude and

sign of the genotype and environment scores for the

interaction axis (Table 1). Therefore, low scores

(close to zero) represent the most stable genotypes

and environments. In Table 1, genotypes G13, G14,

G22, G11, G4, G8, and G5 showed scores closest to

zero, so they are the most stable genotype in this

data set.

Table 3 shows the singular values of the six

principal components (PCs), the variations

explained, and the information ratio (IR) and the

values of the information ratio (IR), proportion

explained, and singular values of the six principal

components (PCs). According to the information

ratio (IR) of the four components (Table 2), only the

first two PCs contain patterns (;

). Therefore, the biplot is considered

adequate to represent the patterns in the data. The

interaction variance explained by PC1 and PC2 is

44.52% and 24.93% respectively. The abscissa of

the biplot shows the CP1 scores and the ordinate

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shows the CP2 scores of the genotypes and

environments.

Table 1. Coordinates of the abscissa axes (PC1) and

coordinates (PC2) for the GGE biplot graphs

Genotype

GGE Biplot

PC1

PC2

-0.89

-1.46

0.09

0.12

-0.93

0.60

0.03

0.26

-0.07

-0.33

-0.06

-0.32

2.89

0.95

0.08

-0.21

-2.21

0.71

G10

-0.04

-0.24

G11

-0.64

1.76

G12

0.08

-1.37

G13

-0.01

-0.22

G14

-0.05

-0.18

G15

2.45

-2.19

G16

0.30

-2.43

G17

-2.21

-0.87

G18

-1.15

-0.54

G19

1.40

0.92

G20

-2.34

1.65

G21

0.56

-2.71

G22

0.06

0.28

Table 2. Coordinates of the abscissa axes (PC1) and

coordinates (PC2) for the GGE biplot graphs

Environments

GGE biplot

PC1

PC2

BEN

-4.54

-3.23

ELV

-1.45

-0.64

REV

-2.65

-1.94

EVO

-0.87

2.31

TAV

-0.3

-0.58

BEJ

-0.33

1.40

BENAVILA-BEN; ELVAS-ELV; REVILHEIRA-RER; ÉVORA-

EVO; TAVIRA- TAV; BEJA-BEJ

Table 3. Singular value, proportion explained, and

information ratio (IR) of the six principal

components (PCs)

Singular value

Variance

explained

(%)

21.18

44.52

2.14

18.30

24.93

1.06

17. 67

22.45

0.92

15.54

17.66

0.82

12.32

15.89

0.65

11.69

13.27

0.50

A key aspect of biplot analysis, where the

straight line extending from the origin of the biplot

to the positioning of an environment or genotype is

termed the environment vector or genotype vector.

These vectors facilitate the visualization of specific

interactions between genotypes and environments,

such as the performance of each genotype in various

environments. The interpretative guideline is as

follows: (1) a genotype's performance in an

environment is deemed better than the mean if the

angle between the genotype vector and the

environment vector is less than 90º; (2) conversely,

it is considered worse than the mean if the angle

exceeds 90º (since the cosine of an obtuse angle is

negative) and an angle approximately equal to 90º

indicates performance close to the mean.

3.2 Selection of Genotypes with Stable

Performance Across Mega-environment

The mega-environment analysis was based on the

GGE biplot in Figure 1. The vertices of the polygon

are formed by the genotypes: G1, G6, G6, G9, G11,

G15, and G20. The 22 genotypes were formed into

six groups by the lines coming from the origin of the

biplot, and the four mega-environments were also

formed by (i) ELV; (ii) REV and TAV; (iii) BEN

and BEJ; (iv) EVO.

Fig. 1: GGE Biplot for corn yield data (Kg/ha) and

genotypes that performed best in six different

environments

Genotypes G11 and G20 are the vertexes of the

sector in which the mega-environment is formed by

the BEJ and BEN environments, so they are the

genotypes that perform best in this environment;

G15 is the vertex in the sector in which ELV is

placed, so it is the most adapted genotype in this

environment (Figure 1). G9 is the vertex in the

sector in which EVO is placed, so it is the most

adapted genotype in this environment. Os genótipos

G6 e G1 colocados nos ambientes TAV e VER são

o mais adaptados nestes ambientes. In the sectors of

G5 and G2, which do not contain environments, this

means that these genotypes were not productive in

any environment, i.e. these genotypes are the worst

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in terms of productivity in some or all

environments.

