Copyright: Weikai Yan


CONTENT
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•	Install GGEbiplot

•	Introduction to Biplot Analysis

•	Get Started with GGEbiplot

•	Function Organization in GGEbiplot

•	Functions for Various Types of Data

Important New Additions:

•	Experimental Design Master

•	Multi-Trait Selection Master

•	Breeder's Toolbox (new)

•	Versatile ANOVA Panel

•	Database management

____________________________________________________________

Author: Dr. Weikai Yan

Web:  www.ggebiplot.com

Contact: support@ggebiplot.com


Entry stratification

Entry stratification consists of a groups of functions for generating subsets of entries based on their placement in the current biplot. The following options are available:

•	Entries with long vectors

When this function is invoked, GGEbiplot first calculates the vector length of all Entries, and then ask the user to set a criterion, against which Entries with LONGER vectors are selected and retained. A new biplot based on the retained subset of Entries will be generated. Entries with longer vectors are more responsive to the (change of) Testers.

•	Entries with short vectors

When this function is invoked, GGEbiplot first calculates the vector length of all Entries, and then ask the user to set a criterion, against which Entries with SHORTER vectors are selected and retained. A new biplot based on the retained subset of Entries will be generated. Entries with shorter vectors are less responsive to the (change of) Testers. If the Testers are of the same units, then less responsive entries can be regarded as more "stable".

•	Entries within a sector

When this function is invoked, GGEbiplot first calculates the angles of the Entries, relative to an arbitrary reference point. It then asks the user to set a sector defined by an upper and a lower bound; all Entries WITHIN this sector will then be retained and a new biplot generated.

•	Entries outside a sector

When this function is invoked, GGEbiplot first calculates the angles of the Entries, relative to an arbitrary reference point. It then asks the user to set a sector defined by an upper and a lower bound; all Entries OUTSIDE this sector will then be retained and a new biplot generated.

•	Vertex entries only

When this function is invoked, GGEbiplot first identify entries that would be the vertex entries in the which-won-where view, and then generates a biplot based on these entries. The vertex entries are the most responsive entries. They are either winners or losers relative to some or all of the testers.

•	Entries with high means

When this function is invoked, GGEbiplot first calculates the means of the entries across all Testers and the entry means are scaled to 1 (maximum) to 0 (minimum). It then asks the user to set a criterion against which entries with higher means are selected. This function may not make sense when the entry main effect (G) is too small relative to the interaction (GE). When this is the case, there will be wide angles (i.e., negative correlations) between the Testers.

•	Entries with low mean

When this function is invoked, GGEbiplot first calculates the means of the entries across all Testers and the entry means are scaled to 1 (maximum) to 0 (minimum). It then asks the user to set a criterion based on which entries with lower means are selected. This function may not make sense when the entry main effect (G) is too small relative to the interaction (GE). When this is the case, there will be wide angles (i.e., negative correlations) between the Testers.


Breeder's Toolbox - in addition

This group includes the following applications:

•	Next generation renaming

•	Cross renaming

•	Test line renaming

•	File (picture) renaming


Cross renaming

When making crosses, the crosses many named sequentially. There might be duplicated crosses; there might be reciprocal crosses. This function was designed to identify unique crosses and reciprocal crosses and rename them systematically. Unsuccessful crosses, i.e., crosses without any hybrid seeds, will be omitted. Crosses with the same female by male combination will be given the same cross name; reciprocal crosses will be given the same cross name with a different postfix.


Offspring Naming

In the pedigree breeding method, selections are conducted in each generation, and selected lines are named by adding an extension after the name of the previous line name. This function was designed to automatically add an extension to selected lines. The necessary information required for this process is the line name of previous generation and the number of lines selected for each line. The so generated file can be used directly in printing labels (tags) for the preparation of next seeding.

This function is also useful for printing tags when the same variety (line) are to be grown in multiple pots or experimental units. For this function, you need two columns in your input file, one listing the entries and the other is the number of pots (units) to be planted for each variety.


Install GGEbiplot

Please do the following:

1. Obtain your license (usually you will be given an internet link to download if you have ordered GGEbiplot. Contact to order: support@ggebiplot.com)

2. Download and install the beta GGEbiplot: www.ggebiplot.com/GGEbiplotBeta.msi (about 3.2 MB). Skip this step if you have run GGEbiplot on your computer before.

3. Extract your GGEbiplot file to C:\Program Files\GGEbiplot\ to replace the existing file.

4. Start GGEbiplot from the shortcut on your desktop.

Hint:

1. If your internet speed is slow, please make sure that you have downloaded the full packages, not just part of it.

2. You can get the full package of Help from this location: www.ggebiplot.com/GGEbiplothlp.zip (about 3 MB).


Test line renaming

In a breeding program, the breeding lines may need to be renamed to shorter, more meaningful names when they reach the stage of yield testing. Before that stage they are usually named by their pedigree history. The convention is to give a cross a uique sequential number after a short name for the breeding institution (e.g., OA may stand for "Ottawa Avena"), and breeding lines to be advanced from the same cross is numbered by the cross name plus an extension of "-1", "-2", etc. This function was designed for this purpose.


File (picture) renaming

Modern plant breeders and geneticists who conduct variety trials not only take notes on various traits about each variety (field plot), but also take pictures of each variety at different times, which contain much information about the varieties and complements with the numerical/text data. The pictures are initially numbered by the camera as automatic numbers and the researcher often has to rename them such that the file names are more meaningful. It can be a tedious and painful process if a large number of pictures are to be renamed. For example, one of our oat variety trials contains more than 300 breeding lines and we have taken pictures for each line at several times. This file remaning tool was created to solve this problem.

The New picture names can be created from a file (a field book)), by combining information from different columns in the file. In the above example, the pictures are renamed by combing the Variety name + Plot ID + Variety entry ID in the above example. The user has an option to decide the order of columns to form the name combination. There are no limitations on the number of columns to form a picture name.


Database management

The crop data management system DUDE reported by Tinker and Yan (CJPS 2006) and Yan and Tinker (Agron J 2007) has become a componnent in GGEbiplot software. It has the following key functions:

1. Convert original data into a MS Access based relational database (called COOL or Context Oriented Observation Library);

2. Extract any part of the data from the database and output it in several formats for data summary and analysis; and

3. Unification of heterigeneous data.


Introduction to biplot analysis

Visit www.ggebiplot.com for the most recent development

•	What is a biplot

•	What is a GGE biplot

•	How a biplot is constructed

•	Principles of biplot analysis

•	Questions to be asked before trying to interpret a biplot

•	References on biplot analysis

•	Why use GGEbiplot software

•	Biplot analysis vs. formal statistical analysis


Four questions before trying to interpret a biplot

When a biplot is generated, the following questions must be asked before trying to interpret it:

•	What is the model based on which the biplot is generated

This determines what questions can be asked of the biplot and what cannot. In the above biplot, the model is based on Scaling = 0 (original data not scaled or divided by anything) and Centering = 2 (tester-centered). And, the data was not transformed (Transform = 0). Therefore, this is a "GGE biplot" without scaling. It has all the interpretations of a GGE biplot.

In Depth: Various biplots for analyzing a genotype-by-environment table.

Even More: All possible biplots based on a single two-way table

•	What is the method of singular value partitioning (SVP)

This is crucial if the purpose is to visualize the relationships among entries or that among testers. In the above biplot, SVP = 1 (entry-focused singular value partitioning). It is therefore more appropriate for entry evaluation but not so for tester evaluation. However, the choice of SVP does not affect the interpretation of entry by tester interactions.

•	What is the goodness of fit

The higher the goodness of fit of the biplot, the more true to the data of the biplot patterns. If only a small portion of the variation is explained, the pattern in the data  is either complicated or there is no discernible pattern at all. The above biplot explained 89% of the tester-centered data (G+GE), not of the original data.

In Depth: How many PC's are required to fully explore the data

In Depth: What if two PC's are not adequate for displaying the two-way table

In Depth: Statistical Test of a biplot pattern, which is a related but different issue than Goodness of Fit.

•	Are the axes drawn to scale

If the axes are not drawn to scale, the relationships among the entries and testers are distorted, which can be misleading. A biplot generated by GGEbiplot is always drawn to scale. However, watch out when reading journal papers and see if a biplot is stretched to fit the space. If so, the biplot may be misleading. It will be difficult to judge wether the biplot is drawn to scale if the biplot does not show scales on its axes. One has to be cautious when interpreting such a biplot.

All these pieces of information are explicitly indicated on the top-left corner of the biplot:

1) Goodness of fit: PC1= 63%, PC2 = 26%, Total = 89%;

2) Data transformation method: Transformation = 0;

3) Data Scaling method: Scaling = 0;

4) Data Centering method: Centering = 2;

5) Singular value partitioning: SVP = 1; and

6) Drawn to scale: It is drawn to scale, which is always true in biplots generated by GGEbiplot.


Statistical test of a biplot pattern

Statistical test of a biplot pattern is a different issue from the Goodness of fit of the biplot. When the biplot has high goodness of fit, strong patterns in the biplot may be more likely statistically significant. However, even when the biplot has perfect goodness of fit, the patterns in the biplot may still not be statistically significant. On the other hand, strong pattern in a biplot may still be statistically sound even when the goodness of fit of biplot is poor.

It is important to understand that the goodness of fit or adequacy of the biplot determines whether the biplot is true to the two-way table it tries to approximate, and true to data does not necessarily indicate the statistical significance of a biplot. This goodness of fit bears no relevance to whether the data are replicated or not, whereas statistical test of a biplot pattern must be tested by replicated data or mega-data. When testers are treated as replicates, then the two-way data may be regarded as mega-data, even if there are no replications within each tester.

When a biplot pattern is considered important, a testable hypothesis should be first formulated and then tested using replicated data or multi-year data using conventional statistics. Validation using multi-year data is much more important than anything else.


What if two PC's are not sufficient?
What if two PC's are not sufficient

When more than two PC's are required to approximate the two-way table that is to be studied using biplots (i.e., the two-way table after appropriate transformation, centering, and scaling), what should one do? We offer two suggestions.

1) Try to divide the Testers into subsets based on the pattern in the biplot, and investigate the subsets in separate biplots.  The "Data Stratification" modules provide convenient ways to generate a biplot based on any subset of the Testers. Yan and Tinker (2005) have used this strategy in QTL by environment data analysis.

2) Try to examine a rotating 3-D biplot. Such a biplot should be sufficient to reveal the most important patterns in most cases. However, a static 3-D biplot, i.e., a 3-D biplot printed in a 2-D plane, is usually more misleading than helping.


Missing Value Estimation

GGEbiplot version 7 has an extremely important addition; it is a newly developed procedure to estimate missing values (Yan, submitted, 2012). This prcedure can be briefly described as below. First, the missing cells are filled with 0 (for column centered models used in constructing GGE biplots). This "complete" two-way table is then subjected to SVD, and the missing values are predicted by the resulting first two principal components. These predicted values are then used to fill the missing cells in the original two-way table, and a 2nd SVD estimation is performed and new predicted values for the missing cells obtained. This process is repeated until the predicted values are sufficiently similar to those in the previous iteration. This procdure proves to be effective in estimating missing values and recover data patterns when missing values are fewer than 40% of the data.

The most important application of this procedure is to analyze unbalanced multi-year data in a single biplot.  The use of this procedure enables unbalanced multiple year data to be effectively analyzed in a single biplot. As a result, it allows genotypes not tested in the same environments to be reasonably compared, and genotypes that have not fully tested to be critically evaluated.  This procedure is regarded as extremely important because requiring a balanced two-way table is a key limitation to the use of biplot analysis.

When a unbalaced two-way table is analyzed in a biplot, GGEbiplot will ask the user if the missing values should be estimated. The user can choose 'yes' or 'no' or 'no' first and then 'yes' so that the results can be compared.


How many principal components are needed?
How many principal components are needed

This can be investigated from three perspectives:

I suggest to look at the Information Content of each PC, which is printed in the log file after each biplot is generated, which looks like:

PC	Singular Value	% of Total SS	Information Ratio (IR)
1	19.664	62.9	8.2
2	7.833	10.0	1.3
3	6.912	7.8	1.0
4	4.536	3.3	0.4
5	3.449	1.9	0.2
6	1.628	0.4	0.1

A PC is information rich if its IR is greater than 1. In the above example, it appears 2 PC's are sufficient but there is still some indedent formation in PC3. Experience indicates that usually 3 PC's, thus a 3-D biplot, are sufficient for most datasets.

___________________________________________________

The following may also be relevant for this question:

1) One approach is to examine the biplot based on Scaling = 1, i.e., tester-standard-deviation scaled. In such a biplot, all testers are assumed to be equally important (with a variace of 1). Therefore, all testers should have vectors of equal length if the biplot adequately approximates the tester-standardized data. For the same reason, if there are testers that have considerably shorter vectors than others, then the biplot does not fully display the patterns with regard to these testers.  The following GGE biplot is based on this scaling for the genotype by environment dataset:

It can be seen that all testers have almost the same vector length (except for Tester 3), indicating that the biplot is adequate. However, this may not relevant to the adequacy of a biplot that is based on other sacling method.

2) Another approach is to examine a secondary biplot, in addition to the primary biplot of PC1 vs. PC2, and see if there are any additional, strong patterns. There should be no clear pattern in a secondary biplot if the primary biplot has sufficiently summarized the data. For example, the biplot of PC3 vs. PC4 in this example does not reveal any obvious patterns (Except that Tester 3 seems to be different from all other testers), suggesting that the primary biplot is adequate, the same conclusion as that from the first approach:

3) Along this line, if the primary biplot does not reveal any patterns, that means that there are no patterns in the data, and the question whether the biplot adequately displays the patterns of the data becomes irrelevant, and the searching for patterns should stop.

In depth: What if two PC's are not sufficientlinkID=32}


Information ratio

The Information Content (IC) of a PC is defined as

IC = % SS explained by a PC * Number of Testers

A PC is regarded as needed (information rich) if its IC > 1.


What are patterns?
What are patterns

Biplot analysis is also called "pattern analysis", but what is a 'pattern'

•	A pattern is anything that does not appear to be random; it is any apparent relationship (similarity or dissimilarity) among the entries or among the testers.

•	For a GGE biplot,  two types of pattens are of greatest interest:

1) the differences between genotypes, and

2) the correlations among test enivronments.


Principles of biplot analysis

The biplot has many utilities, and most of these utilities are based on the following two basic principles:

1. The inner-product property

A biplot is a graphical displays of a two-way table. If the two-way table is 100% displayed by the biplot, then there is a strict trimetric relationship between the value of entry i in tester j, Yij,  and their coordinates in the biplot. Verbally, the value of Entry i in Tester j is the vector length of Entry i (OGi) multiplied by the vector length of Tester j (OEj) and multiplied by the cosine of the angle between Entry i and Tester j (aij). This property allows entries to be compared visually within a tester or across testers. No other graphs, including the famous "AMMI1 biplot", have this property.

2. The cosine-correlation equality

When the biplot is based on tester centered data (all GGE biplots are based on this type of centering), and when the tester-focused singular-value partitioning, i.e., SVP =2, is used, the correlation between two testers can be approximated by the cosine of the angle between them:

cos(aj'j) = rj'j

where aij is the angle between tester j' and tester j, and rj'j is the correlation coefficient between tester j' and tester j.

Examine the angles among the three testers in the following biplot and compare with the correlation coefficients among them in the table below. The correspondence should be near-perfect as the biplot explained 97% of the total variation.

The correlation coefficients among the three testers are:

Correlation	TESTER5	TESTER7	TESTER8
TESTER5	1	0.912	-0.177
TESTER7	0.912	1	-0.406
TESTER8	-0.177	-0.406	1

Important: The cosine-correlation equality principle is valid only when the above two conditions are met and when the biplot is drawn to scale.

For a more complete reference on the principles of biplot analysis, see, for example, Yan and Tinker (2006).

Indepth: Singular value decomposition


References on biplot analysis

Books

•	Weikai Yan and Manjit S. Kang, 2003. GGE Biplot Analysis: A Graphical Tool for Breeders, Geneticists, and Agronomists. CRC Press.

Journal papers

•	Gabriel, K. R. 1971. The biplot graphic display of matrices with application to principal component analysis. Biometrika 58:453-467.  --The origination of biplots

•	Gabriel, K. R. 2002. Goodness of fit of biplots and correspondence analysis. Biometrika 89:423-436.

•	Kroonenberg P. M. 1995. Introduction to biplots for GxE tables. Department of Mathematics, Research Report 51. Australia: Univ. of Queensland. (<http://three-mode.leidenuniv.nl/document/biplot.pdf>)

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•	Yan, W., Judith Frégeau-Reid, Denis Pageau, Richard Martin, Jennifer Mitchell-Fetch, Mark Etienne, John Rowsell, Peter Scott, Mike Price,  Brad de Haan, Allan Cummiskey, Julie Lajeunesse, Julie Durand, and Ellen Sparry. Identifying Essential Test Locations for Oat Breeding in Eastern Canada. Crop Sci. 2010 50: 504-515. Open Access article.

•	Yan, W., Holland, J.B. 2010. A HERITABILITY-ADJUSTED GGE BIPLOT FOR TEST ENVIRONMENT EVALUATION. Euphytica. 171  (3):355-369. Open Access article.

•	Yan, W. and Frégeau-Reid, J.A. 2008. Breeding Line Selection Based on Multiple Traits. Crop Science, 48, pp. 417-423.

•	Yan, W., Manjit S. Kang, Baoluo Ma, Sheila Woods, and Paul L. Cornelius. 2007. GGE Biplot vs. AMMI Analysis of Genotype-by-Environment Data. Crop Sci 47: 643-653.

•	Yan, W., and N. A. Tinker. 2006. Biplot analysis of multi-environment trial data: principles and applications. Can. J. Plant Sci. 86: 623-645.

•	Yan, W., and N. A. Tinker. 2005. An integrated system of biplot analysis for displaying, interpreting, and exploring genotype by environment interactions. Crop Science.

•	Yan, W., N. A. Tinker, and D. Falk. 2005. QTL identification, mega-environment classification, and strategy development for marker-based selection using biplots. J. Crop Improvement 14 (1/2): 299 - 324.

•	Yan, W and N. A. Tinker. 2005. A biplot approach to the investigation of QTL-by-environment patterns. Molecular Breeding 15:31-43.

•	Yan, W. and L. A. Hunt. 2003. Biplot analysis of multi-environment trial data. p.289-303. In: M. S. Kang (ed.) "Quantitative Genetics, Genomics, and Plant Breeding", CAB International, Wallingford, Oxon, UK.

•	Yan, W. 2002. Singular value partitioning for biplot analysis of multi-environment trial data. Agron J 94:990-996.

•	Yan, W. and D. E. Falk. 2002. Biplot analysis of host-by-pathogen interaction. Plant Disease 86:1396-1401.

•	Yan, W. and I. Rajcan. 2002. Biplot evaluation of test sites and trait relations of soybean in Ontario. Crop Sci. 42(1): 11-20.

•	Yan, W. and L. A. Hunt. 2002. Biplot analysis of diallel data. Crop Sci. 42(1): 21-30.

•	Yan, W. 2001. GGEbiplot--a Windows application for graphical analysis of multi-environment trial data and other types of two-way data. Agron J 93(5): 1111-1118.

•	Yan, W., Q. Sheng, Y. G. Hu and L. A. Hunt. 2001. GGE biplot-an ideal method for graphical analysis of genotype by environment interaction. Acta Agronomica Sinica 27:21-28. (In Chinese)

•	Yan, W. and L. A. Hunt, Q. Sheng, and Z. Szlavnics. 2000. Cultivar evaluation and mega-environment investigation based on GGE biplot. Crop Sci. 40(3): 597-605. (Free: http://crop.scijournals.org/cgi/reprint/40/3/597.pdf)


What is a biplot

Biplot was first proposed by Gabriel (1971)  as a graphical tool to display results from principal component analysis (PCA). It is a scatter plot that graphically displays a rank-2 matrix by both the rows (entries) and the columns (testers). Here "bi" means "both". A rank-2 matrix is a matrix that results from multiplying a matrix with 2 columns by a matrix with 2 rows. Most two-way tables are not a rank-2 matrix. However, if a two-way table can be sufficiently approximated by a rank-2 matrix, then it can be effectively investigated using a 2-D biplot. Likewise, a 3-D biplot can be used to effectively investigate a rank-3 matrix. Below is a 2-D biplot based on the genotype by environment data on this page:

In a biplot generated by GGEbiplot, the entries are by default labeled in title case and blue color, and the testers in upper case and red color. The accurate position of an entry or a tester is at the beginning of the label.

In Depth:

•	How a two-way table is displayed in a biplot

•	What are the utilities of a biplot

•	What do you have to know before trying to interpret a biplot


Singular value decomposition

Singular value decomposition (SVD) is a complicated mathematical technique for breaking down a matrix. Please consult a mathematical book for in depth knowledge.

SVD can be viewed from two perspectives:

1) SVD decomposes a matrix or a two-way table into three parts: The singular values, the row eigenvectors, and the column eigenvectors. The singular value matrix is a r-dimension diagonal matrix, the row eigenvector matrix is a r-column matrix, and the column eigenvector matrix is a r-row matrix, with 'rank' r = min (n,m), n being the number of rows, and m the number of columns of the original matrix.  Or

2) SVD decomposes a matrix into r = min(m,n) principal components (PC). For this reason SVD is sometimes referred to as principal component analysis (PCA), although this is not accurate. The PC's are so ranked that the information (variation) explained by each PC are: PC1 > PC2 > PC3 >...>PCr. Each PC consists of a singular value, an entry vector of m elements, and a tester vector of n elements.

Note: A 2-D biplot displays the first two PC's, PC1 vs. PC2, and a 3D-biplot displays the first 3 PC's, i.e., PC1, PC2, and PC3. Biplots other than PC1 vs. PC2 are also possible in GGEbiplot, although they have limited use. Usually, a biplot refers to a 2-D biplot of PC1 vs. PC2.

In Depth:

SVD may be regarded as the reverse process of matrix multiplication. Two matrices can be multiplied if the number of columns in the first matrix equals to the number of rows in the second matrix. For example, a matrix A(n,r) and a matrix B(r, m) can be multiplied to produce a matrix C(n, m):

A(n,r) * B(r,m) => C(n,m).

SVD is to decompose C so that A and B can be recovered. In reality, there is no unique solution in this "reverse" process. An analogy may help understand this. We know 4 * 6 = 24. However, when 24 is decomposed, we can have 24 => 3 * 8, 24 => 2 * 12, 24 => 4 * 6, and so on. There is a unique solution if 24 is decomposed into three parts: 24 => 2 * 4 * 3, with 2 and 3 as prime numbers (that cannot be further divided).

Similarly, C can not be uniquely decomposed to recover A and B. However, it can be uniquely decomposed into three parts:

C(n,m) => a(n,r) * l(r) * b(r,m).

The middle part l(r) is called singular values. To display C in a biplot, l(r) has to be partitioned between a and b such that

A'(n,r)  = a(n,r) * l(r)^alpha,

and

B'(r,m) = b(r,m) * l(r)^beta,

with

alpha + beta = 1, with alpha and beta are non-negative.

This is the so-called singular value partitioning, which is one of the important issues in biplot analysis.


Organization of functions in GGEbiplot

Functions in GGEbiplot are organized under the following main menus:

The menus can be put into the following categories:

INPUT/OUTPUT

•	Data File Format

•	Data Format

•	FILE

DATA ARRANGMENT/SUBSETTING

•	4-WAY DATA

•	ANYWAY

•	DATA

MODEL SELECTION

•	Models

•	S.V.P (Singular Value Partitioning)

•	BIPLOTS

BIPLOT APPEARANCE

•	VIEW

•	FORMAT

BIPLOT TOOLS

•	BIPLOT ANALYSIS

Other multivariate Analyses

•	ASSOCIATION ANALYSIS

•	VBV BIPLOT

CONVENTIONAL STATISTICS

•	ACCESSORIES (including a Breeder's Kit)

•	ANOVA

EXPERIMENTAL DESIGNS

•	Experimental Designs

•	Breeder's Toolbox

HELP

•	Open this Help File

•	GGEbiplot Online

•	Open Log File


Why use GGEbiplot software

Graphical display is desirable, if not necessary, for a better understanding of large datasets with complex interconnectedness and interactions.  Hence the saying "One picture is worth of a thousand words." The GGE biplot methodology is by far the most powerful "picture" for visualizing two-way tables, particularly those from agricultural and life science researches. Biplot display can help understand any two-way table that has more than 2 rows and 2 columns.

There is no argument among researchers whether biplot is useful in understanding their data. What limits the use of biplot by researchers is the availability of  user-friendly software. There are many macros in all major statistical packages, such as SAS, GenStat, R, etc.. However, GGEbiplot is by far the most powerful and user-friendly for biplot analysis. GGEbiplot is developed for all researchers, especially for those who are not particularly trained in statistics and computer application.

