Contact: wyan@ggebiplot.com Biplot Analysis of Multi-Environment Trial Data Weikai Yan May 2006.

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Contact: wyan@ggebiplot.com

Biplot AnalysisBiplot Analysis of of Multi-Environment Trial DataMulti-Environment Trial Data

Weikai YanMay 2006

Weikai Yan2006

Multi-Environment Trials Multi-Environment Trials (MET)(MET)

• MET are essential

• MET are expensive

• MET data are valuable

• MET data are not fully used

Weikai Yan2006

Why biplot analysis?Why biplot analysis?

• Biplot analysis can help understand MET data– Graphically, – Effectively, – Conveniently

Weikai Yan2006

OutlineOutline

• Multi-environment trial (MET) data• Basics of biplot analysis• Biplot analysis of G-by-E data• Biplot analysis of G-by-T data• Better understanding of MET data• Conclusions

Multi-environment Multi-environment trial datatrial data

Weikai Yan2006

MET data is MET data is a genotype-environment-a genotype-environment-

trait trait (G-E-T) 3-way table(G-E-T) 3-way table

• Multiple Genotypes

• Multiple Environments

• Multiple Traits

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A G-E-T 3-way table A G-E-T 3-way table contains contains

many 2-way tablesmany 2-way tables• G by E: for each trait

• G by T (trait): in each environment; across environments

• E by T: for each genotype; across genotypes

G-E-T data >> G-E data

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A G-E-T 3-way table isA G-E-T 3-way table isan extended 2-way tablean extended 2-way table

• G by V:– each E-T combination as a variable (V)

• P by T: – each G-E combination as a phenotype

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A G-E-T 3-way table implies A G-E-T 3-way table implies informative 2-way tablesinformative 2-way tables

• Association by environment 2-way tables– Associations:

• among traits• between traits and genetic markers

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Goals of MET data analysis Goals of MET data analysis

• Short-term goals: – Variety evaluation

• Response to the environment (G x E)• Trait profiles (G x T)

• Long-term goals: – To understand

• the target environment (G x E)• the test environments (G x E)• the crop (G x T)• the genotype x environment interaction (A x T)

Basics of biplot Basics of biplot analysisanalysis

Most two-way tables can be visually studied using biplots

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Origin of biplotOrigin of biplot

Gabriel (1971) One of the most

important advances in data analysis in recent decades

Currently… > 50,000 web pages Numerous academic

publications Included in most

statistical analysis packages

Still a very new technique to most scientists

Prof. Ruben Gabriel, “The founder of biplot”Courtesy of Prof. Purificación Galindo

University of Salamanca, Spain

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What is a biplot?What is a biplot?

• “Biplot” = “bi” + “plot”– “plot”

• scatter plot of two rows OR of two columns, or• scatter plot summarizing the rows OR the columns

– “bi” • BOTH rows AND columns

• 1 biplot >> 2 plots

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Mathematical definition of Mathematical definition of a Biplota Biplot

Graphical display of matrix multiplicationGraphical display of matrix multiplication

“Inner product property”– Pij =OAi*OBj*cosij

– Implies the product matrix

A(4, 2) B(2, 3) P(4, 3)

121284

151262

Matrix multiplication

-4 -3 -2 -1 0 1 2 3 4 5

cos =0.8944

P11 = 5*4.472*0.8944 = 20

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Practical definition of a Practical definition of a biplotbiplot

“Any two-way table can be analyzed using a 2D-biplot as soon as it can be sufficiently approximated by a rank-2 matrix.” (Gabriel, 1971)

121284

151262

G-by-E table

Matrix decomposition

-4 -3 -2 -1 0 1 2 3 4 5

P(4, 3) G(3, 2) E(2, 3)

(Now 3D-biplots are also possible…)

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Singular Value Decomposition Singular Value Decomposition (SVD) & (SVD) &

Singular Value Partitioning (SVP) Singular Value Partitioning (SVP)

kkjkik

(0 ≤ f ≤ 1)

“Singular values”Matrix characterising the rows

Matrix characterising the columns

SVD = PCA?