3.3 Stability and Adaptability Analysis

The GGE biplot, especially when focusing on mean

versus stability, provides a comprehensive view of

genotype performance in different environments.

This information is valuable for plant breeders and

researchers in selecting genotypes that not only

perform well on the mean but also demonstrate

stability across diverse cultivation conditions. In the

GGE biplot, the environmental axis (EAM) is

represented by a straight line with a single arrow

passing through the origin of the biplot. This line

points towards the mean environment and indicates

the direction of higher mean performance for the

genotypes. When another line with two arrows

passes through the origin of the biplot and is

perpendicular to the EAM, it means that this line

represents greater performance variability (less

stability) in both directions, with the arrows

indicating the extent of this variability. Therefore,

Genotypes closer to the EAM are considered to have

stable performance across environments, and

genotypes located along the line perpendicular to

the EAM may exhibit higher variability in

performance, indicating lower stability, [18], [19].

Figure 2 shows the productivity of the genotypes

using the EAM. The small circle represents the

"mean environment" and is defined by the mean

coordinates of all the biplot test environments. In

this biplot, the singular values are completely

compartmentalized for the genotype scores.

The 22 genotypes can be classified according to

their mean productivity, thus, G15 > G12> G19 >

G8 > G11 >G4>G13>... > G10 > overall mean > G9

> G3 >... >G2 > G6; G15 was highly unstable, with

lower than expected yields in the BEJ and BEN

environment, producing relatively well in the ELV

environment; G8 proved to be stable with slightly

above-mean productivity. G11 was not stable but

performed well compared to other genotypes in

terms of mean productivity, although it produced

well in the BEN and BEJ environments; G8 was

stable and showed slightly above mean productivity;

G5 showed above mean productivity (fourth best)

and was the most stable (Figure 2). It should be

noted that if the biplot explains only a proportion of

the total variation, some stable genotypes may not

be truly stable, and their variations cannot be fully

explained in this biplot.

An ideotype, represented in Figure 2, can be a

point on the MAS (center of the concentric circles),

in the positive direction, and has a vector length

equal to the longest vectors of the genotypes on the

positive side of the MAS, i.e., highest mean

performance (Figure 3). Therefore, the genotypes

located closest to the center of the concentric circles

are preferable to the others. The best genotypes

were G13 and G2, although G15, G11, G20, G10,

and G18 showed higher mean yields, they were not

stable. Genotype G22 was more stable than G15.

The least recommended genotype was G6 while

genotype 14 proved to be highly stable, but this does

not mean that its yield was good, it just means that

its relative performance was consistent, but far from

being an ideal genotype. Genotypes G14, G3, and

G22 proved to be highly stable, which doesn't mean

that they had good yields, it just means that their

performance was consistent and therefore not ideal.

According to [20], it only makes sense to talk about

stability when it is associated with the mean

performance of genotypes. Thus, a stable genotype

is only preferable when it has a high mean

performance.

Fig. 2: Genotype Plus Genotype-Environment

Interaction biplot

The representativeness of the environments can

be gauged by considering the angles they form with

the EAM, i.e. environments that have smaller angles

with the EAM will be the most representative. Of

the environments shown in Figure 2, those that form

a smaller angle with the EAM are the TAV and

REV environments, which can be considered ideal

for selecting adapted genotypes. The EVO, BEM

and ELV environments can be useful for selecting

unstable genotypes if a single mega-environment is

considered.

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Fig. 3: The GGE Biplot with the environment-mean

axis (EAM) to classify the genotypes about the

ideotype (in the center of the concentric circles)

3.4 Joint Regression Analysis and the Zigzag

Algorithm

The zigzag algorithm, proposed by [20], is designed

for the joint regression analysis of data in a two-way

array with I rows and b columns. This algorithm is

an extension of the joint regression model but with a

modification in the computation of environmental

indexes. Considering that data is organized in a two-

way array with I rows (genotypes) and b columns

(blocks or replicates), and assuming that 

represents a continuous response variable (e.g.,

yield) for genotype i in block j, the joint regression

model can be described as follows:

 󰇛󰇜 (10)