•	GGEbiplot is user-friendly. GGEbiplot was developed by a plant breeder for his own work. All functions are self-explanatory and all operations are through a mouse pointer. Although GGEbiplot comes with a complete help file, few users bother to use it as the functions are so self-evident. GGEbiplot was praised as "a computer game for scientists."

•	GGEbiplot is powerful.

•	GGEbiplot graphically addresses most, if not all, questions the researcher is likely to ask of a two-way table.  For example, for a genotype-by-environment two-way dataset, GGEbiplot displays the "which-won-where" pattern and addresses the issue of mega-environment differentiation; displays both mean performance and stability of tested varieties and ranks them on an integrated index; displays both discriminating ability and representativeness of the test environments and ranks them on an integrated usefuless index; and displays the genetic correlations among test environments and similarities among genotypes. See Functions for genotype-by-environment table for a more exhausted list of functions.

•	GGEbiplot also performs conventional statistics analysis such as analysis of variance (ANOVA) based on different experimental designs, correlation analysis, multivariate regression,  covariate analysis, etc., the results of which are both tabulated and displayed graphically.

•	GGEbiplot allows QTL mapping based on phenotypic data from a single or multiple environments.

•	GGEbiplot has powerful data manipulation tools, allowing analyzing any possible subset of the original dataset at run-time.

•	GGEbiplot allows the user to modify the size, style, and color of the labels, lines, and background of the biplot to the user's preference. It generates biplots ready for publication and presentation.

•	GGEbiplot also produces numerical outputs from the original data to any subset.

•	GGEbiplot reads a 3- or 4-way dataset at once and produces all possible 2-way tables and corresponding biplots. It also reads data in free format, such as SAS would read.

•	GGEbiplot also has many specified functions for plant breeder's and should meet the needs of most researchers.

•	More good reasons!

•	GGEbiplot is efficient. It takes GGEbiplot about one second from reading data to generating and displaying an information-rich, well-labeled, publication-quality biplot. Although there are many macros for biplot analysis, GGEbiplot is literally a million times more efficient.


Experimental Designs using GGEbiplot

There are three most common experimental designs in plant breeding:

1. Randomized Complete Blocks (RCBD)

To be used when the number of entries is small, e.g., smaller than 30.

2. Incomplete Blocks (ICBD)

To be used when the number of entires to be tested is large, e.g., larger than 30, and up to a few hundreds.

3. Augmented Design (AD)

To be used when the number of entries is large and there is no replication.

All these designs can be easily implemented using the "General Experimental Designer"

This interface is set up for a general Row-Column Design, which is the recommended design to maintain field spatial variation information. The general form is an Incomplete Blocks Design, i.e.,, with the entries devidied into severeal incomplete blocks in each replication.Normal ICBD do not contain controls in each incomplete block. However, This Designer allows checks in each (incomplete) block, although they are not mandatory. The checks can be placed randomly or systematically. This Designer can make plans for many test sites at the same time.

The Row-Column Design is a general design, and the commonly used designs can be regarded as a specific case of it:

•	Row-Column Design = RCBD where there is a single block within a replicate.

•	Row-Column Design = ICBD where checks do not appear in every block.

•	Row-Column Design = AD when there is only a single replicate, while controls are presented in each block.


Biplot analysis vs. formal statistical analysis

The above figure is presented to make the point that biplot analysis and statistical analysis are complementary in data analysis. From obtaining data to making a decision involves 3 steps: 1) data visualization, 2) pattern discovery and hypothesis generation, and 3) hypothesis test.

Biplot analysis is very useful in Steps 1 and 2, while conventional statistical analysis is critical in Step 3. This last step is not always necessary, however. It is necessary only when the pattern is not clear enough and when a critical decision is to be made over vague patterns. Biplot analysis and formal statistical analysis are complementary and should be used as such.  For the analysis of any data that can be arranged into a two-way table, biplot analysis should be the first choice.


What is a GGE biplot

The term "GGE biplot" first appeared in Yan et al. (2000). It refers to a biplot that displays the G and GE of a genotype-by-environment data. The key property of a GGE biplot is that it is based on Tester-Centered data, whereby the tester (environment) main effects (E) are removed, and the entry main effect (G) and the entry by tester interaction (GE) are retained and combined. Therefore, a biplot based on tester-centered data contains only G+GE, shortened as GGE.

The GGE biplot methodology was developed for graphical analysis of genotype by environment data or multi-environment trials (MET) data. However, it is a rather generic method that can be applied to many other types of two-way dada. GGE biplot is the most commonly used and most useful type of biplot. The term "biplot" in this document refers to GGE biplot unless it is explicitly specified.

Read More: Other types of biplots.


Functions for VARIOUS types of data

It is not possible to list all the data types that GGEbiplot can work on, but here are some of the typical types:

•	Simple two-way tables

•	Combined two-way tables

•	Compound two-way tables

•	Three-way tables (Example file)

•	Four-way tables (Example file)

•	Any other types of data


Simple two-way tables

The following two-way tables are common in plant breeding and agricultural research:

•	Genotype-by-environment table (of a trait)

•	Genotype-by-trait tables

•	Genotype-by-marker tables

•	QTL-effect-by-environment tables

•	Gene-by-chip tables (Gene expression data)

•	Host-genotype by-pathogen strain tables

•	Inbred-by-tester tables

•	Diallel cross data

•	Location-by-factor tables

•	Multiple regression problems


Functions for diallel cross data

Diallel cross is a widely used genetic design that is made of all possible crosses (combinations) among a set of parents (inbreds or varieties). Diallel cross data are used to study the inheritance of a trait among the set of parents. For quantitative traits, general and specific combining ability (GCA and SCA) are the common statistics to characterize the parents and their combinations.

A diallel cross dataset is a two-way table, each parent being both an entry and a tester. The parent per se values are the diagonal elements of the two-way table. The following can be achieved through GGE biplot analysis of a diallel cross dataset:

Evaluation of parents as entries

•	To rank parents based on their specific combining ability (F1 performance) with a given parent

•	To show the specifc combining ability of a given parent with other parents

•	To rank parents based on their general combining ability (GCA) (mean across its F1's)

•	To visualize both the general (GCA) and specific (SCA) combining ability of each parent

•	To show the best mating partner of each parent

•	To show the best (F1) combinations

•	To reveal heterotic groups

Evaluation of parents as testers

•	To show similarity among parents as testers

•	To show discriminating ability and representativeness of the parents as testers

•	To rank parents relative to the "ideal" tester

Note: If you are only interested in understanding the entry-by-tester interactions, biplot based on double-centered data, should be used. However, It is not clear how such analysis can be of any use. In fact, they may be misleading if inpterpreted incorrectly.

Also See: Diallel data analysis.


Functions for genotype-by-(marker+environmnet) tables
Functions for genotype-by-(marker+environment) tables

A genotype-by-(marker+environment) table is the combination of a genotype-by-marker table and a genotype-by-environment table of a trait for the same set of genotypes. Therefore, in addition to the functions for 1) a genotype by environment table and 2) a genotype by marker table, such data also have the following unique utilities.

•	QTL identification - across multiple environments

•	Investigation of QTL by environment interactions

•	QTL identification - within single environments

Hint: The genotype by (marker + environment) data can be viewed as a genotype by (marker + trait) table if the same trait measured in different environments is treated as different traits.


Functions for genotype-by-(trait+environment) tables

A genotype-by-(trait+environment) table is the combination of 2 two-way tables: a genotype-by-trait table and a genotype-by-environment for a target trait for the same set of genotypes.

The above snapshot shows a portion of a genotype by (trait+environment) table. This table contains 145 rows (of barley doubled-haploid lines) and 46 columns for genetic values (mean values across environments) of 21 explanatory traits plus data for the target trait (yield) measured in 25 environments. The highlighted part is the yield data from different environments.

A genotype by (trait+environment) table can be analyzed as a genotype by trait table as well as a genotype by environment table.

In addition, it can be used to identify traits that can explain and, therefore, be used in exploring genotype by environment interactions of the target trait. Follow the links below to see how this can be done using GGEbiplot in the following steps:

1. Generating a trait-effect by environment biplot

2. Interpreting a trait-effect by environment biplot

3. Mega-environment classification based on a trait-effect by environment biplot

4. Exploring genotype-by-environment interactions from a trait-effect by environment biplot

Note: A 'trait' can also be a genetic marker. In such cases, the trait effects are marker or QTL effects.


Functions for gene-by-chip tables (gene expresson data)
Functions for gene-by-chip tables (gene expression data)

Gene expression data from microarray studies are often represented as geno by chip (subject) two-way tables. In such studies, the purpose is not to see which gene has higher or lower level of overall expressions. Rather, it is to see which genes respond to the treatments or subjects (chips) more strongly. Therefore, the double-centered biplot is appropriate for analysing such data. In fact, this is probably the only useful way of this type of biplot.

Some data transformation might be necessary before conducting this type of biplot analysis.


Functions for Inbred-by-tester tables

The identification of inbreds with superior general combining ability (GCA) is an important component for breeding heterosis, e.g., in corn or maize breeding. Another equally important issue is to identify superior testers that can be used to effectively identify high GCA inbreds. For these purposes, a set of inbreds are often crossed with a set of testers, resulting an inbred-by-tester two way table.

All functions of GGE biplot analysis for a genotype-by-environment two-way table are applicable to this type of data if treating inbreds as genotypes and testers as environments.


Functions for genotype-by-marker tables

Biplot analysis of genotype by marker data using GGEbiplot can be used to:

•	Identify markers that are associated with a particular marker

•	Generate a pseudo linear map among all or a subset of the markers

•	Visualize the "genetic constitution" or marker profile of each individual in the mapping population.


Functions for location-by-factor tables

Analyzing such data is similar to analyzing genotype-by-trait tables, but replace genotypes with locations and traits with factors. Such data can be used to study the similarities among locations with respect to their physical factors.

Note: this classification of environments is different from the test environment evaluation in a GGE biplot analysis in that the purpose of test environment evaluation is to identify test environments that are superior in evaluating genotypes. This biplot is more relevant to soil and environmental researchers as opposed to plant breeders and geneticists.


GCA and SCA of parents - diallel analysis

From Biplot Tools, click Mean vs. Stability. The following will appear on the biplot:

•	a small red circle indicating the position of the average tester;

•	a thick red line that passes through the biplot origin and the average tester, referred to as the average-tester axis (ATA);

•	a red arrow pointing to the average tester from the biplot origin;

•	a thick blue line that passes through the biplot origin and is perpendicular to the average-tester axis;

•	a set of parallel lines that project each entry to the ATA; and

•	two blue arrows on the thick blue line, pointing outwards from the biplot origin.

Interpretations on GCA:

•	The entries are ranked based on their GCA along the ATA with the arrow pointing to a greater GCA value;

•	The blue line separates entries with above-average GCA (on the same side of the 'average tester' relative to the biplot origin) from those with above-average PCA (on the opposite of the 'average tester');

Caution: The length of the average tester vector (the distance from biplot origin to the average tester), relative to the biplot size, is a measure of the relative importance of the entry main effect (GCA) vs. entry by tester interaction (SCA). The longer this distance, the more important is the GCA, and the more meaningful the ranking of parents based on GCA. At the extreme, a zero average tester vector means there is no GCA and therefore ranking on the ATA, i.e., selection based on GCA, is meaningless.

Interpretations on SCA:

•	A longer projection to the thick blue line represents a greater positive SCA of an entry (in blue) WITH the testers (in red) on the same side of the ATA.

•	A longer projection to the thick blue line represents a greater negative SCA of an entry (in blue) WITH the testers (in red) on the opposite side of the ATA.

Caution: When the biplot explains only a small portion of the total variation, the interpretation of the SCA may not be accurate because some of the SCA are not displayed.

Identifying heterotic groups:

•	Contrasting groups of entries relative to the ATA may be regarded as different heterotic groups.

Note:

•	If the biplot explains only a small proportion of the total variation, this interpretation may not be accurate for short-vectored parents.

•	In a diallel cross dataset, each genotype is both an entry and a tester. GCA and SCA are properties of "entries", not "testers".

•	The testers can also be evaluated for their ability to identify entries with high GCA, analogous to the test enivironment evaluation in genotype by environment data analysis.


Rank entries based on specific combining ability with a tester - diallel analysis

See Rank entries relative to any given tester.

Interpretations:

The parents (entries) are ranked along the vector of the chosen tester, with the arrow pointing to a higher SCA with the selected tester.


Show the specific combining ability of a parent

See Show the specific combining ability of an entry.

Interpretations:

•	The testers are ranked based on their SCA with the selected entry.


Rank entries based on general combining ability - diallel analysis

See Mean performance and Stability of each entry across testers

Special interpretations:

•	The average-tester coordination or ATC abscissa (= average tester axis, or ATA) points to higher GCA.


Marker (QTL) effect-by-environment biplot - QQE biplot

A marker (or QTL) by environment table contains effects of each marker (QTL) on the trait of interest in each environment.

The appropriate biplot for such data should be based on SVD of original data without centering. The biplot is called a QTL-by-environment biplot or QQE biplot.

The following information can be visualized in a QQE biplot:

•	The relative importance of a marker or QTL on the target trait;

•	The mean and stability of the effect of a QTL;

•	The interactions between QTL and environments--the essence of a QQE biplot; and

•	Mega-environment based on QTL by environment interactions, which can lead to the development of appropriate strategies for marker-based selection. That is, a QQE biplot can lead to recommendations thereby specific allele combinations of the QTLs are selected for each mega-environment to maximize the expression of the target trait in that mega-environment.

A complete example of QQE biplot analysis is presented in Yan and Tinker (2005).


The relative importance of a marker or QTL

This interpretation is specific to a QQE biplot:

Under the main menu Biplot Tools, click Entry relations. Lines that connect the markers or QTL to the biplot origin will appear, which are called marker or QTL vectors.

The length of the vectors is a measure of the magnitude of the marker or QTL effect on the target trait. A long vector indicates that the marker or QTL has large effects in at least some of the environments. A marker or QTL located near the biplot origin has only small effects on the trait in any of the environments.

Note:

•	The vector of a QTL that has consistent effects across environments may appear longer than it actually is. This might be a good feature of the QQE biplot for identifying QTL with stable effects.


What are mega-environments

A mega-environment (ME) is defined as a group of environments (more accurately, locations) that share the same (type of) winning varieties. A meaningful ME should be a group of locations that consistently share the same (or similar) winning varieties across years. Therefore, data from multiple years for the same set of genotypes are essential to arrive at a decisive conclusion whether the environments can be divided into mega-environments; data from a single year can  only be used to suggest possible existence of different ME.

Also read: Mega-environment vs. which-won-where


Linear map among markers

Under Biplot Tools, click Show linear map of the testers. A linear map among markers in the biplot will shown on the right of the biplot.

The distance between the testers are in degrees, keeping in mind that cos90 = 0 meaning unrelated (completely independent), cos180 = -1 (completely linked but in repulsive), and cos0 = 1 (completed linked). For example, 1mwg889b and 1brz seem to be independently inherited, even though both are located on the same chromosome. 1mwg626 and 1mwg511 have an angle of around 5 degrees, which may be 'translated" to about 2-3 cM. The linear map is a rough mimic of a genetic map.


Identify markers closely linked with a target marker

•	Under the main menu Association, select Find Associated Variables...

•	You will be prompted to set a probability threshold for an association to be considered as statistically significant. Put a number between 0 and 0.5, exclusive.

•	 Then, a variable selection tool will appear, allowing you to select the target variable (a genetic marker in this case). Select one, click the Find button.

•	A biplot will appear, with the target variable (marker) (with a larger font-size) plus those markers or traits that meet your criterion of being associated.

•	Marker vectors are shown to facilitate visualization of the relationships among markers/variables.

•	This biplot will be based on tester-focused SVP, which supports the interpretation that the cosine of the angle between two markers approximates the correlation between them. Under a perfect goodness of fit, 90 degree means independent, acute angles mean positive correlation (linked), and obtuse angles mean negative correlation (repulsively linked)

•	A rough linear map can be invoked by clicking Linear Map of Testers. The range of 0-90 degree may roughly correspond to the range of 0-50 cent Morgan.

The following biplots provide two examples, in which the markers are in red and genotypes in blue:


Average Tester

The 'average tester' in the context of biplot analysis is defined by the PC1 and PC2 scores averaged across PC1 and PC2 scores of all testers. This concept was first proposed in Yan (2001). Applications of this concept include:

•	Visualize the mean performance of the entries across testers;

•	Visualize the mean performance and stability of the entries;

•	Identify the ideal entries;

•	Identify the ideal testers;

•	Visualize the GCA and SCA of each of the parents in diallel cross data analysis.


Diallel Data Analysis

Diallel crosses have been used in genetic research to determine the inheritance of a trait among a set of genotypes and to identify superior parents for hybrid or cultivar development. Conventional analysis of diallel data is limited to partitioning the total variation due to the crosses into general combining ability (GCA) of each parent and specific combining ability (SCA) of each cross. The SCA effects are just residuals not explained by the GCA effects; they are cross-specific and do not help in understanding the parents.

The biplot approach of diallel data analysis allows better understanding of the parents, in addition to being graphical. For a given set of data, the following information can be easily visualized:

1) the GCA effects of the parents,

2) the SCA effects of the parents;

3) the heterotic groups;

4) the best crosses; and

5) the best testers for identifying parents with high GCA.

Treating a diallel data as an entry by tester two-way table, Points 1, 2, 3 above can be achieved by the Mean vs. Stability view of the GGE biplot; Point 4 can be achieved by the which-won-where view of the GGE biplot, and Point 5 can be achieved by the Discrimination vs. representativeness view of the GGE biplot.

Also see: Functions for diallel data analysis.


Average-tester axis

"Average Tester Axis" or ATA is the vector that passes through the average tester and the biplot origin, with the arrow pointing to higher mean values of the entries. This is an important concept in biplot analysis first appearred in Yan (2001). It can help visualize:

1. The mean vs. stability of the entries

2. The discrimination vs. representativeness of the testers

NOTE BUT:

This view is useful only when the gneotpye main effect (G) is sizable, say, greater or equal to 20% of the total G + GE. When G is too small, it would make no sense to select based on it. A small G to G+GE ratio is reflected by a short Average Tester Vector, i.e., distance from the biplot origin and the placement of the Average Tester. In such cases the Which-Won-Where  function is more meaningful.


QTL identification - single trait in multiple environments

Note: This module requires a genotype-by-(marker+environment) two-way table.

QTL identification based on phenotypic data from multiple environments is one example of covariate effect analysis; another example is the analysis of a genotype by (trait+environment) two-way table.

This analysis involves the following steps, which are automated in GGEbiplot:

•	Generating a Marker-by-Environment table consisting of Marker-Trait Associations in each Environment, i.e., marker effects on the trait in each environment.

•	Screening for relevant markers

•	Estimating a Type I error through permutation

•	Determining the number of QTL, and

•	Investigating mega-environments based on the Q+QE pattern

In depth:

This QTL mapping process involves two steps:

1) Individual marker screening (point statistics)

2) Global threshold (an LR or log ratio for QTL regression or a vector length) for marker screening

NB:

•	Large effect QTL represented by very closely linked markers can be missed if the LR is used as a global error control method, because the markers would have very small LR, even if they have very large effects. An extreme example would be two markers at exactly the same spot. Remove one marker would result in little reductions in the residual SS and, therefore, a zero LR.

•	When using the vector length as a criterion in QQE analysis (i.e., identifying QTL data on data from multiple environments), one should be aware that a large number of closely linked markers tend to have longer vectors whereas unlinked markers tend to have shorter vectors, relative to their true effects. However, this is less of a problem than QTL identification based on data from a single environment.


Generating the marker-by-environment table of marker-effects

Previous: QTL identification based on phenotypic data from multiple environments.

Under the main menu VBV Biplot, click Covariate-effect biplot, and select r-Matrix (or b-matrix). An input box will appear, asking for the number of markers in the combined two-way table. In our case of the barley H-T mapping data, there are 127 markers:

GGEbiplot detects that there are a total of 152 columns, and therefore uses the first 127 columns as explanatory variables and the rest 25 as response variables to generate a table of 127 * 25 correlations, each of which is the Pearson correlation between the relevant marker and the target trait in the relevant environment. When this is done, the researcher will be notified that the data structure is changed:

Note: In the current version of GGEbiplot, the data format is more flexible. When the function is invoked, a form like the one will appear and the user is prompted to separate the variables into explanatory variables and the response variables. So in the input table, the response variables and the eplanatory variables can be in any order.

Next: Screening for relevant markers


Screening for relevant markers

Previous: generating a marker-by-environment two-way table of marker effects

GGEbiplot will then prompt for a probability level, which will be used to delete markers that are not associated with the target trait in any of the environments. Here we supply 0.001:

Note that the probability level is arbitrary and is determined by the user. In general, the level can be more stringent if there are many markers, which will reduce the work later on without losing important information. GGEbiplot will screen the markers using this criterion and notify the user how many markers have been removed:

Next: Type I error concentric circles


Type I error concentric circles

Previous: Screening for relevant markers.

GGEbiplot now is ready to generate the marker by environment biplot. Before that, it will ask if you want to add a set of type I error threshold circles on the biplot, which can serve as an experiment-wise error control:

If you select 'No', GGEbiplot will just generate and display the marker by environment biplot. If the selection is "Yes', you will be asked for the number of permutations to generate the threshold circles:

GGEbiplot remembers previous permutations, and will ask if you want to use existing permutation results or to do it over again:

When permutation is complete, GGEbiplot will ask for a probability level of the threshold circles.

The following biplot is a marker-effect by environment biplot with concentric circles of 0.05:

Note that there are five concentric circles, labeled as 0.05--, 0.05-, 0.05, 0.05+, and 0.05++. This gives a range of 5% type I error. A range is needed as the biplot is not an exact display of the marker-by-environment two-way table. You can choose the least stringent (inner most circle) or the most stringent circle (the outer most circle) to delete markers with short vectors (small effects). There should be no major difference in the identification of QTL; any difference is only about the inclusion or exclusion of minor QTL. We here remove all markers based on the most stringent circle:

Next: Determining the number of QTL.


Determining the number of QTL

Previous: Type I error concentric circles

The following biplot shows that the markers fall into 5 distinct groups. This suggests that at least 5 QTL are involved in determining the yield difference between the two parents (Harrington and Tr306). Since the linkage groups are known, as indicated by the number preceding the marker names, it is clear that more than 5 QTL are involved, distributed in chromosomes 1, 3, 4, 5, 6 and 7, respectively:

If each QTL is represented by its nearest marker, the marker-by-environment biplot becomes:

which may be called as QTL-by-environment biplot or QQE biplot. Markers abg472 and abg715 were excluded because they were linked on chromosome 4 as shown in the biplot; it was suspected that they were included in the biplot above because of their linkage with marker mwg655c, which had the strongest association with yield.

What can we see from this QQE biplot

•	First, the importance of a QTL to yield is measured by the length of its vector. Thus, 7mwg502 > 4mwg655c > 7mwg914 > 3mwg838 > 1mwg626 > 6bcd269 > 5abc261.

•	Second, in terms of QTL by environment relationships, 7mwg914, 6bcd269 and 5abc261 are similar, whereas 7mwg502 and 4mwg655c are similar.

•	Third, the environments seem to fall into two distinct groups.

Next: Mega-environment classification based on the QQE pattern


Mega-environment classification based on the QQE pattern

Previous: Determining the number of QTL

From the above biplot, it is already obvious that the environments fall into two groups: the Eastern (on the right) and the Western (on the left) mega-environments. Change the singular value partitioning (SVP) to environment-focused (SVP =2), which is appropriate for visualizing the similarities among test environments, this pattern becomes even more obvious:

This mega-environment classification is strikingly similar to that based on the GGE biplot of the yield data per se, suggesting that the GGE pattern is well explained by the QQE pattern, which in turn suggests that barley yield can be effectively manipulated by selecting for these QTL.

Marker assisted selection in the Eastern mega-environment

The following QQE biplot contains only environments from the eastern provinces of Canada: Manitoba, Ontario, Quebec, and Prince Edward Island (Note that Manitoba is usually regarded as belonging to Western rather than Eastern Canada but in this particular barley dataset, the locations representing Manitoba were more similar to test locations in Eastern Canada than to those in Western Canada). It can be seen that the Harrington alleles of mwg502 and mwg655c, and the Tr306 alleles of mwg914, bcd269, and abc261 had positive associations with barley yield in all environments in this mega-environment. Therefore selecting for appropriate alleles of these 5 QTL should improve barley yield in this mega-environment. QTL represented by mwg838 and mwg626 had inconsistent effects on yield and therefore can not be used in marker assisted selection for this mega-environment.

Note: The cosine of the angle between the vector of a QTL and that of an environment determines the direction of the effect of that QTL in the environment.

Marker assisted selection in the Western mega-environment

The following QQE biplot contains only environments from the western mega-environment, including Canadian provinces Alberta and Saskatchewan and some states of the USA, i.e., Alaska, Washington, Montana, and North Dakota. Unlike the Eastern mega-environment, no QTL had consistent effects on yield in this mega-environment. One exception is that the Harrington allele of the QTL represented by mwg838 had negative effects in some of the environments within this mega-environment and, therefore, may be useful in marker-assisted selection.