The ‘rank’ of Y, i.e., the minimum number of PC required to fully represent Y

Rows scores

Column scores

BiplotPlot Plot

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Biplot interpretations Biplot interpretations

Inner-product property Interpretations based on biplots with f = 1

approximates YYT, the distance matrix Similarity/dissimilarity among row (genotype) factors

Interpretations based on biplots with f = 0 approximates YTY, the variance matrix Similarity/dissimilarity among column (environment)

factors

Combined use of f = 0 and f = 1

(Gabriel, 2002 Biometrika; Yan, 2002, Agron J; Built in the GGEbiplot software)

fkikij baY

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Biplot analysis is… Biplot analysis is…

to use biplots to display– a two-way data per se (Y), – its distance matrix (YYT), and– its variance matrix (YTY)

so that– relationships among rows, – relationships among columns, and– interactions between rows and columns

can be graphically visualized.

Weikai Yan2006

Data centeringData centering prior to prior to biplot analysisbiplot analysis

• The general linear model for a G-by-E data set (P) – P = M + G + E + GE

• Possible two-way “tables” (Y):• Y = P = M + G + E + GE —original data: QQE biplot

• Y = P – M = G + E + GE —global-centered (PCA)

• Y = P – M – E = G + GE —column-centered: GGE biplot

• Y = P – M – G = E + GE —row-centered

• Y = P – M – G – E = GE —double-centered: GE biplot

All models are useful, depending on the research objectives (built in GGEbiplot)

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Data scalingData scaling prior to prior to biplot analysisbiplot analysis

• Different GGE biplots• Yij = (i + ij)/sj

• Sj = 1 no scaling

• Sj = (s.d.)j all environments are equally important

• Sj = (s.e.)j heterogeneity among environments is removed

(built in GGEbiplot)

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Four questions must be Four questions must be askedasked

before trying to interpret a biplotbefore trying to interpret a biplot1. What is the model?

How the data were centered and scaled?What are we looking at?

2. What is the goodness of fit?How confident are we about what we see?What if the data is fitted poorly?

3. How singular values are partitioned?What questions can be asked?

4. Are the axes drawn to scale?Are the patterns artifacts?

(All are addressed explicitly in GGEbiplot)

Biplot Analysis ofBiplot Analysis ofG-by-E dataG-by-E data

MEGA-MEGA-ENVIRONMENTENVIRONMENT

ANALYSISANALYSIS

TESTTESTENVIRONMENTENVIRONMENTEVALUATIONEVALUATION

GENOTYPEGENOTYPEEVALUATIONEVALUATION

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Sample G-by-E dataSample G-by-E data(Yield data of 18 genotypes in 9 environments, 1993, Ontario, Canada)(Yield data of 18 genotypes in 9 environments, 1993, Ontario, Canada)

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Before trying to interpret a Before trying to interpret a biplot…biplot…

1. Model selection?Centering = 2 (“G+GE”)

Scaling =0

2. Goodness of fit?78%.

3. Singular value partitioning?

SVP = 2 (environment-

metric)

4. Draw to scale?Yes.

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G By E data analysisG By E data analysis

ANALYSISANALYSIS

• Mega-environment is a group of geographical locations that share the same (set of) best genotypes consistently across years.

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Relationships among Relationships among environmentsenvironments

The “Environment-vector” view• Angle vs.

correlation• The angles

among test environments

• Environment grouping

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“Which-won-where”

(Crossover GE is GE that caused genotype rank changes and different “winners” in different test environments)

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Are there meaningful Are there meaningful crossover GE?crossover GE?

The “which-won-where” view

(Crossover GE is GE that caused genotype rank changes and different “winners” in different test environments)

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Are the Are the crossover patternscrossover patterns* * repeatable?repeatable?

• If YES…– The target environment can be divided into multiple

mega-environments– GE can be exploited by selecting for each mega-

environment– GE G

• If NO…– The target environment CANNOT be divided into

multiple mega-environments– GE CANNOT be exploited – GE must be avoided by testing across locations and

• *Not the environment-grouping patterns• Mega-environment is a group of geographical locations that share the same (set of) best genotypes consistently across years.• Multi-year data are needed

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Classify your target Classify your target environment intoenvironment into

one of three categoriesone of three categoriesWith Crossover GE No Crossover

Repeatable (2) Multiple MEsSelect for specifically adapted genotypes for each ME

(1) Single simple MEA single test location, single year suffices to select a single best variety

Not repeatable (3) Single complex MESelect for generally adapted genotypes across the whole regions across multiple years

ME: mega-environment

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ANALYSISANALYSIS

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Discriminating ability and Discriminating ability and representativenessrepresentativeness

Vector length: discriminating abilityAngle to the AE: representativeness

Average-environment axis

Average environment

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Ideal test environments:Ideal test environments:discriminating and discriminating and

representativerepresentative

Ideal testenvironment

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Classify each test environment Classify each test environment into into

one of three categories one of three categories

• For each “good” or “useful” test environment: is it essential?