(󰇜

 is the grand mean

 are the genotype and environment

main effects

 󰇛󰇜 is the interaction and  is the

residual

The yield vectors are assumed to be independent,

normal, and homoscedastic, and genotype i is

present in block j. The use of environmental indices

for individual blocks instead of one per environment

has improved the accuracy of the analysis of

randomized block experiments. This results in K

experiments, each with b blocks, leading to Kb

supporting points per regression instead of K points

used by the classical Finlay-Wilkinson joint

regression model. When we intend to estimate some

stability parameters that will allow us to make

comparisons between varieties it’s possible to use

the sub-model:

󰇛󰇜 , (11)

where  is a linear regression coefficient for the i-th

genotype and  a deviation (unexplained 

interaction). The JRA model can be written as:

 󰇟󰇠󰇟󰇠  (12)

󰇟󰇠



where  comprises both the unexplained 

interaction and the experimental error. We assume

fixed genotypic and environmental effects and

random residual term. This model doesn’t consider

the block effects because instead, it uses the blocks

as environments. This type of design involves

randomized blocks, where treatments correspond to

J genotypes, and are commonly used in experiments

to reduce variability by blocking sources of

variation that are not of primary interest. The

environmental index is measured by the mean yield

within each block of the experimental design. This

index represents the environmental conditions in

which the experiment is conducted. For each of the

J genotypes, a linear regression of yield on

environmental indexes is performed. This way the

relationship between the yield of each genotype and

the environmental conditions are studied.

3.5  Environmental Indexes and the

Zigzag Algorith

Considering the model given in (13), where

󰇱



,

the environmental index corresponds to blocks

instead of environments, b represents the number of

blocks,  is a continuous response (e.g. yield) for

genotype i in block j if present, and the pairs (,

),  are the regression coefficients (the

slopes), for the I genotypes. To estimate the

regression coefficients and environmental indexes,

we wish to minimize the function:

 (13)

  







 ,

where  is the weight of genotype i in block j. If

the genotype is absent, we take  . When the

genotype occurs, we take  .

The weights  are allowed to differ from block to

block. To overcome the difficulty of estimating the

model parameters, the zigzag algorithm, is very

efficient in finding the estimates of (, ), 

 and . This flexibility of the

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algorithm allows express of the differences in the

representativeness of blocks. If there are multiple

blocks in the same location, their weights are set to

be the same. This implies that within a given

location, all blocks are considered equally

representative. The weights are simplified to binary

values: 1 and 0. The use of binary weights implies a

simple weighting scheme where some observations

are given full weight (1) while others are completely

disregarded (0). The decision to use binary weights

(1 and 0) occurs when are a lack of information

about the relevance or importance of individual

blocks. In situations where the significance of

blocks is unknown, using binary weights can be a

straightforward and pragmatic approach. The binary

weighting simplifies the implementation of the

algorithm and might be suitable when detailed

information about block importance is not available,

and their use does not require additional estimation

of weight values for each block, making the

algorithm computationally less demanding. This

algorithm extends the joint regression model for

analyzing data in a two-way array. It modifies the

computation of environmental indexes, aiming to

improve the precision of the analysis compared to

the classical Finlay-Wilkinson model. The zigzag

algorithm can be described as follows

(i) Calculate initial values for the

environmental indexes  within the

interval 󰇟󰇠, where 

󰇛󰇜 and 

󰇛󰇜.

(ii) Repeat the following steps until successive

sums of squares of weighted residuals differ

by less than a fixed value:

 Minimize the loss function concerning

 and  iteratively.

 Minimize the functions 

 for each   to

obtain a new vector of environment

indexes 󰇛

󰇜

 Standardize the vector of environmental

indexes to keep the range unchanged,



󰇛

󰇜, 

󰇛

󰇜

 Update the environmental indexes to

obtain the vector .

󰇛



󰇜





󰇛



󰇜







(iii) Check whether successive sums of squares

of weighted residuals differ by less than a

fixed value.

(iv) At the end of each iteration, standardize the

adjusted environmental indexes to ensure

the range does not change from iteration to

iteration.

(v) The procedure is carried out until the goal

function stabilizes.

The environmental indexes adjusted in this way

are called  environmental indexes, because the 

norm was used.

3.6 Upper Contour

When two regression lines representing genotypes

intersect, it indicates that one genotype performs

better under higher environmental indexes, while

the other is preferable for lower environmental

indexes. This suggests a trade-off or specialization

of genotypes based on environmental conditions.