Note: Other utilities of the QQE biplot.


Generate a triat-association by environment biplot
Generating a Trait-Association by Environment biplot

The trait-assocaition biplot is also referred to as a covariate-effect biplot. This module requires a combined genotype-by-(trait+environment) two-way table or a genotype-by-environment-by-trait three-way dataset, from which such a combined table can be automatically constrcuted using GGEbiplot.

Steps of generating a Covariate-Effect biplot

This analysis involves the following steps, which are automated in GGEbiplot, known as the covariate-effect biplot module under the main menu of VBV Biplot (VBV refers to be 'Variable-By-Variable' biplot; it is also referred to as Canonical Biplot sometimes although there are some differences).

•	Under the main menu VBV biplot, click Covariate-effect biplot based on..., and select r-matrix.

•	In the older version of GGEbiplot, the data have to be arranged such that the columns of explanatory variables are followed by response variables (as the above example) and the user has to know the number of explanatory variables in the data. When the function is invoked, the user will first be asked for the number of explanatory variables in the table so that GGEbiplot knows how to divide the variables into explanatory variables and response variables. The number of explanatory traits in our data is 21. (Under the cover, GGEbiplot will calculate a 21 * 25 trait by environment table of correlations coefficients. Since the correlation coefficients represent the effects of each trait on yield, this table is referred to as a covariate-effect table.)

•	In the newer version of GGEbiplot, the data format is more flexible. When the function is invoked, a form like this will appear and the user is asked to separate the variables into N explanatory variables and the M response variables.

•	The researcher is then asked to provide a probability threshold, which will be used to remove traits that are not significantly associated with the target trait in all the environments. Here we give the value 0.01.

•	Traits that did not associate with the target trait at the probability level in at least one of the environments will be removed from the covariate-effect table. The user will be noticed about the number of traits that are removed:

•	The user will then be asked if he wants to permutate an experiment-wise Type I error threshold for further deletion of explanatory traits:

•	If Yes is selected, then the user will be asked how many permutations to conduct:

•	If the user has done permutations for the same dataset and the same setting, which should have been stored in the hard disc, he will be asked if he wants to use the existing results or do it over again:

•	When the permutation is complete, the user will be asked for a probability level, which will be put in the biplot as threshold circles. Here we supply 0.01.

•	When the calculation is complete, a Covariate-Effect by Environment biplot will finally show up, together with the threshold circles.

Continue to read: Interpreting the Covariate-Effect by Environment biplot

In Depth: How permutation is conducted


How permutation is conducted

To be written.


Interpreting the trait-effect by environment biplot

Previous topic: generating a covariate-effect by environment biplot.

•	Note that there are 5 circles labeled as 0.01, 0.01+, 0.01++, 0.01-, and 0.01-- . Multiple circles are presented because the biplot does not explain 100% of the total variation. Therefore, the threshold is a range rather than a precise level. The length of the vector of a trait, or of a simulated observation, is determined by the magnitude of the trait's effect on the target trait. However, it is also affected by two factors: 1) the similarity of the trait with other traits, and 2) the consistency of the trait's effect across environments. More specifically, if a trait is similar to many other traits, its vector tend to be longer, because the similarity forms a strong pattern, which is likely to be captured by one of the first few PC's. For the same reason, if the effect of a trait is relatively consistent across environments, which also represents a strong pattern, it is more likely to be captured by one of the first few PC's. Therefore, the threshold circle with the shortest vector is more appropriate to select any traits that have a strong effect in any environment, whereas the threshold circle with the longest vector is more appropriate for selecting traits with strong effects in most environments. It is up to the user to decide which circle is used to remove "unimportant" explanatory traits. The difference due to these factors in our example is whether the traits PM and PMGROUP (both are powdery mildew scores) are excluded or included. Here we remove all the traits that fall within the circles, and the biplot becomes:

(BUT NOTE: The new version shows only one circle to simplify interpretation)

•	The length of the vector of a trait measures the importance of the trait in explaining the GGE (genotype main effect plus genotype by environment interaction) of the target trait. The concentric circles help visualize this. Thus, the importance of the traits in determining barley yield in this dataset are:

Kernel Weight > Lodging >= Days to Heading > Test weight = Plumpness > Days to Maturity =Plump kernel weight = protein content > XFI.

•	The angles between traits measure the similarity of the traits in their effects on the target traits. Thus, kernel weight was similar to a cluster of traits including plumpness and test weight, but was very different from lodging and days to heading, in terms of correlation with yield.

•	To simplify the discussion that will follow, the traits with relatively short vectors can be further removed so the covariate-effect biplot looks like:

Continue to read: Mega-environment classification


Randomized Complete Blocks Design

See Experimental Designs using GGEbiplot.


Mega-environment classification based on a covariate-effect biplot

Previous topic: Interpreting the trait-effect by environment biplot

You might have noticed that the environments in the above biplots fall into two distinct clusters, although these biplots are not ideal for classifying environments. For this purpose, the singular values should be partitioned entirely to the environment scores, i.e., SVP =2 should be used so that the biplot looks like:

Now the 25 environments more clearly fall into two groups. The group on the right contains all environments from Ontario (ON), Prince Edward Island (PE), Manitoba (MB), and one of the two Quebec (QC) environments. Since the environments in this group are located mostly in the eastern part of Canada, it may be regarded as the Eastern mega-environment, although it also contains two environments from Saskatchewan (SK), and one environment from Montana (MO, should be shortened as MT).

The group on the left contains all environments from Alberta (AB), most environments from SK, and environments from some USA states, including Washington (WA), North Dakota (ND), and Alaska (AK). One exception is that it includes a QC environment. Therefore, this group may be called the Western mega-environment.

Interestingly, this mega-environment classification is very similar to that based on the GGE biplot of yield, indicating that the GGE of yield can be effectively explained by the covariate effect by environment interactions and, therefore, exploited by indirect selection for these explanatory traits.

Continue to read: Exploiting GE through selection for explanatory traits


Mega-Environment  vs. Which-Won-Where

Dividing the target environment into meaningful mega-environments and deploying different cultivars for different mega-environments is the only way to utilize positive GE and avoid negative GE and the sole purpose for genotype by environment interaction analysis (Yan et al., 2007, Crop Sci). A mega-environment is defined as a group of locations that consistently share the same best cultivar(s) (Yan and Rajcan, 2002). This definition involves several essential elements:

1)  Different mega-environments are defined by different winning cultivars, noting that different genotypes can be equally adapted to the same mega-environment and that a mega-environment may need different types of genotypes to stablize the overall production;

2) The concept of mega-environment is based on geographical locations; and

3)  The cultivar-location interaction pattern should be repeatable across years.

All three aspects are essential for the declaration of different mega-environments. The following biplot based on the multi-environment trials (MET) data of barley yield illustrates two points: 1) A mega-environment may have more than one winning genotype, and 2) even if there exists a universal winner (line "829"), it is still possible, and beneficial, to divide the target environments into meaningful mega-environments.

In the above biplot, each dot represents a genotype for clarity.


Avoiding/Exploiting GE through selection for explanatory traits

Previous topic: Mega-environment classification based on a trait-effect biplot

Comparison between the GGE biplot and the trait-effect by environment biplot indicates that four traits, namely, kernel weight (KW), lodging, days to heading (Headingd), and days to maturity (Maturity), adequately explained the GGE pattern of yield. This implied that yield may be effectively manipulated by selecting for these four traits.

The only way to avoid negative GE and to exploit positive GE is to divide the target environment into meaningful mega-environments and to select and use specifically adapted varieties.

Selection strategy for the Eastern mega-environment

The following biplot contains environments only from the eastern Canadian provinces ON, QC, PEI, and MB:

Interpretation: A trait has positive associations on yield in an environments if the trait and the environment have an acute angle in the biplot. Thus, kernel weight (KW) had positive associations with yield in all or most environments in the Eastern mega-environment, implying that selecting for greater kernel weight in eastern Canada should lead to improvement in barley yield, at least for this Harrington * Tr306 population. On the contrary, lodging had negative associations with yield in all eastern environments, so did days to heading. Therefore, selecting for greater kernel weight, better lodging resistance, and early heading should improve barley yield in eastern Canada, at least in this breeding population. Days to maturity, however, did not have consistent associations with yield in eastern Canada. Therefore, this trait cannot be used in breeding for higher yield in eastern Canada.

Selection strategy for the Western mega-environment

The following biplot includes only the Western locations (Alberta, Saskatchewan, Washington, Montana, and Alaska).

Interpretation: First, the environments took both positive and negative values for both axes, implying that the trait-yield relations varied dramatically among environments within this mega-environment. Consequently, no trait had positive associations with yield in all environments. Secondly, environments from the same locations (Alberta and Saskatchewan) were scattered across the biplot, indicating that the environments cannot be further divided into repeatable sub-groups. Therefore, the Western mega-environment is a single mega-environment with large unpredictable trait-yield relations. As a result, it does not seem feasible to improve yield by indirect selection for any of the traits; selection for yield per se in multiple environments is the only way to improve yield in this mega-environment.


Incomplete Blocks Design

See Experimental Designs.


Row-Column Design

Row-Column design is a general form of incomplete design that is considered more desirable for accounting field spatial variations. IThe Incomplete blocks design and the Augmented design are special forms of the row-column design.

Return to: Experimental Designs.


Covariate-effect by environment biplot

See QTL-effect by environment biplot, but read QTL as "explanatory traits".

Compare Associations-with-a-trait by Environment biplot and Trait-Association by Environment biplot.


Augmented Design

From the Data form, click Experimental Design.


Row-Nursery Design

This is specifically designed for Observation Plots in the ECORC oat breeding program. It is very similar to Augmented Design except for two things:

1) The entries are not randomized, and

2) The checks are at the ends of each block rather than in the middle.


What are Entries and Testers

Entry and Tester are the row and column factors of a two-way table.  An entry is a level of the entry factor, and a tester is a level of the tester factor. Entries are to be evaluated, and testers are used to evaluate the entries.

•	In genotype by environment two-way tables, genotypes are entries and environments are testers.

•	In genotype by trait two-way tables, genotypes are entries and traits are testers.

•	In genotype by genetic marker tables, genotypes are entries and markers are testers.

•	In a multivariate analysis, observations are entries and variables are testers.

Although entries are conventionally presented as rows and testers as columns in a two-way table, this is not a requirement in data input for the GGEbiplot software, as the data can be easily transposed.


Transpose a two-way table

Under Data, Click Entry/Tester Switch Roles. The two-way table will be transposed and the correspondent biplot displayed.

Note that although there is no option for "Tester-centered models" under Models, entry-centered models will be changed to respective tester-centered models through this function.


Functions for genotype by environment data

There are many different models for biplot analysis of a genotype-by-environment two-way data, and different models have different interpretations and utilities in understanding the data.

For the purpose of genotype evaluation, GGE biplots , i.e., biplots based on singular value decomposition of environment-centered should be used. Such environment-centered data can be further scaled in different ways, leading to different GGE biplots.

A GGE biplot can be used not only in evaluating the genotypes, but also in evaluating the test environments for their usefulness/effectiveness in genotype evaluation. Follow the links below for a complete example of biplot analysis of genotype-by-environment two-way data (read entries as genotypes and testers as environments):

Biplot analysis of genotype by environment data include three major aspects (see Yan et al., 2007, Crop Science, 47: 643-653)

•	Genotype evaluation based on a GGE biplot

•	Test environment evaluation based on a GGE biplot

•	Mega-environment analysis

Note: If you are only interested in understanding the genotype-by-environment interactions, the double-centered model should be used. However, I really don't see any use of this model in genotype evaluation or test environment evaluation or in plant breeding in general. In fact, in my opinion, publications that are devoted to the analysis of GE only are more misleading than helping.


Genotype evaluation based on a GGE biplot

A GGE biplot allows the genotypes be evaluated from the many different perspectives:

•	Rank genotypes based on performance in any given environment

•	Discard inferior genotypes based on performance in any given environment

•	Rank genotypes based on performance in any TWO environments

•	Show the specific adaptation of a genotype in individual environments

•	* Rank genotypes based on mean performance across all environments

•	Rank genotypes based on both mean performance and stability (rank genotypes relative to the "ideal" genotype)

•	Compare TWO genotypes for performance in individual environments

•	Compare any THREE genotypes in individual environments

•	Compare all genotypes with a user-specified check

•	* Show the best genotype(s) for individual environments (which-won-where)

•	Plot any TWO genotypes across test environments

* Most useful and used functions.


Test environment evaluation based on the GGE biplot

A GGE biplot allows the test environments to be evaluated from the following perspectives in terms of genotype evaluation:

•	Show relationships among test environments

•	Identify redundant environments or the need for new test environments

•	Group environments based on the genotype-by-environment interaction pattern

•	Group environments based on winning genotypes

•	Show the discriminating ability and representativeness of the test environments

•	Rank test environments relative to the "ideal" test environment

•	Show the correlation matrix among test environments

The following is not based on biplot but it is also a useful tool for comparing test environments:

•	Plot any TWO test environments across the genotypes

NOTE: please do not confuse "Test environment evaluation" with "Environment Evaluation". The purpose of "test environment evaluaiton" is to identify environments that are suitable for selecting/evaluating genotypes. Test environment evaluation depends on G+GE and therefore GGE biplots are appropriate. The purpose of "Environment evaluation" is to identify environments that are have high productivity, which is determined by E (the environment main effect).  Some argue that the AMMI analysis is more appropriate because it includes E and separates G from GE. However, these are really pseudo issues to plant breeders and geneticists.


Rank genotypes based on performance in any given environment

See Rank entries based on any given tester. For interpretation, replace "entries" with "genotypes" and "testers" with "environments".


Rank entries based on any given tester

•	Under the main menu of Biplot Tools, click Examine a tester.

•	An entry/tester selecting pane will appear on the top of the biplot:

•	Select a tester, e.g., "E1", from the drop-down box and click the Rank button, the biplot will become as follows:

•	The following changes are made to the biplot:

1. All testers except the selected tester ("TEST1") are re-labeled as "E" to make the selected tester stands out.

2. A red line is drawn to pass through the biplot origin and the selected "TEST1", with an arrow pointing to it from the biplot origin. This red line is referred to as the "TEST1 Axis".

3. A blue line is drawn to pass through the biplot origin and to be perpendicular to the TEST1 Axis, referred to as the perpendicular line.

Interpretations:

•	The entries are ranked based on their performance with respect to TEST1 along the TEST1 Axis with greater values in the direction of the red arrow: Entry7 > Entry5 > Entry2 > Entry4 > Entry1 > Entry6 > Entry3.

•	Entries on the same side of the perpendicular line as TEST1 had higher-than-average values with respect to TEST1; Entries on the other side of the perpendicular line as TEST1 had lower-than-average values with respect to TEST1; and entries on the perpendicular line had exactly the average value with respect to TEST1. Thus,  we have: Entry7 > Entry5 > Entry2 > average >= Entry4 > Entry1 > Entry6 > Entry3. Here Entry4 is only slightly lower than the average.

•	The entries can be ranked this way relative to any other testers.


Functions for genotype-by-trait tables

A genotype-by-trait two-way table can be used in 1) evaluating the genotypes in terms of their trait profiles and 2) studying the relationships among traits. The most appropriate model for such data is Model 2 or 3, i.e., biplots based on SVD of standardized data using tester standard deviations (Scaling =1) or tester standard errors (Scaling = 3) so that the units (and any heterogeneities in case of model 3) are removed and different traits are put into the same scale.

Reminder: in GGEbiplot genotypes are referred to as entries and traits as testers.

Functions for genotype evaluation include:

•	Rank genotypes based on any trait

•	Discard genotypes based on any trait

•	Rank genotypes based on any two traits

•	Show the relative strength and weakness of a genotype

•	Compare two genotypes for various traits

•	Compare any three genotypes for various traits

•	Rank genotypes based on similarity to a specified check

•	Show genotype(s) with the highest values for various traits

•	Plot any two genotypes for various traits

Functions for trait evaluation include:

•	Show the relationship among various traits

•	Identify redundantly measured traits

•	Plot any two traits across the genotypes

•	Show the correlation matrix among traits


Discard genotypes based on performance in any environment

See Discarding entries based on a tester, but read entries as genotypes and testers as environments.


Rank genotypes based on performance in any two environments

See Rank entries based on two testers but read entries as genotypes and testers as environment.


Independent Culling based on a biplot

This function have been removed in the newer version of GGEbiplot. Instead, the Selection Based on Multiple Traits is recommended.

From the main menu Biplot Analysis, click Independent Culling. A tester selecting pane will appear above the biplot.

Click the drop-down arrow by the textbox to select the tester (environment or trait or variable) based on which the culling is to be conducted and click the Discard button. The following will appear on the biplot:

o	A blue oval that circles the selected tester,

o	A red line that passes through the biplot origin and the beginning of the label of the tester,

o	A red arrow pointing to the beginning of the label,

o	A blue line that passes the biplot origin and is perpendicular to the red line, and

o	A set of 10 lightly colored lines parallel to the blue line, which divides the entries into 10 grades.

At the same time, an input box will appear, asking for the culling intensity. Put a desired number between 0 and 100, then click OK.

Genotypes with values below the specified percentage relative to the maximum value of the selected tester (environment or trait) will be deleted from the biplot, and the blue line will be relocated to mark the discarding position.

The deleted entries can be called back by clicking the Recall button on the tester selecting pane.

This process can be repeated for other testers.

Note: The reliability of this method is determined by the goodness of fit of the biplot. See this page for the traditional independent culling method.


Rank entries based on two testers

Under the main menu Biplot Tools, click Combine two testers. A tester selecting pane containing two drop-down box will appear on the top of the biplot.

Select the two testers (tester1 and tester2, for example) to be joined and click the Combine button. The biplot will look like:

Interpretation:

•	The two selected testers are in full names and other testers are displayed as "E";

•	The two selected testers are connected by a thin red line, and the mid-point of the red line is indicated by a small circle, which may be called the "combined tester";

•	A thick red line is drawn, which passes through the biplot origin and the combined tester, referred to as the "combined tester axis";

•	The entries are ranked along with the combined tester axis, with the arrow pointing to greater values. Thus, based on tester1 and tester2, the rank of the entries is: Entry7 > Entry5> Entry2> Entry1> Entry4> Entry6> Entry3; and

The thick blue line passes the biplot origin and is perpendicular to the combined tester axis. It indicates the position of the grand mean based on the two testers. Thus, we have Entry7 > Entry5> Entry2> AVERAGE> Entry1> Entry4> Entry6> Entry3.


Functions for Genotype by (Marker+Trait) data

A genotype-by-(marker+trait) table is made of 2 two-way tables: a genotype-by-marker table and a genotype-by-trait table (of genotypic or phenotypic values). Therefore, in addition to the functions for 1) a genotype by trait table  and 2) a genotype by marker table.

The following snapshot is part of a barley QTL mapping dataset, in which the rows are 145 barley doubled haploids, and the columns are 127 genetic markers plus 22 agronomic/quality traits. Only part of the data is shown. The highlighted columns are traits, and the un-highlighted are markers. (Note that the milling values are indicated by "-99" in this case, although it can be anything, preferably leaving them empty if it is an MS Excel file.)

We will use this dataset to illustrate the following functions for this type of data:

•	QTL identification-single trait,

•	QTL identification - multiple traits, and

•	Pleiotropic effects of a QTL


Pleiotrpoic effects of a marker
Pleiotropic effects of a marker

This functions requires a genotype by marker + trait table.

Under the main menu Association, click Find Associated Variables. An input box will appear, asking for a probability level to be considered as associated. Input a number, say, 0.001, and click OK.

The variable selection penal will appear on the top of the biplot. Select a marker (represent a QTL or gene) from the drop-down box, say, 7MWG502, and click Find.

The following biplot, or plot rather, as the genotypes were hidden, will appear:

Under the main menu Data, click Run Any Subset..., and select By Deleting Testers. A tester selection device will appear. Remove all markers except 7MWG502 and click the Find button again, the biplot will become:

The biplot shows that the marker 7MWG502 is associated, at the 0.001 level, with four more or less independent groups of traits:

•	positive association with Height, Kernel Weight, Plump Kernel Weight, Yield, and days to Maturity;

•	positive association with Test Weight, Protein content, and kernel plumpness;

•	negative association with lodging, and

•	XFI (fine extraction).

The Pleiotropic effect of genes represented by this marker is therefore very obvious.

Note: Some trait relations in the two biplots on this page are not consistent. This is because the biplot explains only 50% of the total variation. In such cases, the relationships seen from the biplot are only qualitatively correct (positive vs. negative and strong vs. loose), but the correlations among traits (how to get themlinkID=840}) may not be exactly visualized from the angles.


Show the specific adaptation of a genotype

See Rank the testers based on the relative performance of an entry, but read entries as genotypes and testers as environments.


Rank genotypes based on mean performance across all environments

See Mean performance and stability of entries across testers, but read entries as genotypes and testers as environments.


Rank genotypes based on both mean performance and stability

See Rank entries relative to the "ideal" entry, but read entries as genotypes and testers as environments.

Also see Mean vs. Stability of the Entries.


Mean Performance vs. Stability of entries

From Biplot Tools, click Mean vs. Stability Coordination. The following will appear on the biplot:

•	a small red circle indicating the position of the average tester, which is defined by the average PC1 and PC2 scores across all testers. This average tester can be regarded as a virtual tester;

•	a thick red line that passes through the biplot origin and the average tester, referred to as the average-tester axis (ATA) or Average Tester Coordination (ATC) Abscissa;

•	a red arrow pointing to the average tester from the biplot origin;

•	a thick blue line that passes through the biplot origin and is perpendicular to the ATA;

•	two blue arrows on the thick blue line, pointing outwards from the biplot origin (called ATC Ordinate); and

•	a set of lines parallel to the thick blue line, which start from the marker of the entries and project to the ATA.

Interpretations on Mean Performance:

•	Based on their mean performance across all testers, the entries are ranked along the ATA, with the arrow pointing to a greater value. Thus the rank of the entries is:

Entry7 > Entry5 > Entry2 > Entry1 > Entry4 > Entry6 > Entry3.

•	The blue line separates entries with below-average means from those with above-average means. Thus the rank is:

Entry7 > Entry5 > Entry2 > Grand Mean >= Entry1 > Entry4 > Entry6 > Entry3.

•	This view is called Average-Tester-Coordination (ATC) view or Mean vs. Stability view for genotype-by-environment data; ATC is also called AEC (average-environment coordination) for GGE biplots.

Caution: The vector length of the average tester (the distance from biplot origin and the average tester marker), relative to the biplot size, is a measure of the relative importance of the entry main effect (G) vs. the entry by tester interaction (GE). The longer it is, the more important is G and the more meaningful the selection based on mean performance. At extremes, a zero average tester vector means G = 0 and therefore the selection based on mean performance is meaningless.

Interpretations on Stability:

•	A longer projection to the blue line, regardless of the direction, represents a greater tendency of entry by tester interaction for the entry, which means more variable and less stable across testers.

Caution: When the biplot explains only a small portion of the total variation, some entries may not be as stable as they appear, because they may have greater values in PC3 or PC4.

In Depth: Here "stability" is defined by GE alone. Such a stability index can only be used as a modification of the genotype main effect in genotype evaluation. Stability is meaningless if it is detached from the genotype main effects. Some researchers devoted their research solely on GE and tried to identify genes for stability. This is very misleading because there is no such thing as stability genes: high stability means performing consistenly well when associated with high mean performance and consistely poorly when associated with low mean performance. How could the two opposite things be controlled by the same genes

Also see Mean performance AND stability of entries


Mean and Stability of entries

From Biplot Tools, click Rank Entries/Testers…

Select With ref To an "Ideal" Entry. The following will appear on the biplot:

•	A red circle indicating the average tester;

•	A red line that passes through the biplot origin and the average tester, referred to as the "Average Tester Axis" (ATA) or ATC Abscissa;

•	A red arrow pointing to the "Ideal" entry, which is defined as an entry that is best for all testers ;

•	A Blue line with two arrows pointing outwards from the biplot origin (called the ATC ordinate);

•	A set of 5 concentric circles with the "ideal" entry being the concentric center.

•	The testers are represented by "E" to indicate that they are treated as "random" samples of the testers and to focus on the entries.

Note: If the biplot was not in the Entry-focused singular value partitioning (i.e., SVP =1), this partitioning method will be automatically applied.

Interpretation: The closer an entry is placed to this 'ideal" entry, the more desirable it is judged on both mean performance and stability. The ranking of the entries in this example is:  Entry7 > Entry5 > Entry2 > Entry1 > Entry6 > Entry4 > Entry3.

Compare: Mean vs. Stability of Entries.

Warning: If the testers are located on the both sides of the ATC ordinate, this interpretation will not be valid, because the entry main effect is not too small to be meaningful.