Discriminative Not discriminative

Representative (2) Good for selecting (more

important)

(1) Useless

Not representative

(3) Useful for culling (less important)

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Vector length = discrimination Vector length = discrimination = GE = GE1 + GE2= GE = GE1 + GE2

Contribution toProportionateGE

Contribution toNon-proportionateGE

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ANALYSISANALYSIS

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Vector length = GGE = G Vector length = GGE = G + GE+ GE

Contribution To GE(instability)

Contribution To G (mean performance)

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Mean vs. StabilityMean vs. Stability

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Genotype ranking on Genotype ranking on bothboth MEAN MEAN andand STABILITYSTABILITY

“The idealgenotype”

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Genotype classification Genotype classification

Stability

High mean performance

Low mean performance

High stability Generally adapted

(VERY GOOD)

Bad everywhere

(VERY BAD)

Low stability Specifically Adapted

(GOOD)

Bad somewhere

Are there stability genes?!

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G x E data analysis G x E data analysis summarysummary

• 1) Mega-environment analysis• 2) Test environment evaluation• 3) Genotype evaluation

Important comments:– (2) and (3) are meaningful only for a single mega-environment– Any stability analysis is meaningful only for a single mega-

environment– Any stability index can be used only as a modifier to the ranking

based on mean performance

Other ways to view Other ways to view a GGE biplota GGE biplot

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Inner-product propertyInner-product property

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Ranking on a single Ranking on a single environmentenvironment

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Ranking on two Ranking on two environmentsenvironments

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Relative adaptation of a Relative adaptation of a genotypegenotype

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Compare any two genotypesCompare any two genotypes

Biplot analysis of Biplot analysis of Genotype by trait Genotype by trait

datadata

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Objectives of G By T data Objectives of G By T data analysisanalysis

• Genotype evaluation based on trait profiles

• Relationship among breeding objectives

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Data of 4 traits for 19 Data of 4 traits for 19 covered oat varieties covered oat varieties

(Ontario 2004)(Ontario 2004)

(Background info: High yield, high groat, high protein, and low oil are desirable for milling oats)

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Relationships among Relationships among traitstraits

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Trait profile of each Trait profile of each genotypegenotype

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Trait profile of a Trait profile of a genotypegenotype

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Trait profile comparison Trait profile comparison between two genotypesbetween two genotypes

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Genotype ranking based Genotype ranking based on a traiton a trait

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Parent selection based on trait Parent selection based on trait profilesprofiles

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Independent cullingIndependent culling

Fuller understanding Fuller understanding of MET data of MET data

MET data are more informative than you thought

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A A G-E-TG-E-T 3-way dataset 3-way dataset contains various 2-way contains various 2-way

tablestables• G by E data• G by T data• E by T data:

– for each genotype; all genotypes• G by V data:

– each E-T as a variable (V)• P by T data:

– each G-E as a phenotype (P)• Genetic association by environment data• Trait association by environment data

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Genetic-covariate by Genetic-covariate by environment biplotenvironment biplot

(QTL by environment biplot)(QTL by environment biplot)

BarleyGenomicsData

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Trait-association by Trait-association by environment biplotenvironment biplot

OatMETData

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Four-way data analysisFour-way data analysis

• Year…

Conclusions Conclusions

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Conclusion (1)Conclusion (1)

• “GGE biplot analysis” is an effective tool for G by E data analysis to achieve understandings about….

1. the target environment,

2. the test environments, and

3. the genotypes

4. stability analysis is useful only to a single mega-environment

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• “GGE biplot analysis” is an effective tool for G by T data analysis to achieve understandings about….

1. the interconnected plant system,

2. positively correlated traits

3. negatively correlated traits

4. the strength and weakness of the genotypes

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• “Biplot analysis” is an effective tool for other two-way table analysis

–Marker by environment–QTL by environment–Gene by treatment–Diallel cross–…

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• Biplot analysis can be VERY EASY…– From reading data to displaying the biplot: 2 seconds– Displaying any of the perspectives of a biplot and

changing from one to another: 1 second– Displaying the biplot for any subset: 1 second– Learning how to use the software and interpret

biplots: 30 minutes– Everything can be just one mouse-click away

Thank youThank youContact: Weikai Yan: wyan@ggebiplot.com

web: www.ggebiplot.com

Contact: wyan@ggebiplot.com Biplot Analysis of Multi-Environment Trial Data Weikai Yan May 2006.

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