The upper contour of the JRA is described as a

concave polygonal shape, formed by segments of

adjusted regression lines. This contour contains the

higher adjusted yields for environmental indexes.

Each segment corresponds to a range of

environmental indexes in which a particular

genotype will have the maximum adjusted yield.

These genotypes are called "dominant" and are

considered for selection. The Genotypes within the

upper contour of the JRA are considered dominant

and exhibit the maximum adjusted yield for specific

ranges of environmental indexes. These dominant

genotypes are candidates for selection in breeding

programs. The non-dominant genotypes are

compared with the dominant ones to assess if they

are dominated across the entire range of adjusted

environmental indexes. If a genotype is found to be

dominant across the entire range, it can be safely

discarded from the breeding program. Figure 1

shows the graphical representation of the dominated

genotypes (G19, G15, G14, G7, and G8), together

with the ranges of dominance in the durum wheat

population. The abbreviations for the 6

environments are placed on the axis of the

environmental indexes (REV; EVO; BEN; ELV;

BEJ; TAV).

Table 4 and Table 5 present a comparison

between the method based on the joint regression

model i.e., the regression analysis of the mean yield

of individual genotypes on the overall mean of the

trial and the double minimization algorithm. The

estimates of intercept slope and the coefficients of

determination and the zigzag and double

minimization algorithms are also analyzed. To best

compare these procedures, it is important to analyze

the slopes and coefficients of determination, and the

results show that the results are practically the same

regarding the ordering of the genotypes per slope.

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The coefficients of determination are mainly

similar, moreover, the zigzag and double

minimization completely agree and may be seen as

the most suited for regression analysis of complete

randomized blocks because of their convergence to

the minimum of the loss function (14).

Fig. 4: Upper contour with the five dominant

genotypes in the durum wheat population

Table 4. Adjusted regression coefficients and

coefficients of determination, as evaluated by the

two procedures and two algorithms

Finlay and Wilkinson (1963)

Genotype

Intercept

Slope



-0.612

1.529

0.981

-0.456

1.454

0.874

-0.351

1.376

0.843

0.067

1.397

0.910

-0.140

1.254

0.865

-0.160

1.243

0.871

-0.363

1.240

0.976

0.456

1.231

0.887

-0.643

1.161

0.998

G10

0.773

1.153

0.867

G11

0.854

1.140

0.823

G12

-0.234

1.113

0.890

G13

-0.525

1.081

0.993

G14

0.354

1.056

0.843

G15

-0.871

1.053

0.854

G16

0.892

1.022

0.889

G17

-0.076

0.985

0.841

G18

0.242

0.872

0.776

G19

-0.127

0.645

0.753

G20

0.425

0.560

0.614

G21

-0.077

0.542

0.611

G22

1.284

0.501

0.607

The sums of the sums of squares of residuals and

the double minimization algorithm were calculated,

267.4 and 212.6 respectively, showing that the

zigzag and double minimization algorithms as better

results since the algorithms induce lower sums of

the sums of squares of residuals. From these results,

it’s possible to conclude that all the obtained

environmental indexes are highly correlated

(minimum of 0.988). At this point is important to

note that the coefficient of correlation is 1 when the

algorithms are applied, being slightly higher than

those found using other method. After establishing

the upper contour, non-dominant genotypes should

be compared with the dominant ones. The genotypes

whose regression lines contribute to the upper

contour are the so-called dominant genotypes. Each

of them is associated with a range of variation in the

environmental index.