The relative performance of an entry relative to individual testers

From Biplot Tools, click Examine an Entry, an entry selection pane will appear above the biplot.

Click the Tester button, it will be changed to Entry, and the drop-down box will be filled with entries.

Click the drop-down arrow by the textbox to select the entry (genotype) to be examined and click the button Rank. The biplot will be changed to the following:

 The following will appear on the biplot:

o	A frame around selected entry (entry1, this case), with all other entries represented by "o" (or other symbols);

o	A red line passing through the biplot origin and the selected entry, which is referred to as  "Entry1 axis",

o	A red arrow pointing to the selected entry from the biplot origin,

o	A blue line passing through the biplot origin and perpendicular to the red line, and

Interpretation:

o	Based on the relative performance of the selected entry, the testers are ranked along the Entry1 axis, with the arrow pointing to a better relative adaptation of Entry1:

Test7 > Test5 > Test2 > Test3 > Test1 >= Test4 >= Test9 > Test6 >= Test8.

o	The blue line separates testers in which Entry1 had above-average values from those in the entry1 had below-average values:

Test7 > Test5 > Test2 > average > Test3 > Test1 >= Test4 >= Test9 > Test6 >= Test8. So Entry1 is most adapted to Test7, Test5, and Test2, but is not well adpated to the other testers.

Note: The distance between a tester and the Entry1 axis has no meaning (but compare with the mean vs. stability of entries view).

Caution: if the vector of the genotype is short (i.e., when the genotype is positioned close to the biplot origin), relative to the biplot dimension, this interpretation may not be meaningful or accurate, because this genotype tends to have similar relative performances in all environments.


Compare two genotypes for performance in various environments

See Compare two entries, but read entries as genotypes and testers as environments.

Note: The comparison of two genotypes based on the GGE biplot may not be accurate, depending on how much information regarding the compared genotypes is explained by the biplot. For more reliable comparison use one of these two functions:

1. Plot any TWO genotypes across the test environments

2. Compare THREE genotypes


Compare among any three genotypes

See Compare among any three entries, but read entries as genotypes and testers as environments.

Note: since the biplot would explain 100% of the information on the three genotypes, this function is more accurate than the function of Compare TWO genotypes based on the biplot. Therefore this function is recommended even if your purpose is to compare two entries (genotypes).


Compare two entries with regard to various testers

•	Under the main menu Biplot Analysis, click Compare Two Entries. A comparison device with two drop-down text boxes will appear above the biplot.

•	Select the two entries (entry5 and entry7, for example) to be compared from the two boxes, and click the Compare button. The biplot will become:

The two entries are connected by a red line (the connector), and a blue line is drawn, which passes through the biplot origin and is perpendicular to the connector, called equalizer. All other entries are represented only by a "c".

Interpretations:

•	The equalizer divides the biplot into two parts;

•	Entry5 and Entry7 would be equally good for all testers that are located on the equalizer; in the above biplot, the two entries seem to be quite similar wth regard to Tester2, which is almost on the equalizer;

•	Entry5 is better for testers that are on its side of the equalizer, Testers 5 and 7;

•	Similarly, Entry7 is better for testers that are on its side of the equalizer, i.e., Testers 3, 1, 4, 9, 6, and 8.

The above principle also applies when all testers are fall on the same side of the equalizer:

In the following biplot, Entry5 has larger values than Entry7 for all testers:

In the following biplot, Entry6 has larger values than Entry3 for all testers except Tester8:

Note:

1. This method of comparing two entries is the basis of the famous which-won-where function in GGE biplot analysis.

2. The comparison of two genotypes based on the GGE biplot may not be accurate, depending on how much information regarding the compared genotypes is explained by the biplot. For more reliable comparison use one of these two functions:

•	Plot any TWO genotypes across the test environments

•	Compare THREE genotypes


QTL identification - multiple traits in a single environment

This function requires a genotype by (marker+trait) two-way table. The analysis is similar to generating a covariable-effect biplot except that the explanatory variables are genetic markers and the response variables are various traits.

The analysis involves the following steps:

•	Under the main menu VBV Biplot, click Covariate-effect biplot based on..., then select r-matrix.

•	You will be asked for the number of markers so that GGEbiplot knows how many markers and traits are there in the genotype by marker+trait table. GGEbiplot will then generate a marker-by-trait table, each element of which is the Pearson correlation between a marker and a trait across genotypes.  (In the current version, you will be asked to separate the markers from the environments instead).

•	You will then be asked to set up a probability level, which will be used to remove irrelevant markers, i.e. those that had no significant associations with any of the traits.

•	You will be asked if you want to put a Type I error control on the biplot. This will be useful for you to remove markers with short vectors (therefore less important) from the biplot.

•	Generate a marker-by-trait biplot using Centering = 0, that is, no centering should be applied for this type of biplots.

The identification of QTL for 6 traits are illustrated below (achieving each of the biplots is just a mouse-click away):

1) The original marker by trait biplot with all markers:

2) Marker by trait biplot when markers with short vectors are removed:

3) Marker by trait biplot in which each possible QTL is represented by two linked markers:

4) Marker by trait biplot in which each possible QTL is represented by a single marker:

These biplots may be referred to as QQT biplot, in contrast to the QQE biplot for barley yield

NOTE: This biplot is more meaningful when the target traits are somehow inter-related.

In Depth:

Why the No-Centering option is used in the QQT or QQE biplotThis is because the values in the r-matrix are "effects" of the traits/markers on the target trait in each environment, and the purpose of the the QQT or QQE biplot is to visualize these effects. Any centering will distort the effects and any biplot on distorted data will be misleading.


Compare among any three entries

Under Data Manipulation, click Run any subset by .... Deleting entries, a subset tool will appear.

Select the three entries you want to compare (Entries 5, 6, and 7, for example):

A new biplot will appear, containing only the three selected entries. Under Biplot Analysis, select Which-won-where or what, the following biplot will appear:

Interpretations:

•	Entry5 is the best for Testers 5 and 7,

•	Entry7 is the best for Testers 6, 8, 3, 1, and 9,

•	Entries 5 and 7 do not differ much for Testers 2 and 4, as they are nearly on the equalizer between the two entries.

•	Entry6 is inperior to Entry5 in all the testers.

•	Entry6 is inferior to Entry7 in all the testers.

NOTE: Since the biplot explain 100% of the variation about these three entries, it allows accurate visual comparison among the entries.


Functions for host by pathogen data

Biplots are a powerful tool for studying host genotype by pathogen strain data. We will use the following data to illustrate the use of biplots. This dataset contains mean response scores of 13 barley groups of varieties to 9 barley net blotch strain or strain groups (data from Gupta and Loughman, Plant Disease 82:960-966, 2001).

•	Evaluate the host genotypes for general and specific resistance

•	Evaluate the pathogen strains for general and specific virulence, and

•	Evaluate the interaction between the host genotypes and the pathogen strains


Interaction between host genotypes and pathogen strains

The following biplot is most appropriate for visualizing host by pathogen interactions:

The biplot is based on model 4 (double-centered) so that the main effects of the host and the pathogens are removed; anything you see is host by pathogen interaction, or specific virulence or vertical resistance.

This biplot shows the following:

•	 The strains fall into three distinct groups: S-qnb851, S48+S78, and other strains.

•	 These strains have strong specific virulence:  S-qnb851 and S48+S78.

•	 These genotypes have strong vertical resistance: H8, H31, H22, and H-Betzes.

•	  S48 and S78 are particularly virulent to H8 and H31, H10, but not to others.

•	 S-qnb851is particularly virulent to H22 but not virulent to H-betzes and H31.

•	 H16, H32, H33, H30, H34 have no vertical resistance: they are either resistant (H16, H32) or susceptible (H30) to all strains (see resistance of genotypes).

To see the original data for the biplot, follow Functions for host by pathogen data

NOTE BUT: The use of the double-centered biplot is not recommended in all cases except for gene expression data. In most cases, G+GE must be considered together to make any sense in terms of entry evaluation or tester evaluation.


Evaluate pathogen strains for their general and specific virulence

This is a function when the biplot is used in visual analysis of host genotype by pathogen strain two-way data. For the evaluation of the virulence of the pathogen strains, the following biplot is most informative:

Note this biplot has the following settings (compare with host genotype evaluation):

1. host genotypes are treated as testers and pathogen strains as entries;

2. tester-centered model (centering = 2);

3. entry-focused scaling (SVP =1);

4. the average-tester coordination view.

Interpretations:

•	The red lines points to greater general virulence of the strains. Therefore, the general virulence rank is:

S48 > S78 >>> Average > other strains.

•	The virulence of S-qnb85i is highly unstable (i.e., specific): It is highly virulent to host H22 but much less so to H-Betzes. This suggests strong specific virulence or vertical resistance.

•	The strains fall into three distinct groups: S48+S78; S-qnb85i, and others.

•	Host H22, H8, and H-Betzes are best testers to differentiate the strains.


Evaluate host genotypes for their general and specific resistance

This is a function when the biplot is used in visual analysis of host genotype by pathogen strain two-way data. For the evaluation of host genotypes, the following biplot is most informative:

Note this biplot have the following settings (compare with pathogen race evaluation):

1. host genotypes are treated as entries and pathogen strains as testers;

2. tester-centered model (centering = 2);

3. entry-focused scaling (SVP =1); and

4. the average-tester coordination view

Interpretations:

•	The red lines points to larger response scores (higher susceptibility). Therefore, the general susceptibility rank is:

H30 > H-Betzes > ... > H33 = Average > ... > H10 = H32 > H16. H30 is most susceptible, and H16 the most resistant.

•	H8 and H31 are very unstable: They are highly susceptible to strains S48 and S78 but are resistant to other strains. This suggests strong specific (vertical) resistance.

•	The strains fall into two distinct groups: S48 and S78 as one group vs. the others as another group.


Compare all genotypes with a user-specified check

See Compare all entries with a check, and read "entries" as "genotypes".


Show the best genotypes for each environment

See Which-won-where, and read entries as genotypes and testers as environments.


Show relationship among test environments

See Show relationship among testers, but read testers as test environments.


Identify redundant environments or the need for more test environments

Click Show relationship among testers.

Redundant test environments are suggested:

1. When there is a strong association between two enivironments (indicated by long vectors and a small angle)

2. When the asociation is consistent across years

Need for more test environments (within a single mega-environment) are suggested if:

1. There are large gaps among test environments,

2. This is consistent across years, and

Note But: obvious gaps among environments may be strong evidence of the existence of different mega-environments.


Mega-environment classification based on the GGE pattern

The GGE biplot for a trait, e.g., yield, can be used to visually investigate whether the test environments can be divided into meaningful mega-environments. For example, the following GGE biplot is based on the yield data from a barley mapping dataset, using environment-focused SVP.

It can be seen that the 25 environments fall into two groups of environments. The group on the right contains all environments from Ontario (ON), Prince Edward Island (PE), Manitoba (MB), and one of the two Quebec (QC) environments. Since the environments in this group are located mostly in the eastern part of Canada, it may be regarded as the Eastern mega-environment, although it also contains two environments from Saskatchewan (SK), and one environment from Montana (MO, should be shortened as MT). The group on the left contains all environments from Alberta (AB), most environments from SK, and environments from some US states, including Washington (WA), North Dakota (ND), and Alaska (AK). One exception is that it includes a QC environment. Therefore, this group may be called the Western mega-environment.

Also see:

What are mega-environments{linkID=280}

Mega-environment vs. which-won-where

Mega-environment classification based on the covariate-effect biplot


Group environments based on winning genotypes

See Which-Won-Where.


Discrimitiveness vs. representativeness of test environments
Discrimination vs. representativeness of test environments

See Discrimination vs. representativeness of testers


Show relationship among testers

From Biplot Analysis, click Relation among Testers, and the vectors of the testers will show on the biplot. If the biplot was not in the tester-focused singular value partitioning (i.e., SVP = 2), this method will be automatically applied, which may change the biplot shape.

Interpretation:

•	When the data is sufficiently approximated by the biplot (How do I know it is sufficientlinkID=35}), the cosine of the angle between the vectors of two testers approximates the correlation coefficients between them.

•	Two testers are positively correlated if the angle between their vectors is <90°;

•	Two testers are negatively correlated if the angle between their vectors is >90°;

•	Two testers are independent if the angle between them is 90°.

•	0° means r =1.

•	180° means r = -1.

•	Testers with longer vectors are more discriminating of the entries; those with short vectors are less discriminating; those located at the biplot origin are not discriminating.

•	Caution: These interpretations are valid only for GGE biplots that are not sacled by within tesster standard-deviations)

Applications

•	Closely associated testers may suggest redundant testers (e.g., test environments)

•	Closely associated testers may be used in identifying variables (traits or makers) that can be used in indirect selection for a target variable.

•	Large gaps between testers may suggest that additional testers are needed to fully sample the target population. Alternatively, it may indicate that the testers fall in groups.

•	The largest angle in a biplot may be used as an indicator of the size of entry by tester interaction. The more the number of negative correlations, the larger the interaction between the entries and the testers.

Note:

•	When the biplot explain only a small portion of the total variation, say, <50%, the angles may not exactly reflect the correlations among the testers.

•	The accuracy of the relationships among testers as shown in the biplot can be examined by requesting a correlation matrix among testers.


Discrimitiveness and representativeness of test environments
Discrimination and representativeness of test environments

See Rank testers relative to the "ideal" tester, but read entries as genotypes and testers as environments.


Plot any two test environments across genotypes

See Plot any two testers across entries.


Rank all entries relative to a check

Under the main menu of Biplot Tools, click Rank Entries/Testers, and select With ref. to an entry/tester. An entry/tester selecting panel will appear on the top of the biplot.

If the drop-down box is labeled "Tester", then click the Tester button, it will be changed to "Entry".

Select an entry to be used as a check or reference, e.g., entry5, and then click the Compare button. The biplot will look like:

In this biplot, Entry5 is used as a reference, and concentric circles are drawn, with Entry5 as the center. The closer an entry is to the concentric center, the similar it is to Entry5.

The testers are represented as "E" to focus on the entries; full names of the entries and the testers can always be called back.


Correlation matrix among the environments

See Print the correlation matrix among testers.


Plot any two testers across entries

Under the main menu of Accessories, select Data Plotting, and select Two Testers. A Tester Selection Pane, containing two drop-down text boxes, will appear on the top of the biplot. Select two testers, say, Tester5 and Tester7, and click the button of Plot 2 Testers. The biplot will be changed to a mono-plot containing only entries:

This plot indicates the following:

•	Testers 5 and 7 are highly correlated, with r = 0.912;

•	As a result, most entries are around the equality line.

•	Entry4 is better for tester5, while entry3 is better for tester7.


Plot any two entries across testers

Under the main menu of Accessories, select Data Plotting, and select Two Entries. A Tester Selection Pane, containing two drop-down text boxes, will appear on the top of the biplot. Select two entries, say, entry3 and entry4, and click the button of Plot 2 Entries. The biplot will be changed to a mono-plot containing only the testers:

This plot shows the following:

•	Entry3 has higher values than entry 4 for testers 3 and 7; entry4 has higher values for other testers.

•	The correlation between entry3 and entry4 across the testers is 0.3.


Functions for multiple regression data

Multiple regression data are typically observation by variable two-way tables, in which one variable is the dependent variable and others are independent variables.

•	 GGEbiplot allows the user to select any of the variables as the dependent variable and treat other variables as independent variables. See Multiple linear regression.

•	 GGEbiplot allows the user to manually delete irrelevant variables before conducting multiple linear regression. See Delete Any Testers.

•	 GGEbiplot allows the user to automatically delete variables that are not significantly associated with the dependent variable before conducting multiple linear regression. See Find Associated Variables.

•	 GGEbiplot can remove variables that are only indirectly associated with the dependent variable from the multiple linear model. This function is available only after the multiple regression function has been evoked. See Remove collinear variables in multiple regression.

•	 GGEbiplot can search and include any two-factor interaction (epistasis in case of QTL mapping) terms between the independent variables in the regression model. This function is available only after the multiple regression function has been evoked. See Include any interaction terms in the model.

•	 GGEbiplot can include a quadratic term in the regression model. This function is available only after the multiple regression function has been evoked. See Include a quadratic term in the model.

Note: QTL identification is a special case of multivariate linear regression, in which a target trait is the dependent variable and genetic markers are independent variables.


Multiple linear regression

We use the barley QTL mapping data to illustrate this function. After deleting unwanted traits and removing irrelevant markers, we are ready to conduct multiple linear regression. (Note: these previous steps are not mandatory, although they simplifies the analysis.)

Under the main menu of Association, click Multiple Linear Regression, and select Conduct Regression. A dependent variable selector will appear on the top of the biplot.

Select the variable that you want to use as the dependent variable, here KW in this example, and click the Regress button. If the number of variables (testers) plus 2 is greater than the number of observations (entries), you will get the following message:

In our case, we had 145 observations, many more than the number of variables. So this was not a problem and multiple regression was conducted as requested. The regression results were printed to the log file, and the biplot was changed to the following:

There is much to say about this "biplot" (actually, the above image is not a biplot; it is just a "plot" of the testers, because the genotypes are hidden so that a better view about the "testers" can be achieved).

•	The testers include the target trait KW (framed and with a big font) and the markers that are associated with it at the selected 0.05 probability level.

•	All the markers together explained 72% of the total KW variation.

•	On the background, the regression results are printed. These results are also printed to the log file. They can be cleared from the biplot by clicking Refresh under the main menu View.

•	Groups of markers suggest independent QTL for KW.

In Depth:

GGEbiplot has a function to remove markers that are only remotely related to the QTL.


Delete unwanted variables

When the barley mapping data is displayed in a biplot, it looks like:

Note that it is based on tester-standard deviation standardized data so that the units of the traits are removed.

Suppose one wants to identify QTL for kernel weight (KW):

•	Under the main menu of Data Manipulation, click Run any subset by ... and select Deleting any testers. A form with two text list boxes will appear, which allows moving the testers between the two boxed freely. Now move all traits other than KW in the box "To Remove":

•	Click OK, the a biplot will appear, which contains only the markers plus KW:

In depth: Why double-haploid "31" stands out from other double-haploids because this genotype has a value of '0'  for all markers on chromosome 6, which are located on the other side of the biplot origin.


Remove collinear independent variables in multiple regression

This function becomes activated after multiple regression is completed.

•	Under Association, click Multiple Regression, and select Delete Variables on Log Ratio. An input box will appear, asking for a threshold value. The default is set to 2.5, which is roughly equivalent to 5% of probability.

•	Click OK, markers whose log ratios are smaller than 2.5 will be removed from the regression model. Multiple regression is conducted again with retained markers and the biplot will change to the following:

In the biplot, all retained markers can be said to have "independent" effect on the target trait kernel weight; each represents a different QTL. The number of QTL can actually be counted: it is about 7 to 10. These markers explained 69% of the KW variation.

If one uses a log ratio of 3.5, further markers will be deleted:

Now only 5 markers, representing probably 4 QTL are left. They explained 62% of the KW variation, implying that the 6 markers that were deleted explained only 7% of the KW variation.

In depth:

1) Do the two markers on chromosome 1, VATP57a and MWG626, represent a single QTL or two different QTL since they have independent effects and are closely linked with a distance of about 5 cM?  It is likely that they are flanking markers of a single QTL.

2) How log ratio is calculated


Include any interaction terms in the model

Applying the "Find Associated Variables" function to the genotype-by-(marker+trait) table, three QTL for barley yield are identified:

The (Harrington allele of) QTL represented by mwg665c and mwg502 have positive effects on yield, whereas that represented by mwg838 has a negative effect. They together explained 25% of the mean yield variation. Note that these QTL are identified based on the "genotype main effect" in this eample, and any genotype-by-environment interactions are excluded by averaging across environments.

Detect and include epistasis terms in the regression and biplot

GGEbiplot provides two options for adding epistasis terms in the linear regression model and the biplot. The first is possible epistasis among selected markers, and the second is possible epistasis among all markers. This function is activated only when multiple regression is performed.

Epistasis among selected markers

Under Association, click Multivariate Analysis, then Include Interaction (Epistasis) terms, and select Between Selected Markers. GGEbiplot will first calculate the residues from the current multiple linear model; it will then construct a new set of "variables", which are all combinations among the selected markers, and then select among these new variables that are significantly associated with the residues.  If any such variable is selected, GGEbiplot will put the selected variables together with the originally selected markers (i.e., the three markers on the above biplot) in the multiple regression, and re-conduct biplot analysis. If not, the following message will appear:

In our case, no significant interaction among selected markers were found, which is quite common in the Harrington * Tr306 barley mapping dataset.

Epistasis among all markers

Under Association, click Multivariate Analysis, click Include Interaction (Epistasis) terms, and select Between All Markers. GGEbiplot will first calculate the residues from the current multiple linear model; it will then construct a new set of "variables", which are possible marker*marker combinations among all markers, and then select among these new variables that are significantly associated with the residues.  If any such variable is selected, GGEbiplot will put the newly selected variables together with the originally selected markers (i.e., the three markers on the above biplot), and re-conduct the biplot analysis. Before showing the new biplot, however, the following message will show:

Unwanted variables include any variables that are not a marker or marker combination. For example, some traits may be associated with the target traits, which should be removed because the purpose is to identify QTL. This can be done following this example. No such variables were selected in our example.

The search for epistasis among all markers identified one interaction that had significant effect on yield: abg613-abg609a * abg378, with abg613 and abg609a closely linked on chromosome 2:

Re-conduct multiple linear regression using yield as the dependent variable reveals that this interaction explained additional 8% of the yield variation:

This finding has not been reported by original researcher of the barley mapping project. It wouldn't it be fascinating to follow up this finding and find out why two markers from different chromosomes interacted to influence barley yield.

Apply the log ratio method to remove redundant or indirectly associated variables leads to the following final results:

To summarize, we found three QTL, represented by markers mwg665c, mwg502, and mwg838, plus one epistasis, abg613 * abg378, which altogether explained 33.5% of yield variation.


Find Associated Markers

After deleting unwanted variables, the GGE biplot will contain only molecular markers plus the target trait kernel weight (KW). To limit markers to those that are significantly associated with KW, do the following:

•	Under the main menu Association, click Find Associated Variables.... You will be asked to supply a probability level to be used to remove less strongly associated markers:

•	Type a value (here 0.01) and click OK.

•	The tester selecting device will appear on the top of the biplot, whereby the target trait can be specified.

•	Select KW from the drop-down list as the target variable, and click Find.

•	Another input box appears, asking if you want to use any variables as covariable:

•	Click No to continue. (What would happen if you click Yes

•	GGEbiplot will calculate the correlation between KW and each of the markers and associated probability, based on which markers that are not significantly associated with KW will be excluded:

Also see: Find Associated Variables


Covariate analysis

This function is built with the association and regression analysis and is used to remove the effects of some variables on a target variable.

•	If you choose to remove the effects of some variables before conducting association analysis, a form with two list boxes will appear, which allows selecting any of the traits/markers as a covariate. Click OK when done.

•	The selected traits/markers will be used as covariate;

•	Multiple linear regression will be conducted using the target trait as dependent variable and the selected variables as independent variables

•	Residuals of the regression will be calculated for each observation (entry), which will then be used as the dependent variable in further regression Analysis.

•	The results of the covariate analysis will be printed to the log file.


Include any quadratic terms in the model

This functions becomes available after multiple linear regression. The following example is the multiple regression results using yield (YLD) as dependent variable. It indicates that kernel weight (KW), lodging, days to heading, and days to maturity were variables significantly associated with yield, and they altogether explained nearly 36% of the yield variation.

Under the main menu Association then Multivariate, click Include Quadratic Terms. The biplot becomes:

A single quadratic term KW*KW is added to the model, which explained an additional 4% yield variation. This indicates that although there is strong linear relationship between KW and yield, extremely large KW may lead to lower yield.

The computing process of adding quadratic terms to the model is similar to that of adding interaction terms to the model.


Log ratio for removing colliearity among variables
Log ratio for removing collinearity among variables

Log Ratio (LG) is a measure of the independent effect of a variable on a dependent variable. It is calculated as:

LG = n * log (SSr/SSf)

where n is the number of observations (entries), SSr is the residual sum of squares from the reduced model, i.e., the multiple regression model with a given variable, SSf is the residual sum of squares of full model, i.e., the model containing all independent variables.

Obviously, the more important the variable, the greater the LG, and the more the number of observations in the regression, the greater the LG. LG is related to the well-known F-test, and a LG of 2.5 is roughly a probability of 0.05.

NOTE BUT:

The LG can be 0 for two markers that are extremely closely located!!!!! In such a case, the QTL represented by these very closely linked markers will not be detected. This is a very serious preblem of this method. IF prevent this from happening, closely linked markers should be removed first so remaining markers are evenly distributed across the genome or chromesomes. The LG method can then be applied. One advantage of suing biplot is that it allows visualization of the marker linkages so that some markers can be removed visually before numerical analysis.