Table 5. Adjusted regression coefficients and

coefficients of determination, as evaluated by the

Zigzag and Double Minimization algorithm

Zigzag and Double Minimization

Genotype

Intercept

Slope



-0.712

1.629

0.881

-0.545

1.542

0.876

-0.343

1.436

0.847

0.070

1.388

0.915

-0.164

1.234

0.869

-0.170

1.267

0.877

-0.623

1.245

0.996

0.546

1.237

0.884

-0.463

1.172

0.992

G10

0.779

1.168

0.874

G11

0.884

1.162

0.826

G12

-0.324

1.110

0.891

G13

-0.528

1.092

0.990

G14

0.356

1.060

0.865

G15

-0.773

1.059

0.859

G16

0.882

1.026

0.898

G17

-0.066

0.988

0.877

G18

0.262

0.873

0.826

G19

-0.173

0.565

0.762

G20

0.456

0.554

0.664

G21

-0.081

0.533

0.641

G22

1.362

0.521

0.619

The other genotypes must be compared with the

dominant genotypes. Table 3 shows the results of

the t-tests, Scheffé multiple comparison tests, [21],

and Bonferroni multiple comparison method used to

select genotypes and to compare the rate

corresponding to the dominant genotype on the

right, or the genotype with the steepest slope, with

the others, applied to our data, [22], [23], [24]. The

section to the right of the upper contour corresponds

to the genotype that responds best in highly

productive situations. The hypothesis that was tested

was the equality of 

 coefficients, whereby the

dominant cultivar will be superior to the others. The

bounds of the environmental indexes 2.34 and 10.43

(Table 4 and Table 5, complete data) are kept

unchanged by the zigzag algorithm and correspond

to the lowest and highest mean yield of all the

blocks.

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Cristina Dias, Carla Santos, João Tiago Mexia

E-ISSN: 2224-3496

1485

Volume 19, 2023

Table 6. Dominant and number of significantly dominated genotypes for JRA, environments where the

genotypes were dominant (JRA), and where the genotypes were winners (GGE). The results are for the

complete data set. Abbreviations for the environments: REV; EVO; BEN; ELV; BEJ; TAV

JRA

GGE

Dominant

Winner

genotype

Range of

dominance

Number of significantly dominated

genotypes

Environments

t test*

t test**

Scheffé*

Bonferroni*

󰇟󰇠

REV

󰇟󰇠

󰇟󰇠

EVO; BEN; ELV;

BEJ

ELV; BEJ;

G11

󰇟󰇠

REV; EVO; ELV;

BEJ; BEN, TAV

REV; ELV; BEN;

ELV; BEJ; TAV

󰇟󰇠

REV; EVO; ELV;

BEJ; TAV, BEN

REV; EVO; BEN;

ELV; BEJ; TAV

G22

󰇟󰇠

REV; EVO; ELV;

BEJ; TAV, BEN

REV; EVO; BEN;

ELV; BEJ; TAV

G14

󰇟󰇠

REV; EVO; ELV;

BEJ; BEN, TAV

REV; ELV; BEN;

ELV; BEJ; TAV

G13

󰇟󰇠

REV; EVO; ELV;

BEJ; TAV, BEN

REV; EVO; BEN;

ELV; BEJ; TAV

4 Conclusion

Figure 4 shows the six environments placed on the

axis of the environmental indexes. The first three

environments, namely Rev, Evo, and Ben, have

higher yield with the genotype G5, and the

remaining eight environments have better

production with the genotypes G13, G14, G22, G2,

G11, G4, G8 and G5. We may conclude that G13,

G14, G22, G2, and G11 win in all environments, G4

wins in environment ELV and BEJ, and G5 wins in

environment REV. Both analyses reveal that the

G13 and G14 are the universal winners (Table 6)

and also are the strongest genotypes regarding yield

production. The main conclusion is that these two

genotypes (G13 and G14) are always dominant for

higher environmental indexes and always win one

mega-environment. The aim was to compare the

final results (i.e. dominant/winner genotypes and

environments where they were dominant/winner)

between JRA and GGE biplot. The main

conclusions were the similarity between the

dominant genotypes in JRA and the winners of the

mega-environments in the GGE biplot analysis. The

results from JRA tend to be more significant than

those from the GGE biplot model in this kind of trial

because the genotypes in the data set have proved to

have strong adaptability.

In our ongoing and upcoming research, we plan

to conduct additional statistical analyses.

Specifically, we aim to compare the models

introduced in this study with other existing models,

including the AMMI model (Additive Main effects

and Multiplicative Interaction). This comparison

will involve assessing our models alongside

established methodologies such as Joint Regression

Analysis, AMMI, and GGE biplot.

Acknowledgement:

This work is funded by national funds through the

FCT - Fundação para a Ciência e a Tecnologia,

I.P., under the scope of the project

UIDB/00297/2020 (Center for Mathematics and

Applications

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WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT

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WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT

DOI: 10.37394/232015.2023.19.134

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed to the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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(Attribution 4.0 International, CC BY 4.0)

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Creative Commons Attribution License 4.0

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WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT

DOI: 10.37394/232015.2023.19.134

Cristina Dias, Carla Santos, João Tiago Mexia

E-ISSN: 2224-3496

1488

Volume 19, 2023