Data formats for analysis using GGEbiplot

GGEbiplot can read the following SEVEN types of data:

1. Two-way data - matrix

2. Two-way data - serial

3. Three-way data - traits in parallel columns

4. Three-way data - traits in a single column

5. Four-way data - traits in parallel columns

6. Four-way data - traits in a single column

7. The universal format: any way


Any-way data input and analyses

This data format is the most flexible of all seven formats GGEbiplot reads. It can have any number of columns and in any order. The only requirement is that different traits or variables are in parallel columns, as is required in many other statistical software packages such as SAS.

Four major groups of functions are associated with this type of data under the main menu "Anyway":

1. Univariate biplot

2. Multivariate biplot

3. Main effects ANOVA, and

4. Four-way data ANOVA


Multivariate Biplot

When the data are read as 3-way parallel, 4-way parallel, or Any-way (multi-way), and when the Multivariate Biplot option under the main menu of "Anyway" is selected, the following form will appear:

This device allows generating all possible types of multi-trait biplots. In the current example, "genotype" was selected as the entries, so a genotype by trait biplot will be generated when this form is closed.

If location and year are selected, then a environment by trait biplot will be generated, with the environment being a location-year combination.

If genotype and location are both selected, then a phenotype by trait biplot will be generated, with the phenotype being a genotype-location combination.

If genotype, location, and year are all selected, then a phenotype by trait biplot will be generated, with a phenotype being defined by a genotype-location-year combination.

All possible types of multivariate biplots can be generated using this device.


ANOVA-Main Effects

When the data are read as 3-way parallel, 4-way parallel, or Anyway parallel, and when the Main-effect ANOVA option under the main menu of "Anyway" is selected, the following form will appear:

This interface allows the researcher to choose any of the traits and to conduct ANOVA with regard this trait for any factors that are selected from the list of factors in the data. In this example, location and genotype are selected as factors. GGEbiplot will treat the genotype*location combinations as treatments and genotype and location as factor to generate and ANOVA table, and to test the significance of the treatment, genotype, and location. GGEbiplot will also conduct multiple comparisons among the treatments, the genotypes, and the environments.

GGEbiplot does the multiple comparison for any factor (e.g., genotype) by calculating an LSD-standardized mean value across other factors for each genotype such that if two genotypes differ by 1.0 or more, they will be considered as statistically different at the 0.05 or 0.01 level of significance.

The results will be written into the log file, which can be opened in Excel by clicking the "Open Log File" under the main menu bar "Help".


Univariate Biplot

A univariate biplot is a genotype by environment biplot regarding a trait, Here the "environments" can be locations, years, year-location combinations, treatments, blocks, etc.

When the data are read as 3-way parallel, 4-way parallel, or Anyway parallel, and when the Univariate Biplot option under the main menu of "Anyway" is selected, the following form will appear:

This device allows generating any possible biplot for a trait or variable. In the current example, a genotype by environment biplot for "trait1" will be generated, with environments being year-location combinations.

Using this device, a genotype by block biplot is also possible for a single environment, which can be used to investigate the consistency of the replications in the test.


Comma-delimited text file (*.csv)

Comma-separated values (csv) is a text file format that is regarded as universal data format, because it can be read by most machine-oriented programs. GGEbiplot can read various data files, and therefore it is not necessary to use csv files. However, if you need to generate a csv file, you can follow the following steps:

1. Open the data file with Microsoft Excel,

2. Make the data to look like one of the 7 data formats,

3. Put 'END" at the end of the header row,

4. From File, click SAVE AS...,

5. Select "Save as type" as "CSV (comma delimited) (*.csv)",

6. Click the SAVE button,

7. When a message box pops out, asking if you want to save to this format, click YES, and

8. Close the file. When asked if you want to save the changes, click NO.


Data file types for GGEbiplot

GGEbiplot can read Six types of data format in the following formats:

1. Microsoft Excel files (*.xls)

2. Microsoft Access files (*.mdb)

3. dbase data base files (*.dbf)

4. comma-separated text files (*.csv)

Hint: How to make a comma delimited text fileFollow these steps.


Vectors

A vector on a biplot is the (virtual) line (or distance) between the biplot origin and the label of an entry or a tester in the biplot. The vector for an entry is called the entry vector, and the vector for a tester the tester vector.


Correlation matrix among testers

Under Accessories, Select Conventional Analysis, and select Correlation matrix among testers. The correlation matrix will be calculated and printed to the log file. When opened with Excel, it will look like:


Two-way Table - matrix

Requirements for this data format are:

•	The first column is for entry names, and each subsequent columns are values for various testers (environment, trait, marker, etc.)

•	The first row is the header, contains a name for the entry column, followed by the names of the tester columns. If the data is in text format, the end of the header row must be indicated by the word "End" (case insensitive).

•	Each of the subsequent rows contains data for an entry, consisting of the entry name and the values for each of the testers with regard to the entry.

•	Missing cell can be indicated by anything (Use the same indicator for all missing cells in the same data sheet), but for text files it can not be left empty.

See the following example, note that the "end" is required only for text files (*.csv).


Two-way table - serial

Requirements for this data format are:

•	The data file should have 4 columns, representing 1) tester (e.g., test environment), 2) block (replicate), 3) entry (e.g., genotype), and 4) measured value for the trait, respectively.

•	The first row is the header, containing the names of the columns, 1) "Tester name", 2) "Block name", 3) "Entry name", and 4) "Trait name" four items. You could put "End" after the trait name, but this is not required.

•	Each subsequent row contains a single observation (a data point), and the 4 columns are for 1) the tester name, 2) the block (replication) name, 3) the entry name, and 4) the measured value, respectively.

•	The data do not have to be balanced in terms of entry by tester combinations. The number of blocks can differ for different testers.

•	There is no need to indicate missing values. If you do, missing cell can be indicated by anything except that it cannot be left empty for text files.

•	The number of missing cells, if any, will be replaced by the respective tester means and the user will be notified once missing cells are detected.

See the following for an example:

Note: This format is a special case of the Three-way data - parallel format in which there is only one trait.


How does GGEbiplot treat missing cells

This depends on the centering method, i.e., model that is used in generating the biplot.

GGEbiplot first calculates a two-way table, then subject the table to singular value decomposition (SVD), and thereby displays the rank-2 approximation of the table in a biplot.

If the data is replicated, the two-way table is calculated by simply averaging across all replications, regardless of the number of replicates. Therefore, if the number of replicates is different for different treatments (entry-tester combinations), the resulting two-way table can be biased. A solution to this is to use REML to generate a genotype by enivronment table of BLUP (best linear unbiased prodictors) and use this table to generate the biplot.  Current GGEbiplot does not have the capability to generate BLUPs. Another software such as SAS may be needed for this purpose.

The number of missing cells reported by GGEbiplot are the number of empty cells in the two-way table, not that in the input table. Before subjecting to SVD, the missing cells are replaced according to the model:

•	Replaced by the Grand Mean if Centering = 0;

•	Replaced by 0 if Centering = 1, 2, or 3.

There are other options for filling missing cells under the menu Data.


Three-way data - parallel

Requirements for this data format are:

•	The data file should have 3 + N columns, representing tester (e.g., test environment), block (replicate), entry (e.g., genotype), and measured values for each of the N traits, respectively.

•	The first row is the header, containing the names of the columns, i.e., "Tester name", "Block name", "Entry name", and names for each of the N traits. You could put "End" after the trait name, but this is not required.

•	Each subsequent row contains tester name, block (replication) name, entry name, and measured value for each trait, respectively.

•	The data do not have to be balanced in terms of entry by tester combinations. The number of blocks can differ for different testers.

•	Missing cells can be indicated by anything. For text files, however, it cannot be left empty.

See the following as an example:

Related: How to contruct a 3-way data table?


How to contruct a 3-way or 4-way data table?
How to contruct a 3-way or 4-way data table

You can do this manually be prepare data from each year and then stack them together, but this is tedious and takes a lot of time. We use a piece of software that we have developed called Data Unification and Data Distillation or DUDE to do this. This Crop Information System was developed by Yan and Tinker (Agronomy Journal 2007). It is easy to use and is freely available from the authors.


Three-way data - serial

Requirements for this data format are:

•	The data file should have 5 columns, representing 1) tester (e.g., test environment), 2) block (replicate), 3) entry (e.g., genotype), 4) trait, and 5) measured value for the trait, respectively.

•	The first row is the header, containing the names of the columns, namely, 1) "Tester name", 2) "Block name", 3) "Entry name", and 4) "Trait name" , and 5) "Value". The end of the header should be indicated by the word "End" if the file is in text format.

•	Each subsequent row contains a single observation (a data point), and the 5 columns are for 1) the tester name, 2) the block (replication) name, 3) the entry name, 4) trait name, and 5) measured values, respectively.

•	The data do not have to be balanced in terms of entry by tester combinations. The number of blocks can differ for different testers.

•	Missing cells can be indicated by anything except that it cannot be left empty for text files.

Here is an example:


Four-way data - parallel

This type of data has the following features:

•	It should have 4 + N columns, with the first 4 columns representing Year, Tester (e.g, location), Block (i.e., replication), Entry (e.g., genotype), and followed by N columns of the traits.

•	The first row is the header, containing the names of the columns, i.e., Year, "location name", "Block name", "Entry name", and names for each of the N traits. You could put "End" after the trait name, but this is not required if it is an Excel or Access file; it is required only if the data file is in csv data format.

•	Each subsequent row contains an observation or entry, wiht values for year, location name, block (replication) name, entry name, and measured value for each of the traits, respectively.

•	The data do not have to be balanced in terms of entry by location combinations. The number of blocks can differ for different testers.

•	Missing cells can be indicated by anything except that it cannot be left empty for text files.

Below is an example data file:

Related: How to contruct a 4-way data table?


Four-way data - serial

•	This data format should have 6 columns, representing 1) year, 2) location, 3) block (replicate), 4) entry (e.g., genotype), 5) trait, and 6) measured value for the trait, respectively.

•	The first row is the header, containing the names of the columns, namely, 1) "Year",  2) "Location name", 3) "Block name", 4) "Entry name", 5) "Trait name" , and 6) measured values. The end of the header should be indicated by the word "End" if file is in text format for a text file. "End" is not needed if the file is an Excel or Access file.

•	Each subsequent row contains a single observation (a data point), and the 6 columns are for 1) year, 2) tester name, 3) block (replication) name, 4) entry name, 5) trait name, and 6) measured value, respectively.

•	The data do not have to be balanced in terms of entry by tester combinations. The number of blocks can differ for different testers.

•	Missing cells can be indicated by anything except that it cannot be left empty for text files.

Example:


Start GGEbiplot

•	Start the program from the Windows Start/Program Files/GGEbiplot if you have GGEbiplot installed. If not, double-click the executable file GGEbiplot.exe or its shortcut.

•	Once started, a window with three buttons, namely, Start, Exit, and Help, will appear:

•	Click the Start button. Another window, the Spreadsheet window,  will appear with the following buttons: Open Data (*.*), Open Data (*.csv), Biplot Analysis, Close, and Exit.

•	If your data is of Excel or Access format

•	Click Open Data (*.*), and an Open File dialog will appear for you to select a file.

•	If the file contains only one table, the table will be opened directly on the Spreadsheet; otherwise you will be prompted to select a table from the list box on the top-left of the Spreadsheet window.

•	The data sheet will be opened:

NOTE: This dataset is used extensively in this manual to demonstrate various functions of GGEbiplot. You could go back to this table to verify the interpretations based on the biplot are accurate.

•	Click the Biplot Analysis button after the table is opened on the spreadsheet.

•	A window will appear, asking you to select among 6 data formats. Select the appropriate data format and click OK.

•	A model selection form will appear for you to select a data Scaling method and a data Centering method:

A biplot based on the data will show on the screen in one or a few seconds, depending on the dataset size.

If your data is of "*.csv":

•	Click the Open Data (*.csv) button. A message box will appear to guide you to open a csv file from the Biplot window, which will appear when you close the message box.

•	Click the File...Open a text data file, and an Open File dialog will appear for you to select a file.

Comments: You will find it surprisingly fast to generate a well labeled, nice-looking, publication quality, fully functional biplot using GGEbiplot; much faster than any other available methods. Furthermore, generating a biplot is just the beginning, rather than the end, of biplot analysis. This is the more compelling reason why you need GGEbiplot for biplot analysis. My estimation is, and I think your experience will confirm, that using the GGEbiplot is one million times more efficient than using a good SAS macro in terms of both biplot construction and analysis.


Main Menu FILE

The following functions are organized under File:

•	New job

•	Open a text data file

•	Save image as

•	Print

•	Close

•	Exit


Save biplot image

Under File, click Save image as ...

This saves the current view of the biplot into an image file of "jpeg"  (jpg) or "bmp" (JPG files are preferred as BMP files take too much computer memory.

In addition, there are other methods that can be used to save the biplot image:

1.	Save the image by "printing" it into a pdf-file if you have installed an Acrobat Adobe writer.

2.	Save it as a picture by 1) hitting the "PrtSc" key on your keyboard; 2) pasting it to a photo editor such as Microsoft Photo Editor, Paint, etc., and 3) cutting the useful part, paste it to a new window, and save it to a file.

Hint:

•	A pdf-file has much higher resolution than a picture file, but the latter can be easily inserted into other Microsoft Office documents. You need to have Adobe PDF Writer to print a biplot to a pdf file. Alternatively, you can use an Apple LaserWriter to print the biplot into a "*.eps" file, which is a wrapped pdf file and can be inserted into other Windows documents.

•	The Apple LaserWriter printer is inherent in your Windows. You can set it up by adding printers from the Control Panel.


Open a text data file

Under File, click Open a text data file.

•	Saves the log file to a comma separated text file (*.csv) file so that it can be opened automatically using Excel,

•	Starts the open file dialog so that you can select a comma separated text file (*.csv).

Note:

•	Although Excel files are preferred, sometimes csv files are necessary. This is true when the data has more than 255 columns, which are beyond the capacity of Excel or MS Access.


Print biplot

Under File, click Print.

•	Prints the current view of the biplot using any printer connected to the computer.

•	If the computer is installed with Adobe Writer, the current image can also be "printed" to a pdf file. A pdf file has a much better resolution than a bitmap file; it also has some editing capacities. This is the preferred way to save a biplot.

•	Another way to print into a pdf file is to use an Apple Laser printer, which can be installed from the Windows' "Add Printer" options. Howeer, the quality of the file is not as good as that from ADOBE Writer.


Close

Under File, click Close. The following will happen:

•	Saves the log file to a comma separated text file (*.csv) so that it can be opened using Excel when double-clicked,

•	Closes the biplot window.

Note: The biplot window can also be closed by checking at the top right corner. However, When this is done, the log file will not be renamed to a csv file, which can be opened with MS Excel.


New project

Under File, click New Job.

This function:

•	Saves the log file to a comma separated text file (*.csv) so that it can be opened using Excel when double-clicked,

•	Closes the current biplot window, and

•	Starts the open file dialog to allow you open a new data file or open a new table from the opened Excel or Access file.


Exit GGEbiplot

Under File, click Exit.

•	Saves the log file to a comma separated text file (*.csv) file so that it can be opened automatically using MS Excel,

•	Exits GGEbiplot.


Main Menu VIEW

This group of functions allows modifying the appearance of the biplot.

Relabeling

•	Label Testers with anything you like, allows labeling the testers with anything you type from the keyboard.

•	Label Entries with anything you like, allows labeling the entries with anything you type from the keyboard.

Shortening the labels

•	Testers with FIRST few Letters, allows labeling the testers with FIRST few letters of their original names.

•	Entries with FIRST few Letters, allows labeling the entries with the FIRST few letters of their original names.

•	Testers with LAST few Letters, allows labeling the testers with LAST few letters of their original names.

•	Entries with LAST few Letters, allows labeling the entries with LAST few letters of their original names.

•	Testers WITHOUT First few Letters, allows labeling the testers WITHOUT FIRST few letters of their original names.

•	Entries WITHOUT First few Letters, allows labeling the entries WITHOUT FIRST few letters of their original names.

•	Testers WITHOUT First few Letters, allows labeling the testers WITHOUT LAST few letters of their original names.

•	Entries WITHOUT First few Letters, allows labeling the entries WITHOUT LAST few letters of their original names.

Show/Hide labels

•	Full Name, shows the full names of the entry and tester labels.

•	Index-Legend, shows the entries and/or testers as indices and open at the same time a Legend window that connects the indices and the full names.

•	Show Testers only

•	Show Entries only

•	Show Both (entries and testers)

Show/Hide other information

•	Refresh, removes the concentric circles or noises on the background of the biplot.

•	Show/Hide Title

•	Show/Hide Logo

•	Show/Hide Guidelines

•	Show/Hide Data Source


Testers with FIRST N Letters

Under View, click Label with first N letters..., and select Testers. An input box will appear, allowing you to specify "N". When closed, the testers will be represented by their first N letters. No change will be made if the tester name is shorter than N letters.


Main Menu 4-WAY DATA

GGEbiplot provided options to dissect a year-location-genotype-trait 4-way table or a environment-by-genotype-by-trait 3-way data into the following two-way tables and corresponding biplots:

Genotype-by-environment biplots

•	Genotype-by-year biplot for any trait

•	Genotype-by-location biplot for any trait

•	genotype-by-environment (year-location) biplot for any trait

Genotype-by-trait biplots

•	Genotype-by-trait biplot in any environment (year-location)

•	genotype-by-trait biplot across selected environments

•	genotype-by-trait biplot across all environments

Trait Association By Environment (TABE) biplots

•	Associations among selected traits

•	Associations with a trait

VBV biplots

•	Covariate-effect biplot based on genotypic values

Environment-by-trait biplots

•	Year-by-trait biplot

•	Location-by-trait-biplot

•	Environment-by-trait biplot

Compound two-way biplots

•	Genotype*environment-by-trait biplot

•	Genotype-by-environment*trait biplot

Other biplots

Year-by-location biplot for any trait

For 3-way and 4-way data with traits in parallel columns, these biplots and ANOVA options are also available.

Note: For 3-way datasets or 4-way datasets with fewer than 2 years, the year-related two-way biplot options will not be available because a biplot with the year as one way requires at least 3 years.


Genotype-by-year*location table for a trait

Under the main menu 4-Way Data, click Geno by Yr-Loc table for one trait.

A trait selection device will appear on the top of the biplot. Select a trait from the drop-down list and click Get Biplot. The genotype by yr-loc biplot for this trait will appear.

Note:

•	Here each year-location combination is treated as an environment, and the interpretation is exactly the same as for a genotype-by-environment biplot.

Hint:

•	A genotype by yr-loc biplot allows studying the repeatability of the genotype by location interaction patterns across years, which is important for dividing the target environment into meaningful mega-environments. For example, the following GGE biplot for the yield data of 13 wheat varieties tested in Ontario, Canada, across 1996-1998,

can be displayed as a genotype by year biplot

or a genotype by location biplot:

They are exactly the same biplot and differ only by the way of displaying the labels. Comparing these biplots reveals that the year factor is more important than the location factor for the genotype-by-environment interaction patterns because the 1996 environments are more or less distinct from the 1997 and 1998 environments, whereas the locations tend to scatter across the biplots. Some locations such as NN (Narin),  RN (Ridgetown) and WE (Woodslee) seem to be relatively repeatable across years whereas others such as EA (Elora), OA (Ottawa) and WK (Woodstock) more variable.

Related topic: mega-environment identification.


Genetype-by-year table for a trait
Genotype-by-year table for a trait

•	Under the main menu 4-WAY, click Geno by yr table for any trait. A trait selection device will appear on the top of the biplot, allowing selecting one of the traits in the 4-way dataset.

•	Select a trait and click Get Biplot. The biplot based on the genotype by year data for that trait will appear. The genotype by year table for the trait is constructed by averaging across all locations in each year for the selected trait.

•	The interpretation of the biplot is exactly the same as for a genotype-by-environment biplot, but read years as "environments".


Genotype-by-trait table in an environment

•	Under the main menu 4-WAY, click Geno by trait table in any environment. An environment selection device will appear on the top of the biplot, which allows selecting one of the environments in the 4-way dataset. The environments are year-location combinations if different years are involved..

•	Select an environment and click GT Biplot. A biplot will appear, which is based on the genotype by trait two-way data in the selected environment.

See this page for interpreting a genotype-by-trait biplot.

In depth:

Comparing genotype by trait biplots in contrasting environments may reveal important information on the genotype by environment interactions in the data.

The following example is based on the 2003 Eastern Canada Spring Oat Trials data.

Comparing the genotype by trait biplot in Woodstock NS

with that in Hebertville QC

reveals that plant height may be responsible for the observed genotype by environment interaction for oat yield, because it had contrasting effects on yield in the two environments.

Covariate-effect biplot analysis is a more systematic approach for identifying explanatory traits that can explain the observed genotype-by-environment interactions regarding a trait of interest.


Genotype by location table from 4-way data

•	Under the main menu 4-WAY, click Geno by loc table for any trait. A trait selection device will appear on the top of the biplot, allowing selecting one of the traits in the 4-way dataset.

•	Select a trait and click Get Biplot. The biplot based on the genotype by location data for that trait will appear. The genotype by location table for the trait is constructed by averaging across all years for each location for the selected trait.

•	The interpretation of the biplot is exactly the same as for a genotype-by-environment biplot, but read locations as "environments".


Genotype-by-trait table across environments

If your data are read as a 3-way or 4-way data, under the main menu of 4-WAY, click Geno by trait table ...in all environments. A genotype by trait biplot will appear, which is based on a genotype by trait two-way table averaged across all environments.

See this page for interpreting a genotype-by-trait biplot.


Genotype-by-(trait+environment) tables from 4-way data

See generating a covariate effect biplot from a three-way or four-way dataset.


Associations (among traits) by environment biplot
Relationship among traits in individual environments

Under the main menu 4-Way, click Trait Association by Environment biplot - Associations among selected traits.

A trait selection form containing two list boxes will appear, allowing you select traits whose interrelationships are to be investigated.

GGEbiplot will first make a list of "relations" or 'Associations', i.e., pairs of selected traits. If N traits are selected, there will be N(N-1)/2 possible relations.

GGEbiplot will then generate a trait association by environment two-way table, which consists of correlation coefficients between each pair of traits in each of the environments.

A biplot based on this table, without any data transformation or centering, will appear. The following biplot is based on the 2003 Eastern oat screening trials, from which several agronomic traits were selected:

The biplot shows that

1) Seven relations (pairs of traits) (in blue) were significant at the 1% level in at least one of the five environments (in red). The relations with long vectors were stronger than those with shorter vectors. Thus, the relation between Days to Heading and Days to Maturity was strongest, as one would expect, followed by the relation between Days to Heading and Plant Height. These two relations are positive in all environments, as indicated by the angles between their vectors relative to those of the environments.

2) On the opposite, the relation between days to heading and yield seem to be negative in all environments.

3) Most interesting is the relation between plant height and yield, which was positive in HERBERT, NORMAND, and PRINCEWILLE, but negative in PEI and WOODSTOCK. This can be interpreted as that plant height was an important cause for the observed genotype by environment for oat yield.


Trait Association By Environment (TABE) biplot

For 3-way or 4-way data, GGEbiplot offers two types of TABE Biplots:

1. Association (among traits) by environment biplot, and

2. Association (with a trait) by environment biplot


Environment-by-trait tables from a 4-way data
Environment-by-trait tables from a 4-way dataset

GGEbiplot generates three types of environment-by-trait tables and corresponding biplots out of a year-location-genotype-trait 4-way table:

•	Year-by-trait table

•	Location-by-trait-table

•	(Year*location)-by-trait table

Such tables and biplots might be more useful to production agronomists and plant physiologists to understand:

1) Environmental correlations among traits, and

2) Environments that are good for producing particular parts (traits) of the crop (yield, oil, protein, etc.).


Main Menu BIPLOT TOOLS

This is the core of GGEbiplot, which allows a biplot to be visualized from all possible perspectives:

Functions for entry evaluation

•	* Which-won-where

•	* Mean and stability

•	Rank entries based on any given tester

•	Discarding entries based on a tester

•	Rank entries based on a tester

•	The relative performance of an entry relative to individual testers

•	Rank entries relative to the "ideal" entry

•	Compare two entries

•	Compare all entries with a check

Functions for tester evaluation

•	* Show relationship among testers

•	Identify redundant environments or need for more test environments

•	Group environments based on the genotype-by-environment interaction pattern

•	Group environments based on winning genotypes

•	Show discriminating ability and representativeness of the test environments

•	Rank test environments relative to the "ideal" test environment

Others

•	Linear map of testers

•	Interrelationship among entries

•	Flip vertically

•	Flip horizontally

•	Switch axes

*Most commonly used functions


Switch biplot axes

Under Biplot Analysis, click Switch axes.

The X and Y axes of the biplot will be switched in response this function. Note that this does not affect the biplot interpretation.


Flip biplot horizontally

Under the main menu of Biplot Analysis, click Flip horizontally. This will not affect the interpretation of the biplot.


Flip biplot vertically

Under the main menu of Biplot Analysis, click Flip vertically. This will not affect the interpretation of the biplot.


Compound two-way tables

A genotype-by-environment-by-trait three-way table can be re-organized (collapsed) into, among others, the following two compound two-way tables:

•	A genotype*environment-by-trait table

This is actually a phenotype-by-trait table, and is most useful in visualizing the phenotypic correlations among traits.

•	A genotype-by-environment*trait table

Such arrangement treats the same trait in different environments as different variables, and is a fuller use of available information for visual classification of the genotypes.

These tables and corresponding biplots can be requested from the main menu 4-Way.


Combined two-way tables

A combined two-way table is one that combines two different two-way tables. They can be one of the following:

•	Genotype-by-(trait+environment) tables

•	Genotype-by-(marker+environment) tables

•	Genotype-by-(marker+trait) tables

Note: The common characteristics of this type of two-way tables is that the variables (columns) are of two types: the explanatory variables and the response variables. The relationships between these two types of variables, as well as those within them, can be investigated using the variable-by-variable (VBV) biplot.


Generate a covariate-effect biplot from three-way or four-way data
Generate a covariate-effect biplot from 3-way or 4-way data

For a 3-way or 4-way dataset, covariate-effect biplots can be generated in two ways, under the 4-Way main menu:

1) Covariate-effect biplot based on genotypic values, which will first generate a genotype by trait+environment two-way table and then utilize the module Covariate-effect biplot based on genotypic values.

2) Covariate-effect biplot based on phenotypic values, which will first generate a covariate-effect table of correlation coefficients, and then display it in a covariate-effect biplot.

Which one is better

A covariate effect biplot based on phenotypic values probably makes more sense, because it looks into the target trait vs. explanatory trait relations in the individual environments. However, since it is common that not all traits are measured in all environments, there could be many missing cells. When this is the case, covariate effect biplot based on genotypic values may be more feasible. The latter takes the same form as the data a QQE biplot is based, in which the explanatory variables are genetic markers ("genotypes"!).

Compare: VBV biplot.


Main Menu MODELS - old

See Functions under MODELS-new for more options

For each two-way tables, biplots can be generated on any of the SIX built-in models:

•	Model 0 generates biplots based on singular value decomposition of the grand-mean centered data.

•	Model 1 generates biplots based on singular value decomposition of tester-centered data. It is used for dataset in which all testers have the same unit, such as a genotype-by-environment table of a single trait. This model may be called "GGE1".

•	Model 2 generates biplots based on singular value decomposition of within-tester standard deviation standardized data. It is mostly used for datasets in which different units have different testers (such as a genotype-by-trait table). It can also be used in analyzing a two-way table of any trait when all testers are assumed to be equally important. This model may be called "GGE2".

•	Model 3 generates biplots based on singular value decomposition of within-tester standard error standardized data. This option is available only when the data are replicated so that the standard error for each tester can be calculated. This model supposedly can remove any heterogeneity among the testers. This model may be called "GGE3".

•	Model 4 generates biplots based on singular value decomposition of entry and tester double-centered data. This mode is effective in the following situations: 1) to study the genotype by environment interaction (GE) of a multi-environment dataset; 2) to study the specific combing ability (SCA) of a diallel cross dataset; and 3) to analyze gene expression data (or microarray data) to identify gene-treatment correspondences.

•	Model 5 generates biplots based on the original two-way table. This model is essential for the so called covariate-effect biplot analysis.

Hint: For GGE biplot analysis, Model 3 is preferred and should be used as soon as feasible. See Yan et al. (2008).

Note: Biplots based on different models are compared through the above links using this dataset:


Principal component analysis

Principal component analysis (PCA) is a well-known multivariate analysis technique aiming at summarizing data with reduced dimensions. Its focus is on the interrelationships among testers (variables) rather than the entries (observations). Therefore, PCA generates a "plot" of testers rather than a "biplot" of both testers and entries.

Mathematically, PCA is a process that decomposes the covariance matrix of a matrix into two parts: eigenvalues and column eigenvectors, whereas Singular Value Decomposition (SVD) decomposes a matrix per se into three parts: singular values, column eigenvectors, and row eigenvectors. The relationships between PCA and SVD lie in that the eigenvalues are the square of the singular values and the column vectors are the same for both.

Gabriel (1971) was the first to propose the use of biplots in displaying the results from PCA, where the term was equivalently used for SVD. In a PCA biplot, the row vectors are loadings on the column eigenvectors (regression coefficients of the rows of the original matrix on the column eigenvectors).

The definition of PCA is rather ambiguous in the literature; it is variably referred to model 0, model 1, or model 2 by different researchers (see references on this). It is more accurate to use SVD after specific centering and scaling methods instead of PCA when referring a particular biplot or analysis.


Main Menu FORMAT

This group of functions allows modification of the biplot image.

Plot Title: allows adding a name for the current biplot.

Plot Size

Color Scheme

Individual label Formatting

Change Color

Change case

Change Font


Change Font

For both entry and tester labels

•	Larger

•	Smaller

•	Change to specific font size

For entry or tester labels

•	Tester Labels

•	Entry Labels

•	Plot Title


Change color

The color of the following components can be changed:

•	Background

•	Lines

•	Tester Labels

•	Entry Labels

•	Plot Title


Main Menu BIPLOTS

The term "biplot" usually refers to the biplot of PC1 vs. PC2. However, there are other types of biplots, which might be useful in helping the analysis (follow this link). In particular, the 3-D biplot module is very useful for a better understanding the data.

In addition to the default biplot of PC1 vs. PC2, which may be referred to as the "primary biplot", GGEbiplot provides biplots of a higher order or secondary biplots:

•	PC3 vs. PC4,

•	PC5 vs. PC6,

•	PC1 vs. PC3,

•	PC2 vs. PC3

For users who are familiar with AMMI (Additive Main effect and Multiplicative Interaction) analysis, GGEbiplot also generates the two most commonly used AMMI biplots:

•	AMMI1 biplot (Main Effect vs. IPC1), and

•	AMM2 biplot (IPC1 vs. IPC2)

In addition, GGEbiplot also provide a 3D-biplot viewer:

•	3D-biplot

Note:

•	A higher order biplot may be useful when patterns are found in the biplot of PC1 and PC2.

•	When the doubled-centered model is used, i.e., when entry and tester main effects are removed from the data before singular value decomposition, the PC's are referred to as IPC's, indicating that the PC's contains only the interaction component of the total variation. However, I don't see any use of this biplot except for gene expression data.


Genotype-by-location table for any trait

If the data are read as a 3-way or 4-way data, under the main menu of 4-Way, click Geno by Loc for one trait.

A trait selection device will appear on the top of the biplot, allowing you to select a trait. Click Get Biplot, the biplot based on the genotype by location table for that trait will appear.

If there are data from more than one year, the genotype by location table will consist of values averaged across years.


Main Menu DATA

This is a group of functions that makes GGEbiplot many times more efficient. Biplots based on any subset of the data can be conveniently generated on the fly without modifying the data file.

•	Entry/Tester Switch Roles.

It transposes the data matrix so that the rows are treated as columns and columns as rows.

•	Filling Missing Cells

•	Derived values

•	Data Subsetting

•	Reverse the Sign of one or all testers.

This is useful for visualization of traits for which the values of a tester are so recorded that greater values means less desirable. Examples are disease scores, days to maturity, plant height, some quality measures, etc.

•	Diallel Data manipulation

•	Entry stratification

•	Tester stratification

•	Back to Previous Subset:

•	Back to Original Data

These modifications will NOT change your data file. You can always go back to the original data matrix.


Fill Missing Cells

GGEbiplot provides 4 options to fill missing cells:

1) For each variable (column), fill the missing cell of an observation (row) using the weighted average of 5 observations (rows) that are most similar to the observation.

2) For each column, fill the missing cells using the mean across all other rows.

3) For each row, fill the missing cells using the mean across all other columns.

4) Fill the missing cells using the average of the respective row mean and of the respective column mean.

Number #2 is the default method of filling missing cells in GGEbiplot, but #1 might be more rational.


Remove reciporical effects - diallel analysis
Remove reciprocal effects

From Data manipulation, click For Diallel Data, and select Remove reciprocal effect.

This function averages pairs of reciprocal crosses and given them the same averaged value.


Derived values

From the main menu Data, click Derived Variables.

This function allows generating new variables from any two of the existing variables by one of the four operations: Add, Subtract, Multiply, and Divide.


Get partially balanced Subset

By removing unbalanced entries

•	From Data, click Get a Partially Balanced Subset by…. and select Removing Entries.

•	An input box will appear, asking for a tolerance rate for the unbalancedness. Put a number between 0 and 100, and click OK.

•	Entries that have unbalancedness greater than the value you put will be deleted, and a biplot based on remaining subset will appear.

By removing unbalanced testers

•	From Data, click Get a Partially Balanced Subset by…. and select Removing Testers.

•	An input box will appear, asking for a tolerance rate for the unbalancedness. Put a number between 0 and 100, and click OK.

•	Testers that have unbalancedness greater than the value you put will be deleted, and a biplot based on remaining subset will appear.

More options for subsetting.


Tester stratification

Test stratification is to generate subset of testers based on their placements in the biplot. The following options are available:

•	Testers with long vectors

When this function is invoked, GGEbiplot first calculate the vector length for each of the testers and express them relative to the longer vector, and then ask the user to set a criterion, against which testers with LONGER vectors are selected and retained. A new biplot based on the selected subset of testers will be generated.

•	Testers with short vectors

When this function is invoked, GGEbiplot first calculate the vector length of all testers and express them relative to the longer vector, and then ask the user to set a criterion, against which testers with SHORTER vectors are selected and retained. A new biplot based on the selected subset of testers will be generated.

•	Testers within a sector

When this function is invoked, GGEbiplot first calculates the angles of the testers, relative to an arbitrary reference point. It then asks the user to set a sector defined by an upper and a lower bound; all testers WITHIN this sector will then be retained and a new biplot generated.

•	Tester outside a sector

When this function is invoked, GGEbiplot first calculates the angles of the testers, relative to an arbitrary reference point. It then asks the user to set a sector defined by a upper and a lower bound; all testers OUTSIDE this sector will then be retained and a new biplot generated.


Subset

•	Get Balanced Subset:

This allows getting a balanced subset of the data by removing entries or testers that have missing cells.

•	Get Partially Balanced Subset:

This allows getting a balanced subset of the data by removing entries or testers that have missing cells.

•	Get any Subset:

This allows visualizing a subset of the original data by removing any of the entries and/or testers.

•	Get a balanced subset with regard to specific entries.

•	  Get a balanced subset with regard to specific testers

For other subset options see:

•	Entry stratification based on the biplot pattern.

•	Test stratification based on the biplot pattern


Diallel Data manipulation

The following data manipulation options are specific to diallel data:

•	Complete The Two-Way Table:

useful when the data file contains a triangle with n(n-1)/2 cells.

•	Delete the Parent Per se Values:

deletes the diagonal values of the two-way table and replace it with respective means.

•	Remove reciprocal effects:

When there are reciprocals, this functions averages the reciprocals.

Related issues: Functions for diallel cross data analysis


Genotype-by-trait table across selected environments

Under the main menu 4-Way, click Geno by Trait table in Some Environments. A form containing two list boxes, labeled as "to Select" and "to De-Select", will appear, allowing you to select any one or any number of the environments (year-location combinations).

After the form is closed, a biplot will appear that is based on the genotype by trait two-way table with values averaged across the selected environments.

See this page for interpreting a genotype-by-trait biplot.


Singular value partitioning

There are three options for Singular Value Partitioning (S.V.P.)

•	Entry-focused: The singular value is entirely partitioned into the entry eigenvectors, i.e, alpha = 1, beta = 0. This SVP is needed for accurate comparison among the entries. This is also called row metric preserving, leading to "JK biplot".

•	Tester-focused: The singular value is entirely partitioned into the tester eigenvectors, i.e, alpha = 0, beta = 1. This SVP is needed for accurate visualization of the relationship among the testers. This is also called column metric preserving, leading to a "GH biplot".

•	Symmetrical: The singular value is symmetrically partitioned into the entry and the tester eigenvectors (alpha = 0.5, beta = 0.5). This SVP is most often used in AMMI analysis and other biplot analysis, but it is not ideal for visualizing either the relationship among entries or that among the testers.

Note:

•	There are numerous other possible ways to partition the singular values, and their interpretations are not defined.

•	All singular value partitioning methods are equally valid in approximating the two-way table and in showing the which-won-where pattern.

Related topics: Singular value decomposition.


Main Menu Association

This group of functions utilizes correlation and multiple regression techniques, assisted by biplots, to study the associations of various variables with a target variable (trait):

Find Associated Variables

•	find variables associated with a target variable

•	Remove less associated variables

Multiple Regression

•	multiple linear regression use any of the variables as dependent variable and all others as independent variables

•	remove variables that are only indirectly associated with the dependent variable

•	add two-variable interaction terms in the model

•	add quadratic terms in the model

Also See: Association and regression analysis


VBV biplot

"VBV biplot" was previously referred variably to QQE biplot, Covariate-effect Biplot, or "Canonical Biplot", etc. However, "variable-by-variable (VBV) biplot" seems to be more accurate for the biplot that GGEbiplot generates to depict the interrelationships between two types of variables.

The term "VBV analysis" is used in this documentation to refer to studies of the interrelationships among two sets of variables, namely, a set of response variables and a set of explanatory variables. This type of analysis may be more appropriately referred to as "variable by variable" (VBV) analysis.

Given a two-way table that contains two sets of variables, this module can do the following:

•	Generate a VBV biplot

•	Remove covariate (explanatory variables) with relatively small effects (on the response variables)

•	Remove covariate (explanatory variables) not significantly (at a user-specified level) associated with the response variables

These functions can be used in the following scenarios:

1) Genetic markers as explanatory variables and phenotypic data of a target trait in different environments as response variables:

•	QTL identification based on phenotypic data from multiple environments

•	studying mega-environments based on QTL by environment interactions

•	studying linkage relations among genetic markers that are used as explanatory variables

2) Genetic values of explanatory traits as explanatory variables and phenotypic data of a target trait in different environments as response variables:

•	identifying explanatory traits that can explain the observed genotype by environment interaction for a target trait

•	studying mega-environments based on covariate-effect by environment interactions

3) Genetic markers as explanatory variables and a set of target traits in one environment or averaged across environments as response variables:

•	QTL identification for multiple traits

Note: This type of analysis is similar to but is different from the so-called canonical analysis. The common point is that both try to study relationships between two sets of variables (response variables and explanatory variables). The difference is that canonical analysis tries to find the linear combinations within each set of variables that correlates the strongest, while covariate-effect analysis tries to identify the most important explanatory variables that are associated with the response variables.


Main Menu ACCESSORIES

GGEbiplot also has many functions that are not part of biplot analysis but are equally useful for understanding research data. They are organized under the menu Accessories:

•	Data Plotting

•	Numerical Output

•	Conventional Analysis

•	Breeder's Kit. Plant breeders: don't miss this, and this toolbox!


Breeder's Kit

The development of the GGE Biplot Analysis and the GGEbiplot software was primarily aimed at facilitating plant breeders to understand their data and to make selection decisions. The following functions are not part of biplot analysis, but they are important tools for plant breeders, particularly the multi-trait selection tool:

•	Multi-trait selection tool

•	Multi-trait selection against checks

•	Match-maker (parent selection for crossing)

•	Mean and Stability Deviation

•	YREM (Superiority Index)

•	r-p-t Converter


Conventional statistics

In addition to biplot analysis, GGEbiplot has the following conventional statistical analyses, which are more user-friendly than most other statistical packages:

•	Multiple regression

•	Analysis of variance

•	Correlation matrix among testers


Data Plotting

Although Data Plotting is not part of biplot analysis, but they are useful and convenient tools for comparing two entries across the testers or comparing two testers across entries:

•	Plot two testers

•	Plot two entries


r-p-t Convertor
r-p-t Converter

This is one of the many accessories in GGEbiplot:

This is a correlation coefficient (r), student-t (t), and probability (P) Converter. For a given degree of freedom (n), knowing any of r, t, and P, the other statistics will immediately known.


Multi-trait selection against checks

When deciding if a new genotype is good enough to release as a new cultivar, they often have to be compared to check or control cultivars, and multiple control cultivars are usually used for different adapation types or for different traits (yield, quality, diseases, etc.).  The following device was designed for this purpose.

1. The first thing you need to do is to select the "traits" (can be environments for a given trait) from your list of "testers" to be used for genotpe evaluation.

2. Check the box if a trait is so measured that a lower value means more desirable.

3. Then you have to decide how to use your check cultivars. You are given 4 options: You can 1) manually choose a check for each trait, or 2) use the best check for each trait (Note: the best check is the one that have the highest or lowest value for the trait if the higher or the lower the better), or 3) use the mean of the checks, or 4) use the poorest check (see the same note for the best check).  Once the selecion of the option is made, you will be prompted to identify the checks. For options 2, 3, or 4, the checks will be automatically selected based on your criteria.  I use #4 more often so that this tool is to do a prescreening rather than a final decision. After this, the Multi-Trait Selection Master should be used to rank and the pre-selected genotypes and make a final decision.

4. Set a bar for each trait

5. Click "RUN".

The output of this function will include the following:

1. The original data table

2. The data relative to the checks

3. The pass or not pass table.

The genotypes will be listed in order of their number of "passes", with the ones that passsed for all the traits listed first. Note that these genotypes may be the most "all-rounded" ones but they are not necessarily the best genotypes. This is why the selection on selection index (Multi-Trait Selection Master) is still needed.


Match-maker (Seleton of parent for crossing)

The Match Maker was designed to identify promising bi-parent crosses based on genotype by environment data for an important trait (such as yield) or genotype by trait data or a combination of environments and traits. It utilizes the Index Selection and Independent Culling concepts and assumes that a cross can potentially combine the strength from both parents. The interface is like the following:

The user is asked to set the weight for each trait so that an Selection Index can be calculated for each potential bi-parent cross. The use is also asked to set a bar for each trait so that the user is noted if a potential cross has defect in any of the traits (breeding objectives). A good cross should have no defect; on top of this, it shuld have high Selection Index. The output is something like the following:

The above table clearly identifies the cross OA1306-1/OA1331-6 as the best cross, better than any others, based on the yield data at multiple locations.


Plot randomization Tool

The function is under the menu Accessories/Breeder's Kit:

This randomization tool allows instant generation of experiment plan. It asks for the number of entries and the number of replications and outputs  a plan in the form of Randomized Complete Blocks.

However, see more advanced Experimental Designs functions.


Main Menu MODELS

A model is defined by the following four aspects:

1) Data Transformation

2) Scaling methods

3) Centering methods

4) Singular value partitioning

Comparison with the classification of models in the older GGEbiplot:


Data Scaling methods

Under the main menu of Models, click Scaled By, and select among the four Scaling methods:

Options of data scaling in GGEbiplot are:

1) No Scaling. Replicated data are not required.

2) Scaled by Tester Standard Deviation (SD): each value is divided by the standard deviation of its corresponding tester (column). This will put all testers roughly the same rang of values. Replicated data are not required.

3) Scaled by Heritabiliy-Adjusted Means, which replaces the "Divided by Tester Means" in earlier versions. Replicated data are required.

4) Scaled by LSD (5%), which replaces the "Divided by Tester Standard Error" in earlier versions. Replicated data are required.

5) Scaled by Heritability-adjusted SD, which is a newly added model. Replicated data are required.

For complete interpretations on GGE biplot based on different scaling methods, see: Yan and Holland. 2010, Euphytica 171(3):355-369.

In Depth: Which model is better

My first choice would be #2, "Scaled by SD", because i) It is the best for visualizing relationships among testers, ii) No replicated is needed, and iii) the adquacy of biplot can be judged by the uniformity of the length of the vectors. If the vectors of all testers are of similar length, then the biplot is adquate. Otherwise the biplot is not adquate in displaying patterns of the short-vectored testers. However, for this model, the vector length of the environments cannot be interpreted as representing the discriminating power of the environments.

When the evaluation of testers is the focus, #5 is most preferred, because the length of the testers in this biplot approximates the heritability of the testers (which is an indication of the discriminating power of the test environments), PROVIDED that the biplot is adequate in displaying the data. Model #4 should be similar to model #5. If replicated data is not available, then #1 can be used in place of #5.


Various Genotype-by-Environment Biplots

Although we have recommended using a GGE biplot in studying genotype-by-environment tables, there are other types of biplots that have been used in studying such data (DeLacy et al., 1996). All biplots can be useful as long as they are interpreted correctly.  Some comparisons of six types of GE-containing biplots are briefly given below.

•	The COMM biplot, based on the completely multiplicative model (i.e., SVD of the original two-way table without centering). This biplot contains E, G, and GE, and is often inflated with the grand mean. This biplot is useful for visualizing the real data but is effective only when the grand-mean is close to 0, as in the case of the covariate-effect biplot. Such a biplot can be generated using GGEbiplot by setting Centering  = 0.

•	The SHMM biplot, based on the shifted multiplicative model (Cornelius et al., 1996). This biplot also displays a mixture of E, G and GE, but can be inflated by the shifting factor. GGEbiplot does NOT generate the SHMM biplot because GGE biplots are more effective in addressing the crossover issue, for which SHMM was specifically developed.

•	The PCA biplot, based on SVD of grand mean-centered (Centering = 1)genotype-by-environment data (Zobel et al., 1988). This biplot contains E, G, and GE, with E often predominating over G and GE.  This biplot is least useful. (NOTE BUT: usually PCA biplot are GGE biplots; the terms used here followed Zobel et al 1988 but it is confusing)

•	The GE biplot or AMMI2 biplot, based on SVD of double-centered (Centering = 3) genotype-by-environment data (Gauch, 1992). Since G and E are removed prior to SVD, it displays GE only. This biplot is useful for visualizing gene expression data (Chapman 2002).

•	The AMMI1 biplot, plotting genotype and environment main effects against interactive PC1 scores (Gauch, 1992). This "biplot" displays E, G and GE but usually explains more variation. It does not have the interpretation of a normal biplot, however.

•	The GGE biplot, based on SVD of environment-centered (Centering = 2) or standardized genotype-by-environment table. As its name suggests, the GGE biplot displays G and GE (Yan et al., 2000).

For more detailed comparison between GGE biplots and AMMI biplots, see Yan et al.(2007, Crop Science).


Data Centering methods

Data Centering is most important factor in a model. GGEbiplot offer 4 data centering options. Under the main menu of Models, click Centered By, and select among the following options:

•	No Centering: the biplot will contain mixed information of P = M + E + G + GE.

•	Grand-mean centered: the biplot will contain mixed information of P - M = E + G + GE.

•	Tester centered: the biplot will contain a mixed information of P - M - E = G + GE, resulting in GGE biplots. Depending on data scaling methods, there can be different types of GGE biplots.

•	Double-centered: the biplot will contain information only on P - M - E - G = GE.

Note: Although entry-centered models are not listed explicitly, they can be easily achieved by transposing the two-way data before choosing a centering method.


Data Transformation

Under the main menu of Models, click Transformation, and select among the following options:

Under the main menu of Models, click Transformation, and select among the four options:

0. No transformation (default),

1. Natural log,

2. Log10, and

3. Square root

The current two-way table will be transformed. To go back to the original untransformed data after the data are transformed, please do the following, depending on how the data were read:

1) If the data were read as a two-way table (example), go to the spreadsheet window and click the "Biplot Analysis" button.

2) Otherwise select the appropriate function under the Anyway or 4-Way menu.


Polynomial regression

Under construction.


Association and Regression Analysis

Find associated variables

•	Find Associated Variables

•	Remove the effects of some covariable

Multiple Regression

•	Multiple linear regression

•	Eliminate collinearity based on Log Ratio

•	Include any Interaction (Epistasis) Terms

•	Include any quadratic terms

Example: QTL identification for a single trait based on phenotypic data a single environment or average across environments.


Year-by-trait table

Under the main menu 4-Way, click Year by trait table.


Location-by-trait-table

Under the main menu 4-Way Data, click Loc by trait table.

Although it is arealy used by plant breeders, such a biplot may be used to select locations that are more suitable in producing a crop of specific quality (Trait).


Environment-by-trait table

This is a groups of functions for 3-way or 4-way data and is organized under the main menu "4-Way". It has the following three options:

1. Environment by trait biplot for any genotype

This may be useful in understanding the particular genotype. This type of study might be more interesting to production agronomists than to breeders, however.

2. Environment by trait biplot for any subset of the genotypes

3. Environment by trait biplot across all genotypes


Genotype*environment-by-trait table

If your data are read as a 3-way or 4-way data, under the main menu 4-Way, click geno*env by trait table.

This function actually generates a biplot based on a phenotype-by-trait table, and is useful to study the phenotypic relationships among traits, across genotype-environment combinations. This type of analysis may be more useful to production agronomists than to plant breeders.


Genotype-by-environment*trait table

If you data are read as a 3-way or 4-way data, under the main menu 4-Way, click geno by env*trait table.

This type of biplot treats a trait measured in different environments as different variables and is useful for characterizing the genotypes. It is therefore most useful for genotype classification.


Covariate-effect biplot based on genotypic values

•	Under the main menu 4-Way, click VBV biplot - Genotypic. A trait selection device will appear on the top of the biplot, allowing the user to select a trait as the target trait.

•	GGEbiplot will generate a genotype by (trait+environment) two-way table. This table contains explanatory trait values for each genotype averaged across environments, and values of target trait for each genotype measured in each environment. These numerical values will be recorded in the log file.

•	GGEbiplot will then trigger the process detailed in Covariate-effect analysis of a genotype by (trait+environment) two-way table.


Association (with a trait) by environment biplot

•	Under the main menu 4-Way Data, click Trait Association by Environment biplot - Associations with a target trait. A trait selection device will appear on the top of the biplot, asking you select a trait as the target trait; other traits will be treated as "explanatory traits".

•	GGEbiplot will generate a trait by environment two-way table, which consists of the correlation coefficients between the target trait and each of the explanatory traits in each of the environments.

•	The table will be printed to the log file and missing correlation coefficients will be replaced with '0'.

•	A biplot will be generated based on this two-way table without any data transformations, i.e., using Model 5:

See Interpretation of a covariate effect biplot.

In depth: This TABE biplot is similar to a covariate-effect by environment biplot in that both are based on a explanatory variable (traits) vs. response variable (target trait measured in various environments) two-way table. The difference is that the explanatory variables in the later are genetic values, i.e., averaged across environments. Genetic values of explanatory traits are equivalent to genetic covariate such as genes, QTL, or genetic markers. In a TABE biplot, it is the phenotypic values of the explanatory traits that are used in calculating the "trait associations".


Year-by-location table for any trait

Under the main menu 4-Way, click Yr by Loc table for one trait. A trait selection device will appear on the top of the biplot, allowing you select a trait. Click Get biplot will show the biplot based on the year by location table of the selected trait, averaged across all genotypes.

What is the use of such biplots? Not sure yet...


Biplot based on grand-mean centered data

From Models, select Centering, then Grand-mean Centered.

For genotype-by-environment data, biplot based on this model would contain E+G+GE:

Note that PC1 explains mostly the variation of tester main effect (E), and PC2 mostly that of the entry main effect (G). Thus, Testers 5, 6, and 8 had obviously higher mean values than testers Testers 9, 3, and 5. Entries 5 and 7 had obviously higher mean values than entries 3, 4, and 6. WhyBecause all testers are on the side of entries 5 and 7! In other words, Entries 5 and 7, in contrast with entries 3, 4 and 6, 'interacted' positively with all the testers.

Although 88% of the total E+G+GE is explained by this biplot, not much GE can be visualized from this biplot. This biplot is therefore not very effective in displaying the GE patterns. Therefore, this biplot is not effective in genotype evaluation, environment evaluation, or revealing genotype by environment interaction patterns. It may be more effective than model 5, however, as the grand mean is at least removed, which represents no variation. Compare with other types of biplots.


Biplot based on tester-centered data

From Models, select Centering, then Tester-centered.

All biplots that are based on tester-centered data is a GGE biplot. A GGE Biplot based on this mode, without any scaling, may be called GGE1 biplot.

This biplot differs from the GGE2 biplot in that the vector length of the testers are directly related to the within tester standard deviation. Therefore it is a measure of discriminating ability of the testers (if the biplot adequately displays the GGE patterns and if the environments are homogeneous). Thus, testers 5 and 7 were more discriminating than others, say, tester8.


Biplot based on tester Stdev standardized data

From Models, select Centering, then Within-tester Stdev scaled.

GGE Biplot (with tester centered) based on this model may be called GGE2 biplot.

Note that the vectors of the testers have almost the same length. This is because the data was scaled by tester standard deviation. The interpretation of the GGE2 biplot is the same as that of the GGE1 biplot except that the vector length of the testers does not represent their discriminating abilities. Instead, the testers were assumed to be equally important in entry evaluation.

This type of GGE biplot has a useful property. It can be used to assess the adequacy of the biplot in explaining the GGE of the data. If all testers have vectors of similar length, it means that the main patterns of the GGE are presented. Otherwise, it indicates that current biplot does not adequately display the GGE patterns and that information on the testers with apparently shorter vectors is not presented in the current biplot. In the latter case, examining additional PC's or subdivision of the data is needed to fully explore the data.


Biplot based on tester Stderr standardized data

From Models, select Scaling, then Within-tester Stderr Scaling.

GGE Biplot based on this model may be called GGE3 biplot. Sample biplot is not available for non-replicated data (e.g. the two-way data on this page) as it is not possible to calculate the standard errors.

The interpretation of a GGE3 biplot is the same as for a GGE1 biplot except that possible heterogeneity among testers are removed. However, it is possible that the importance (discriminating power) of testers with very small stderr values may be over-emphasized.


Biplot based on double-centered data

From Models select Centering, then Double-Centered.

This biplot is usually referred to as the AMMI2 biplot or GE biplot. It displays only the entry-by-tester interaction (GE).

Since entry and tester main effects have been removed, this biplot explains only the interaction GE. It is characterized by entries and testers falling in all directions. This biplot is most effective in displaying the GE patterns but cannot be used in selecting superior entries or testers. You have to use a GGE biplot (models 1- 3) for this purpose. That is, to select superior entries, the entry main effect G must be reconciled with the GE. This is a very important concept; its importance is manifested by the terms "GGE biplot" and the fact this software is called "GGEbiplot".

For example, the above biplot shows that tester 3 and entry 3 interacted positively. But this does NOT necessarily mean that entry 3 is the winner for tester 3. Verify this with the original data table:


Biplot based on the original two-way table

From Models, click Centering, then No Centering.

This biplot displays E+G+GE as well as the Grand mean. The advantage is that the real data is approximated and can be visualized if the biplot displays most of the variation in the data. For example, all the entry-tester combinations are positive as there are no negative values in the original data. The disadvantage is that the grand mean masks other variations so that the variations among testers (E), among entries (G), and particularly, the interactions between the entries and the testers (GE), all appear to be small (note that PC1 explained 99% of the total variation!). This problem becomes more severe as the absolute value of the grand mean becomes greater. This model is useful only when the grand mean is close to 0, such as a covariate-effect tableof correlation coefficients.

From the biplot, we can see that testers 5 and 6 had higher mean values whereas testers 9, 3, and 4 had smaller mean values. Also, it is barely obvious that Entries 5 and 7 had higher mean values than other entries (e.g. Entry3). Some interactions can be seen too, although one cannot be sure if this is real, because PC2 explained only 1% of the total variation. For example, entries 1, 2, 5 and 6 seem to have interacted positively with testers 5 and 7.

It can be seen that the pattern in this biplot is very similar to that of model 0, except the "resolution" is not as good. Also, compare biplots based on other models based on the same data.


Change the title of the biplot

Click Plot Title from Format, an input box will appear.

Type whatever you like, and it will become the new title of the current image.


Change the font of the biplot title

From Format, click Change Font, and select Plot Title. A font selection dialog will appear;

Select the desired font type, size, and style and then click OK.


Change the color of the biplot title

From Format, click Change Color, and select Plot Title. A color selection palette will appear;

Select the desired color and click OK.


Color scheme of the biplot

From Format, click Color Scheme, and select one of the following:

1. Black/White

2. Color Scheme1 (white background and blue and red labels)

3. Color Scheme2 (more colorful)

Note:

1. The color of the various components of the biplot (Entries, testers, titles, etc.) can be changed individually by clicking Change Color.

2. Individual labels can be formatted by clicking Individual Label Formatting.


Change the background color

From Format, click Change Color, and select Background. A color selection palette will appear;

Select the desired color and click OK.


Full names of the entries and testers

Under the main menu View, click Original Names.

When a biplot is generated, full names are displayed by default, which can be displayed in various ways by using functions under the main menu of View.


Change the color of the lines

From Format, click Change Color, and select Lines. A color selection palette will appear;

Select the desired color and click OK.


Show entries and/or testers as indices

For large datasets, the entry and/or tester labels can heavily clustered on the biplot. To avoid this, one way is to click Index-Legend from View and select Entries or Testers. The entries or testers will be presented as indices, and a text file that shows the legends and full names will appear.

There are other functions for changing the appearance of the biplot.


Change the color of the tester labels

From Format, click Change Color, and select Testers. A color selection palette will appear;

Select the desired color and click OK.


Change the font of the tester labels

From Format, click Change Font, and select Testers. A font selection dialog will appear;

Select the desired font type, size, and style and then click OK.


Change the color of the entry labels

From Format, click Change Color, and select Entries. A color selection dialog will appear;

Select the desired color and click OK.


Change the font of the entry labels

From Format, click Change Font, and select Entries. A font selection dialog will appear;

Select the desired font type, size, and style and then click OK.


Change biplot size

From Format, click Plot Size, an input box will appear, asking a number. To enlarge the biplot, put a number greater than 100; to reduce, put a number smaller than 100.


Change font size of the entries and/or testers
Change font size of the entries and testers

From Format, click Change Font, and select Larger or Smaller. The fonts will be 2 points larger or smaller. The font size can also be specified.


Change case

Under Format, click Change Case, and select Entries or Testers, a new form will pop up, and you will be given the option to change the font case of the labels to one of the following:

•	Title case: the first letter of each word is in upper case and others in lowercase,

•	Upper case, or

•	Lower case

Select one and click OK.


Individual label Formatting

Under the main menu Format, reside the following functions:

1. Set format (color, font name, size, italic, bold, or italic bold)

2. Apply format to:

1) single labels, or

2) group of labels

3. Back to Default Format

4. Reverse the last format change


Entry-focused SVP

Under Models/SVP, click 1. Entry-focused.

Entry-focused SVP is appropriate for evaluating the entries. It is NOT appropriate for evaluating the relationships among testers.


Tester-focused SVP

Under SVP, click 2. Tester-focused.

Tester-focused SVP is appropriate for visually evaluating the relationships among testers but is NOT appropriate for visually evaluating the entries.

See SVP for more information.


Symmetrical SVP

Under Models/SVP, click 3. Symmetrical.

•	Symmetrical SVP is not ideal for either entry evaluation or tester evaluation. Unfortunately many people are still use this method, probably due to lack of understanding of this issue of SVP or lack of apprpriate software.

•	All possible SVP are equally valid in visualizing the entry-by-tester interactions, however.


Biplot based on transposed data

From the main menu Data, click Entry/Tester Switch Roles.

Interpretation:

•	The data matrix will be transposed and a biplot based on the transposed data displayed.

•	The transposition does not alter the model used for generating the biplot.

•	If you have a "G+GE" biplot originally, now you have an "E+GE" biplot instead. Clicking Entry/Tester Switch Roles again or clicking "Back to the Original Data" will reverse this.


Get any subset

Under the main menu of Data, click Run any subset by..., and select either

•	Deleting any testers, or

•	Deleting any entries.

A form will appear, allowing freely deleting entries or testers, e.g.:

When the form closed, a biplot based on the subset of entries or testers will show, with all settings in the previous biplot maintained.

This seemingly simple device is not trivial!  It generates a subset of data in the fly and allows you to run ANY subset of the original dataset easily, without going to your data file and saves a lot of time and trouble!!!


Get balanced subset with regard to selected testers or entries

Sometimes you may be interested in only specific entries or testers. When this is the case, a maximum blanced subset can be taken from the original unbalanced data. To do this,

1. Click Data

2. Click Get Subset relative to

3. Select Entries or Testers

You will be allowed to selected only the entries or testers you are most interested in.


Reverse the sign of one or all testers

There are cases when traits are measured such that large values indicate less desirable. In such cases one may want to reverse the sign of these traits. There are two options in GGEbiplot for this purpose:

•	Reverse the sign of one tester

•	Reverse the sign of all testers


Back to Previous Subset

From Data Manipulation, click Back to Previous Subset.

This function is available only for large datasets. The "Previous Subset" refers to a subset that is considerably different from the original two-way data as well as the current subset.


Back to Original Data

Click the follow function will bring up the biplot that includes all the entries and testers in the original two-way table:

•	Under Data Manipulation, click Back to Original Data.


Re-label the entries or testers

When there are many entries and/or testers on the biplot, it can look crowded. To reduce the overlap among entries, or if the names of the entries are not of interest, click View/Free Labeling...Entries, an input box will appear, which allows you to type anything you desire, which will become the new label for all entries. The default label for entries is '+'.

To reduce the overlap among testers or if the names of the testers are not of interest, click View/Free Labeling...Entries, an input box will appear, which allows you to type anything you like, which will become the label for all testers. The default label for testers "x".

Note:

o	The full name of a label will appear if you rest the mouse pointer on the label.

o	The full names can be called back by clicking View/Full Names.


Complete the two-way table of a diallel cross dataset

Click Data/For Diallel Analysis/Complete the Two-way Table,

•	This function is useful when the diallel data file contains only n(n-1)/2 values, where n is the number of parents. The two-way table will be completed upon this call.

•	When this action is successful, a biplot based on the completed two-way table will be displayed. Or a message will be displayed, informing you that you don't have a diallel dataset.


Delete parents per se values - diallel data analysis
Delete parents per se- diallel data analysis

Under the main menu Data, click For Diallel Analysis and then Delete Parents. This function is useful for visualizing the GCA and SCA estimated without parents. You will be noticed if the dataset is not a diallel dataset.


Biplot of PC3 vs. PC4

Click PC3 vs. PC4 under Biplots.

This biplot is useful to reveal patterns in addition to that found in the biplot of PC1 vs. PC2. It is useful only when

1. Patterns have already found in the biplot of PC1 vs. PC2, and

2. PC3 and/or PC4 explain a sizable variation.


Biplot of PC1 vs. PC2

Under Biplots, click PC1 vs. PC2.

This makes a default biplot. Except when explicitly defined, a biplot refers to a biplot f PC1 vs. PC2.


Biplot of PC5 vs. PC6

Under Biplots, Click PC5 vs. PC6.

This biplot is useful only when

1. patterns are found in the biplot of PC3 vs. PC4, and

2. PC5 and/or PC6 explain a sizable variation.


Biplot of PC1 vs. PC3

Under Biplots, click PC1 vs. PC3.

This biplot may be useful in finding additional patterns other than that displayed by a biplot of PC1 vs. PC2.


Biplot of PC2 vs. PC3

Under Biplots, click PC2 vs. PC3.

May be useful in finding additional patterns other than that explained by PC1.


AMMI1 biplot

Under Biplots, click AMMI1 biplot.

Note: The AMMI1 biplot plots main effects of the entries and testers against their IPC1 scores (IPC = Interactive PC). This biplot does not have the two most important properties of a true biplot. Therefore, many interpretations of a biplot do not apply to this AMMI1 biplot.

The two axes of the AMMI1 biplot should be visualized separately. The X-axis represents the mean values of the entries and of the testers, and the unit is the square root of what is in the original data. Thus, from the above AMMI1 biplot,  the entries are ranked on their means across testers as follows:

         Entry7 > Entry5 > Entry2 > Grand Mean > Entry1 > Entry4 > Entry6 > Entry3,

and the testers are ranked on their means across entries as follows:

        Tester5 > Tester6 > Tester8 > Tester2 > Tester1 = Tester7 > Grand Mean > Tester4 > Tester3 > Tester9.

The IPC1 axis represents the interaction scores, thereby specific interactions between the entries and the testers can be visualized. The AMMI1 biplot is effective in summarizing the total variation of the data but it does not show which-won-where, which is one of the important features of GGE biplot analysis (models 1-3).

Note that the units in this biplot is both square root of the original units in the data. This makes the shape of the AMMI1 biplot objective. In conventional AMMI1 biplots, the X-axis is in the units of the original data whereas the Y-axis is in square root of the original data, making the shape of the biplot totally subjective.


3D-biplot

Under the main menu Biplots, click 3D-biplot.

A 3D biplot is made of the first three principal components. It is usually difficult to visualize a 3D biplot on a 2D plane, but a rotating 3D biplot can be very useful when the first two PC's does not explain most of the information.

As shown from this snapshot, the 3D biplot viewer allows you to do the following:

•	Visualize a constantly rotating 3D biplot;

•	Adjust the rotation speed;

•	Freeze the image of an interesting pattern;

•	Rotate any of the three axes while holding the other two;

•	Visualize 2D biplots of PC1 vs PC2, PC1 vs PC3, and PC2 vs PC3; and

•	Save the current view of the 3D biplot into a graphic file, in color or in black-white, which may be included in a publication.


Remove backgorund noise
Remove background noise

Under View, click Refresh.

This will also remove any concentric circles on the biplot image.


Entries with FIRST N Letters

Under View, click Label with first N letters..., and select Entries. An input box will appear, allowing you to specify "N", the number of the letters to be displayed, counted form the first letter. All entries will be labeled with the first N letters.


Testers with LAST N Letters

Under View, click Label with Last N letters..., and select Testers. An input box will appear, allowing you to specify "N". All testers will be labeled with the last few N letters. No change will be made to those testers whose name is shorter than N.


Entries with LAST N Letters

Under View, click Label with Last N letters..., and select Entries. An input box will appear, allowing you to specify "N", the number of the letters to be displayed, counted form the last letter. All entries will be labeled with the last N letters.


Testers WITHOUT First N Letters

Under View, click Label without first N letters..., and select Testers. An input box will appear, allowing you to specify "N". All testers will be labeled without the first few N letters. If a tester's name is shorter than N, then it will be represented by its last letter.


Entries WITHOUT First N Letters

Under View, click Label without first N letters..., and select Entries. An input box will appear, allowing you to specify "N", the number of the letters to be hidden, counted form the first letter. All entries will be labeled without the first N letters.


Testers WITHOUT Last N Letters

Under View, click Label without Last N letters..., and select Testers. An input box will appear, allowing you to specify "N". All testers will be labeled without the last few N letters. If a tester's name is shorter than N, then it will be represented by its first letter.


Entries WITHOUT Last N Letters

Under View, click Label without Last N letters..., and select Entries. An input box will appear, allowing you to specify "N", the number of the letters to be hidden, counted form the first letter. All entries will be labeled without the last N letters.


Show full names

Under View, click Full Name.


Hide testers and show entries only

Under View, click Show Entries Only.


Hide entries and show testers only

Under View, click Show Testers Only.


Show both entries and testers

Under View, click Show Both.


Show or hide the biplot title

Click Show/hide Title from View.

Note: This function may be useful when preparing biplots for formal publication.


Show or hide the logo

Click Show/hide Logo from View.


Show or hide data source

Clicking Show/hide Data Source from View, the path and file name of the dataset that is under analysis will appear or disappear on the top of the biplot.


Show/Hide the Guidelines

Under View, click Show/Hide Guidelines.


Delete irrelevant variables

This function was designed to remove less relevant variables with regard to a target, or dependent, trait or variable. It involves the following steps:

•	Under Multivariate, click Find Associated Testers.

•	An input box will appear, asking for the probability level for a trait to be considered as 'associated' with the target trait. Put a number between 0 and 0.5, and click OK.

•	A panel will appear above the biplot.

•	Click the drop-down arrow by the textbox, select the target trait, and then click Find.

•	An input box will appear, asking you if you want to use any traits/markers as covariates.

•	Click No. (If you click Yes, then you are asking for a Covariate Analysis)

A new biplot will appear that has the following changes:

•	traits/markers that were chosen as covariate and traits/markers that do not meet the association criteria are deleted from the biplot;

•	the target trait is indicated by a larger font and is framed;

•	only trait/marker labels are shown to focus on the relationships among the traits/markers (the entries can be called back by clicking Full Names or Show Both under View);

•	the vectors of the trait/marker (lines from biplot origin to the trait labels) are drawn to facilitate visualization of the relationships among them.

Interpretations when the target variable is a trait and the other variables are genetic markers:

•	Markers that have longer vectors and smaller angles (close to 0° or 180°) with the target trait are more closely associated with the target trait. An acute angle means positive correlation and an obtuse angle means negative correlation.

•	Markers that have short vectors or that have angles close to 90° with the target trait are more loosely associated with the target trait. However, isolated traits or markers may have stronger associations with the target trait than they the angles suggested.

•	Clustered markers may be closely linked and therefore suggest a single QTL.

•	An isolated marker may suggest a QTL.

Here is a real example of QTL identification based on a GGE biplot.


Covariate-effect analysis

Covariate-effect analysis is the analysis of the effects of a set of covariables (covariates) on a set of response variables. For example, the yield of a set of genotypes measured in different environments can be treated as response variables, whereas the genetic values of other traits such as yield components, plant height, maturity, etc.,can be treated as covariables (or explanatory variables), which are used to "explain" the responses of the genotypes to the environments (Yan and Tinker, 2005, Crop Science).  Genetic markers can also be regarded as explanatory "traits" in such analysis (Yan and Tinker 2005, Molecular Breeding).

Covariate-effect analysis is also referred to as variable-by-variable biplot analysis, and it involves the following steps:

•	Calculate the effects (linear correlation or regression coefficient) of each explanatory variable on each of the response variables, which resultsin an explanatory variable by response variable (variable-by-variable or VBV) two-way table of "covariate effects", which is referred to as the "covariate-effect table" or VBV table.

•	Remove explanatory variables that have no significant effects on any of the response variables at a user-specified significance level, resulting in a reduced covariate-effect table.

•	Generate a covariate-effect biplot based on the reduced covariate-effect table.

•	Study the covariate-effect patterns in the biplot, which includes

1) similarities among the explanatory variables in their effects on the response variables,

2) similarities among the response variables in their response to the explanatory variables, and

3) interactions between the two sets of variables.

See QTL identification based on phenotypic data from multiple environments for a full example of this type of analysis.


Generate a variable-by-varaible (VBV) biplot based on a two-way table
Generate a variable-by-variable (VBV) biplot based on a two-way table

Note: This functions requires a combined two-way table. As a common knowledge in biplot analysis, both the number of explanatory variables and the number of response variables have to be greater than 3 for a 2-D biplot.

Under the main menu of VBV Biplot, click Covariate-Effect Biplot, and select r-matrix (or b-matrix).

This is what will happen:

•	In the older version of GGEbiplot, the data have to be so arranged that the columns of explanatory variables are followed by response variables (as the above example) and the user has to know the number of explanatory variables in the data. When the function is invoked, the user will first be asked how many explanatory variables are there in the table. Suppose the number is N, then the first N variables (columns) will be treated as explanatory variables and the other columns (M) will be treated as response variables.

•	In the current version of GGEbiplot, the data format is more flexible. When the function in invoked, a form like the one will appear and the user is asked to separate the variables into N explanatory variables and the M response variables.

•	GGEbiplot will then calculate a correlation coefficient (if r-matrix is selected, or else a regression coefficient) between each of the explanatory variables and each of the response variables, across the observations (rows), leading to a N * M matrix of Pearson correlation coefficients. These coefficients are used as effects of the explanatory variables on the response variables, and therefore referred to as covariate-effect values. This matrix is referred to as a "covariate-effect table" and will be printed to the log file for your reference.

•	A biplot will then be generated based on the covariate-effect table, referred to as a covariable-effect biplot if the explanatory variables are genetic values of some traits or QQE biplot if the explanatory variables are genetic markers.

•	A unique property of this biplot is that it is based on non-centered data so that the true effects can be visualized.


Congruence coefficient

Under Multivariate, click Congruence Coefficient.

Note: Congruence coefficient here is the correlation coefficient between the distance matrices among environments in the GGE biplot and that in the covariate-effect biplot pattern.


Linear Map of Testers

From the main menu Biplot Tools, click Linear map of testers. A linear map of the testers (genetic markers) will appear.

See Linear map among markers for illustration.


Similarity among entries

From Biplot Tools, click Relation among Entries, and the vectors of the entries will appear on the biplot. If the biplot was not in the entry-focused singular value partitioning (i.e., SVP =1), this method will be automatically applied, which may change the shape of the biplot.

Interpretations:

•	Entries with longer vectors are more responsive to the testers, i.e., they have either large main effect (G, i.e., greater or smaller than the average) or large interaction with the testers (GE) (e.g. entries 3 and 7); those with short vectors are less responsive (entries 1 and 2). Entries located at the biplot origin performed intemediately with regard to all testers.

•	Entries are similar in both main effect and interaction with the environments if they are placed closely in the biplot (e.g. entries 2 and 5).


Mean vs. stability of genotypes

Within a single mega-environment, genotypes must be evaluated on both mean performance and stability across environments. A GGE biplot offer a graphical and convenient way for this. See Mean performance vs. stability of entries, but read entries as genotypes and testers as environments.


Discrimitiveness vs. representativeness of testers
Discrimination vs. representativeness of testers

From Biplot Analysis, click Discrimination vs. representativeness. The following will appear on the biplot:

•	The vectors for the testers;

•	A small red circle indicating the position of the average tester, which is defined by the average PC1 and PC2 scores across all environments. This average environment can be regarded as a virtual environment;

•	a thick red line that passes through the biplot origin and the average environment, referred to as the average-tester axis;

•	a red arrow pointing to the average environment from the biplot origin;

Interpretation:

•	The vector length of a tester represents its discriminating ability. Thus testers 5 and 7 are the most discriminating testers.

•	The angle between a tester and the average-tester axis represents the representativeness of the tester: the larger the angle, the less representative the tester. Thus testers 3 is most representative whereas tester 7 the least representative.

•	An ideal tester should be both discriminating of the entries and representative of the average tester. See Rank testers relative to the ideal tester..

In Depth:

This function applies only to GGE biplots with certain data scaling methods.  Particularly, GGE biplot based on Standard Deviation scaled data does NOT have this interpretation.


Compare entries as packages of traits

See compare all entries with a check across testers, but read "testers" as "traits".


Discrimitiveness power and representativeness of testers
Discrimination and representativeness of testers

From Biplot Analysis, click Rank Entries/Testers…

Select With ref. To an "Ideal" Tester. The following will appear on the biplot:

•	a red circle indicating the average tester;

•	a red line that passes through the biplot origin and the average tester (average tester axis);

•	A red arrow pointing to the "Ideal" tester, which is defined as an tester for testing the entries; and

•	a set of 5 concentric circles with the "ideal" tester being the concentric center.

•	The entries are represented by "c" to indicate that they are treated as "random" samples of the entries and this view is to focus on the testers.

Note: If the biplot was not in the tester-focused singular value partitioning, this partitioning method will be automatically applied.

Interpretation:

•	The closer a tester is located to this 'ideal" tester, the more desirable it is judged on both Discrimination and representativeness. The ranking of the testers is:

Tester3 > Tester1 = Tester2 > Tester4 = Tester9 > Tester6 > Tester5 > Tester8 = Tester9.

In Depth:

This function applies only to GGE biplots with certain data scaling methods.  Particularly, GGE biplot based on Standard Deviation scaled data does NOT have this interpretation.


Which genotype won in which environment?
Which genotype won in which environment

See which-won-where, but read entries as genotypes and testers as environments.

Specific remarks:

To the rank-2 approximation (i.e., to the approximation of the biplot),

•	The genotypes at the vertices are the most responsive genotypes; they are either the best or the worst genotypes in one or more environments;

•	The best genotype for an environment is the vertex genotype for the sector in which the environment resides.

•	The vertex genotype is the best genotype for all environments falling in its sector.


Which genotype is the best for which trait

See Which-won-where, but read entries as genotypes and testers as traits.

Special remarks:

•	The genotypes at the vertices are either the best or the worst genotypes for one or more traits; they are the most "unbalanced" genotypes.

•	The genotype with the largest value for a trait is the vertex genotype for the sector in which the trait resides.

•	The vertex genotype is the genotype with the highest values for all traits falling in its sector.


Which genotype is the poorest in what

1) From Data, click Reverse Sign of ... All Testers;

2) From Biplot Tools, click Which Won Where/What.

The interpretation is similar but opposite to that in Which-won-what.


Which-won-where

From Biplot Tools, click Which Won Where/What. The following will appear on the biplot:

•	a polygon that is drawn on entries (entries 5, 7, 4, 3, and 2) located away from the biplot origin such that all other entries are contained within the polygon, and

•	a set of lines that are perpendicular to the sides of the polygon.

•	each perpendicular line is actually the equality line between two testers the polygon side connects (see compare two entries).

Interpretations:

•	In this example, the perpendicular lines divide the biplot into 5 sectors

•	A tester falls inevitably into one of the sectors

•	The vertex entry for (not necessarily in) each sector had the greatest values for all testers falling within that sector. In this example, Entry5 wins for Testers 5 and 7, and Entry7 wins for Testers 1, 3, 4, 6, 8, and 9. Thereby the testers are divided into two groups based on the winners.

•	Entry5 and Entry7 performed similarly with regard to Tester2.

In depth:

•	The choice of a SVP method does not affect the which-won-where pattern.


Find the best entry-tester combinations in a diallel cross

Refer to Which-won-where and apply the following interpretations:

•	An irregular polygon (convex hull) is drawn on entries that are most interactive to the testers in various directions so that all other entries are contained in the polygon;

•	The perpendicular lines to the sides of the convex hull divide the biplot into sectors;

•	The best crosses, i.e., hybrids or entry-tester combinations, are those between the vertex entries and the testers falling within their respective sectors;

•	The vertex genotype is said to be the best mating partner for, or have the highest specific combining ability with, the testers falling within its respective sectors.

Note: 	In a diallel cross dataset, each genotype is both an entry (in blue) and a tester (in red).


QTL identification - single trait in a single environment

This function is applicable to a QTL mapping dataset, i.e., a genotype-by-(marker+trait) two-way table. We will use a real barley QTL mapping data to illustrate the process of QTL identification using GGEbiplot. We will try to identify QTL for kernel weight (KW).

QTL identification is basically a multiple linear regression problem except that a target trait is selected as the dependent variable and all markers are as independent variables. The following steps are suggested for QTL identification using GGEbiplot:

•	Delete irrelevant traits manually.

•	Delete irrelevant markers based on their association with the target trait.

•	Conduct multiple linear regression using the target trait as the dependent variable and the relevant markers as independent variables.

•	Remove markers that are indirectly associated with the target trait (Follow this link) so that only markers that are most closely associated with the target trait and have an "real" effect on the trait are retained in the model (and biplot). Each such remaining marker should represent a putative QTL.

•	It is also possible to include any di-genic interactions (epistasis) in the regression model.


Delete unwanted testers manually

Under the main menu of Data Manipulation, click Run any Subset by… and choose Removing any Testers. A form with two list boxes will appear, one of them contain all the testers. You can freely move the testers between the boxes. Click the OK button when satisfied.

A new biplot will shown, in which only the testers that were selected:

This method can be used to generate a subset that contains any number of the testers.

Here is another example.


Find Associated Variables

This function is applicable to a genotype-by-marker two-way table or a QTL mapping dataset.

•	Under Association, click Find Associated Testers of….

•	An input box will appear, asking for the probability level for a marker to be considered as 'associated' with the target marker. Put a number between 0 and 0.5, and click OK. A panel will appear above the biplot.

•	Click the drop-down arrow by the textbox, and select the target marker, and then click Find.

•	An input box will appear, asking if you want to use any traits/markers as covariate. Since the default answer is No, just click OK.

•	The following changes to the biplot will occur:

•	Traits/markers that do not meet the criteria are deleted;

•	The target marker is indicated by a larger and underlined font;

•	Only trait/marker labels are shown to focus on the relationships among traits/markers (the genotypes are hidden and can be called back by clicking Full Names or Show Both under the main menu View);

•	Trait/marker vectors (lines from biplot origin to the trait labels) are drawn to facilitate visualization of the relations among them.

•	Traits may be manually deleted by clicking Data Manipulation/Run a Subset by…/Removing any Testers.

•	The linear map of the markers can be brought up by clicking Biplot Analysis/Linear Map of Testers.

Interpretation:

•	If the retained markers have a maximum angle near 90°, the markers are from more than one linkage group. You could repeat the process using more stringent criteria to delete less linked markers. Note: there can be more than one linkage group on a single chromosome.

•	Within a linkage group, the order and relative distance among the markers on the biplot and the linear map should be roughly consistent with their positions on the chromosome.

Also see: Find Associated Markers


Find traits associated with a single locus (Pleiotropic effect)

This function is applicable to a QTL mapping dataset, i.e., a genotype-by-(marker+trait) two-way table

•	Under Association, click Find Associated Testers of…,

•	An input box will appear, asking for the probability level for a trait to be considered as 'associated' with the target marker. Put a number between 0 and 0.5, and click OK. A panel will appear above the biplot.

•	Click the drop-down arrow by the textbox, and select the target marker, and then click Find.

•	An input box will appear, asking if you want to use any of the markers or traits as covariates. Take the default value ('no') and click OK.

•	The biplot will be transformed to the following:

•	The target marker is indicated by a larger and underlined font;

•	Only marker/trait labels are shown in the biplot to focus on the relationships among the markers and traits (the genotypes are hidden and can be called back by clicking Show Both from menu View);

•	Vectors (lines from biplot origin to the trait labels) are drawn for all retained markers and traits.

•	A set of concentric circles will appear to facilitate visualizing the relative length of the vectors of the markers/traits. Short-vectored markers/traits may not be adequately represented in the biplot.

•	Unwanted markers can be deleted from the biplot using the function Data/Run a Subset by…/Removing any Testers.

Interpretations:

•	Traits that have longer vectors and smaller angles (close to 0° or 180°) with the target marker are more strongly associated with the target marker. An acute angle means positive association and an obtuse angle negative association.

•	Traits with short vectors may have a weak association with the target trait or other traits.

•	Traits that have shorter vectors or a close to 90° angle with the target marker are less associated with the target marker.

•	Clustered traits are similarly associated with the target marker.


Conduct multiple regression

This function is applicable to any multivariate dataset or two-way table that can be regarded as a multivariate dataset. Follow the following steps:

•	Delete any unwanted independent variables using the function "Data/Run SUBSET by/Removing any Testers";

•	Under Multivariate, clicking Multiple Regression, a panel will appear above the biplot, which allows you to select the dependent variable.

•	Select the dependent variable from the drop-down box and click Regress. The program will try to conduct multiple regression using the selected variable as dependent variable and all others as independent variables.

•	If the number of independent variables are more than the number of observations, a message will appear, telling you that over-parametrization has occurred, and suggesting you to reduce the number of independent variables using "Find Associates Testers of…" function to remove less associated independent variables.

•	When done, run Multivariate /Multiple Regression again. Once the over-parametrization problem is solved, the following will appear on the biplot:

1. The dependent variable in a larger font and underlined;

2. The retained independent variables;

3. Variable vectors to facilitate visualization of the relationships among the variables, particularly those between the dependent variable and the independent variables. For interpretation;

4. Total variation of the dependent variable explained by the multiple regression, on the top of the biplot; and

5. Numerical results, including the partial regression coefficient, t-statistics, type I error probability of the partial regression coefficients, and simple r-square values (variation of the dependent variable explained by each independent variable). This information is also printed to the log file for your reference.

Note: The simple r-square values are more correct when the multiple regression was preceded by the execution of the "find associates" function).

Follow this link for an example: QTL identification based on multiple linear regression.


Delete variables on Log Ratio

Under the main menu Multivariate, click Multiple Regression, and select Delete Variables on Log Ratio. An input box will appear, asking for a threshold level. Put a number and click OK. The default value is 2.5, which is equivalent to a probability level of 0.05.

Here is an example.


Generate a QQE biplot

A QQE biplot can be generated from

•	a genotype by (marker+environment) two-way table, or

•	a QTL by environment two-way table, which can be derived directly by GGEbiplot from a genotype by (marker+environment) two-way table:

This hypothesized example dataset contains the effects of 8 QTL on a trait in 8 environments. The QQE biplot based on this data looks like:

Note that the biplot is based on un-centered data (Centering = 0).

For interpretations, see Interpret a QQE biplot.


Two-way table of covariate-effects

A two-way table of covariate-effects consists of effects of the entries (the row factor in the two-way table) on a target trait in each environment (the column factor in the two-way table). Do not confuse this with a two-way table of measured values such as a genotype-by-environment two-way table of yield or a genotype-by-trait table in an environment or across environments.

The most typical example is a table that contains the effects of a number of QTL (rows) in a number of environments (columns).

Also see: Covariate-effect analysis.


Interpret a QQE biplot

A QQE biplot has the same interpretability as a GGE biplot, except that genotypes are replaced by genetic markers or QTL. All enabled functions are applicable to a QQE biplot, but the following may be more general:

•	Examine the effect of each marker on the trait of interest (i.e., the associations of the marker with the target trait) in various environments.

•	Identify the most important markers in each environment.

•	Examine the mean effect and consistency of a marker on the trait across the environments.

•	Identify markers with relatively consistent effect across environments (QTL represented by these markers are often called QTL for genotype main effect).

•	Identify markers that have contrasting effects in different environments and thereby are responsible for the observed GE interaction for the target trait.

•	Examine the grouping of environments (mega-environment classification) based on the QQE pattern, which may help develop a strategy to uitlize the GE through marker assisted selection.

Here is a complete example of QQE biplot analysis.


Linkage relation among selected markers in a QQE biplot

Under the menu VBV Biplot, click Genetic relations.

•	The relation among markers displayed in a QQE biplot is based on their similarity or dissimilarity in response to the environments. Genetically linked markers are often located together as expected. However, genetically unlinked markers can also be located quite closely if they respond to the environment similarly.

•	To examine the linkage relations among the markers in a QQE biplot, click VBV Biplot/Genetic correlation biplot.

•	Closely linked markers will have long vectors and small angles among them; markers with short vectors and/or large angles tend to be independent of each other. The relationship among markers with short vectors can be more closely examined using secondary biplots (e.g. PC3 vs. PC4 or biplots contains only these markers).

•	QTLs are suggested by an isolated marker or a group of closely linked markers.

Related topics: QTL identification based on data from multiple environments.


Eliminate less important markers from a QQE biplot

•	Under menu VBV Biplot, click Eliminate Short-vectored Markers, an input box will appear, asking for the name of a marker.

•	Markers (covariables, in general) with vectors shorter than that of the marker that is used as a criterion will be eliminated from the biplot. They can always be called back by clicking Data/Back to Previous Subset.

Note:

•	The overall importance of a marker is measured by the length of its vector (the distance between the marker and the biplot origin). A short vector means the marker has relatively small effect on the trait in all or most environments.

•	The relative vector length will change with the singular value partitioning method. The entry-metric partitioning should be used for eliminating markers. This is the default method for this function. Clicking Relation among Testers under Biplot Tools will also bring up the tester-metric scaling, and clicking Relation Among Entries will bring up the entry-focused scaling.


Generate a biplot based on the balanced subset

GGEbiplot can counstract a balanced subset of data from a unbalanced dataset and generate a biplot based on this subset in two ways:

By removing unbalanced entries

From Data Manipulation, click Get a Balanced Subset by…. and select Removing Entries. A biplot based on the balanced subset will appear. A message will be displayed if too few entries are left for biplot analysis.

By removing unbalanced testers

From Data Manipulation, click Get a Balanced Subset by…. and select Removing Testers. A biplot based on the balanced subset will appear. A message will be displayed if too few testers are left for biplot analysis.

Hint:  it is not necessary to manually pull out a balanced subset for biplot analysis. You can put all data that are available and let GGEbiplot do the tedious work for you, and it's good for that!


Biplots based on any subset of data

Delete any of the entries

•	From Data, click Get a Subset by…, and select Removing Entries.

•	A form with two list-boxes will be displayed, with all entries being initially listed in the To Be Retained box and the To-Be-Deleted box empty.

•	You can freely move any of the entries between the two list-boxes.

•	Click OK when finished moving the entries. A new biplot based on the subset of selected entries will be displayed. You will be notified if too few (<3) entries are left for biplot analysis.

Delete any of the testers

From Data, click Get a Subset by…, and select Removing Testers. A window contains two list-boxes will be displayed, with all testers listed in the box To Be Retained and the To Be Deleted box empty.

•	You can freely move any of the testers between the two list-boxes.

•	Click OK when finished moving the testers. A new biplot based on the subset of testers will be displayed. You will be notified if too few testers are left for biplot analysis.

Note: No change will be made if all or none of the entries are deleted.

There are many other ways to subset data in GGEbiplot.


Reverse the sign of a variable

•	From Data, click Reverse Sign of ... and select One Tester, a trait selection device will appear above the biplot.

•	Click the drop-down arrow and select the trait whose sign is to be reversed and click the button Reverse. The biplot will be changed accordingly, and the reversed trait will be underlined.

Note:

•	This action can be repeated to reverse the sign of other traits.

•	The original biplot will be restored by clicking the button Back.

Example: the sign of "E7" in the following biplot is reversed:


Reverse the sign of all testers

From Data, click Reverse Sign of ... and select All Testers.

When the sign of all testers are reversed, the "which-is-the-best-for-what" function can be used to show "which-is-worst-for-what."


Plot two testers across the entries
Plot two testers for all entries

From Accessories, select Data Plotting ... Two Testers. A panel with two text boxes will appear above the biplot.

Select two testers or traits to be compared and click Plot 2 Testers, a scatter plot will appear.

The simple plot of two testers contains the following features:

•	It is the tester-centered data that is displayed, and the guidelines indicate the averages of the two testers, respectively.

•	The X-axis represents the first tester, and the Y-axis represents the second tester.

•	The correlation coefficient between the two testers across the entries is indicated on the upper-left corner.

•	The names of the entries are displayed in the plot. Therefore, the entries can be evaluated based on the two environments or traits.

•	A y = x line (the equality line) indicating the expected trend if the two testers are completely proportional.

Here is an example.


Plot two entries across the testers

From the menu Accessories, select Data Plotting... Two Entries. A panel with two text boxes will appear above the biplot.

Select the entries to be compared and click Plot 2 Entries, a scatter plot will appear. It contains the following features:

•	It is the tester-centered data that is displayed; the guidelines indicate the averages of the testers, respectively.

•	The X-axis represents values of the first entry, and the Y-axis represents values of the second entry.

•	The correlation coefficient between the two entries across the testers is indicated on the upper-left corner.

•	The tester names are displayed in the plot. Therefore, the two entries can be compared for the various testers (environments or traits).

•	A y = x line (the equality line) indicating the expected trend if the two entries are identical with regard to the testers.

Here is an example.


Numerical Output

Tabulate the original data

From Accessories/Numeric Outputs, click Original Data in Matrix. This is useful when the data file is not in a two-way table. GGEbiplot calculates and organize a two-way table for you.

Print the current subset

From Accessories/Numeric Outputs, click Current Subset. The current subset refers to the two-way table that is currently displayed in the biplot.

Print th predicted values based on the current biplot

From Accessories/Numeric Outputs, click Print Predicted Values.

Print the eigenvectors

From Accessories/Numeric Outputs, click Print Eigenvectors.

Print the axis values of the current biplot

From Accessories/Numeric Outputs, click Biplot Axes.

Note: The requested numerical output will be printed to the log file.


ANOVA table

•	The ANOVA table is automatically printed to the log file if the data is replicated. For non-replicated datasets, the relative magnitude of E, G, and GE are printed instead.

•	The relative magnitude of E, G, and GE of the current subset can be requested by clicking Accessories/Numerical Output/Analysis of Variance.

For 3-way, 4-way, or any-way data, the following more advanced ANOVA options are available:

•	ANOVA for 4-way data

•	ANOVA for main effects of any number of factors


ANOVA of 4-Way Data

When the data are read as 3-way parallel, 4-way parallel, or Anyway parallel, and when the 4-way ANOVA option under the main menu of "Anyway" is selected, the following form will appear:

This is a major component of GGEbiplot, that can be used to conduct ANVOA for a year-by-location-by-genotype-by-trait 4-way data (all any 4-way data in a similar data structure) all possible ways. This interface allows the research to specify the replication (block), genotype, location, and year, and specify the experiemental design to be a RCBD (rondomized complete bolcks design), ICBD (incomplete blocks design) or CRD (completely randomized design). Once this is done, the options are possible:

1. Conduct ANOVA for any selected traits or for all traits at one strike,

2. Conduct ANOVA and multiple genotype comparison for each test (year-location combination),

3. Conduct ANOVA for the genotype by location data each each year, and

4. Conduct ANOVA for the genotype by year data by location.

5. Conduct ANOVA based on different experimental designs

Hint: For multi-year multi-location data without replications, the researcher can choose to use the year or the location as replications for genotype comparison.


ANOVA in a single environment (year-location combination)

This output of this function includes the following:

1. ANOVA Table

2. LSD scaled genotype means, which is a convenient way for comparing genotypes within the environment

3. Common statistical and quantitative genetics parameters including:

•	number of reps

•	mean

•	CV%

•	Heritability (H)

•	Standard error (SE)

•	Standard deviation (SD)

•	LSD at 5%

•	Genetic variance

•	Phenotypic variance


Multi-Trait Selection Tool

Under the main menu "Accessories", under "Breeder's Kit", there are several functions, "Multi-trait selection tool" being one of the most important. This is probably the most comprehensive tool for selection based on multiple traits, because it combined Index Selection, independent culling, and independent selection in the same interface and allows selecting and culling on both ends of selected traits.

When this function is selected, the following interface will appear.

The following options are available for the researcher to select (and decide!):

1. Select traits to be selected/culled on;

2. Set a weight for a trait in the range of [-1, +1]. If the higher level of the trait is desirable, then its weight should be positive, otherwise should be negative. No action will be taken on a trait if its weight is set to "0" (i.e., this trait will be ignored in the decision making process);

3. Specify a culling rate on the Selection Index (0 to 100%);

4. Specify if independent selection is to be conducted. If yes, specify a rate of selection (0 to 100%);

5. Specify if independent culling is to be conducted. If yes, specify a rate of culling (0 to 100%); and

6. For each trait, specify if one (high or low) end or both (high and low) ends of the trait are to be selected/culled on. (Example for selecting on both ends:

The results will be pop-up in an Excel spreadsheet when the "Go" button is clicked.

Reference:

Yan and Fregeau-Reid 2008. Crop Science.


Selection Index

A selection Index is a linear combination of a set of traits. Each trait is given a weight between [-1, +1] based on its direction of action (higher levels are favorable or unfavorable) and importance, as judged by the researcher.

Since the selection index is a linear combination of the traits, it has two major properties:

1) it allows multiple traits are selected on the same time, and

2) it allows the trait to compensate for each other such that a genotype may be selected if it has a large Selection Index even though it has obvious defects on some traits.

More: Selection based on multiple traits.


Independent Selection

Independent Selection is a selection strategy whereby a genotype is selected if it has an outstanding level for a single trait (breeding objective), regardless of its levels on other traits. This strategy guarantees that a genotype that might be a donor for some trait will not be discarded for its defects on other traits.

In the multi-trait selection tool of GGEbiplot, Independent Selection on a trait is conducted by considering three pieces of specifications by the researcher:

1. Selection rate (0 to 50%),

2. Weight (relative importance, [-1, +1]) of the trait, and

3. Action direction of the trait (the sign of the weight).

Examples:

If the Selection Rate is 30%, and the weight of the trait is 0.5, then genotypes with a value of 0.85 (=1 -  0.3 * 0.5) or higher for the trait will be selected. If the culling rate is 30%, and the weight is -0.6, then genotypes of a value of 0.18 (=  0.3 * 0.6) or lower will be selected.


Plot of Mean vs. Stability deviation

From Accessories/Breeder's Kit, click Mean and Stability Deviation.

A plot (not biplot) of Mean vs. Stability-deviation will appear for all entries

The stability deviation is the square root of Shukla's stability variance (Shukla, 1972)

Numerical values of the entries will be printed to the log file.

For the following dataset

the plot of mean vs. stability deviation is as follows:

The rank of the entries on mean is clear; entry3 seems to be most "stable" whereas entry4 least stable. Note that Entry3 is consistently low yield!! Therefore, stability is meaningless when considered alone. It becomes meaningful only when attached to the mean performance, a key concept of "GGE biplot analysis".

Comment: Although GGEbiplot generates the plot of the mean vs. stability deviation, the mean and stability view and the mean vs. stability view of the GGE biplot are much more informative if the GGE biplot adequately approximates the GGE of the data.


Independent Culling

Independent Culling is a selection strategy whereby a genotype is culled if it does not meet the requirement for a single trait, regardless of its levels on other traits. This strategy guarantees that a genotype that is selected as a new cultivar/variety has no major defects.

In the multi-trait selection tool of GGEbiplot, Independent culling on a trait is conducted by considering three pieces of specifications by the researcher:

1. Culling rate (0 to 50%),

2. Weight (relative importance, [-1, +1]) of the trait, and

3. Action direction of the trait (the sign of the weight).

For Example: If the culling rate is 30%, and the weight of the trait is 0.5, then genotypes with a value of 0.15 (=  0.3 * 0.5) or lower relatitve to the maximum value of the trait will be discarded. If the culling rate is 30%, and the weight is -0.6, then genotypes of a value of 0.82 (= 1 - 0.3 * 0.6) or higher relative to the maximum will be discarded.


Index Selection

Index selection is a popular selection strategy, in which selection is made on a Selection Index. If the culling rate is 10%, then 10% of the entries with the lowest Selection Index will be discarded and the rest will be retained.

See multi-trait selection tool.


Log file

The log file automatically records the actions of GGEbiplot, which include:

•	ANOVA table of the current two-way table, if the biplot is based on replicated data,

•	ANOVA and multiple comparison results,

•	The current two-way table on which the biplot is based,

•	Variations explained by each of the first 6 principal components (PC), and

•	Any user requested Numerical Outputs.

The log file can be opened by clicking the "Open Log File" option under the main menu bar "Help".


YREM (Superiority Index)

•	From Accessories/Breeder's Kit, click YREM (Superiority Index).

•	YREM refers to "Yield relative to the maximum" first proposed by Weikai Yan (1999). The output is the average YREM of each genotype across environments. It is a [0,1] superiority index, the larger the value of a genotype the more superior it is.