Biplot Analysis of MET Data IITA
Transcript of Biplot Analysis of MET Data IITA
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Contact: [email protected]
Weikai YanMay 2006
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Weikai Yan 2006
Multi-Environment Trials (MET)
MET are essential MET are expensive
MET data are valuable MET data are not fully used
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Why biplot analysis?
Biplot analysis can help understand METdata Graphically, Effectively, Conveniently
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Outline
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
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Contact: [email protected]
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Weikai Yan 2006
MET data isa genotype-environment-trait
(G-E-T) 3-way table Multiple Genotypes
Multiple Environments Multiple Traits
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A G-E-T 3-way table containsmany 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 isan 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
(P)
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A G-E-T 3-way table impliesinformative 2-way tables
Association by environment 2-waytables
Associations: among traits between traits and genetic markers
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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)
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Most two-way tables can be
visually studied using biplots
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Origin of biplot
Gabriel (1971)One of the mostimportant advances indata analysis in recentdecadesCurrently
> 50,000 web pagesNumerous academicpublicationsIncluded in moststatistical analysispackages
Still a very newtechnique to mostscientists
Prof. Ruben Gabriel, The founder of biplot Courtesy of Prof. Purificacin Galindo
University of Salamanca, Spain
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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 a BiplotGraphical display of matrix multiplication
Inner product property P ij =OA i *OBj *cos ij Implies the product matrix
A(4, 2) B(2, 3) P(4, 3)
121284
96103
151262
69201
321
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321
044
313
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341
a
a
a
a
bbb
y
x
bbb
a
a
a
a
y x
Matrix multiplication
-4
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-2
-1
0
1
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5
-4 -3 -2 -1 0 1 2 3 4 5
X
Y
O
A1A2
A3
A4
B1
B2
B3
5.0
cos =0.8944
4.472
P11 = 5*4.472*0.8944 = 20
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Practical definition of a biplotAny 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)
214
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321
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121284
96103
151262
69201
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y
x
eee
g
g
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y x
g
g
g
g
eee
G-by-E table
Matrix decomposition
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-4 -3 -2 -1 0 1 2 3 4 5
X
Y
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G1G2
G3
G4
E1
E2
E3
P(4, 3) G(3, 2) E(2, 3)
(Now 3D- biplots are also possible)
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Singular Value Decomposition (SVD) &Singular Value Partitioning (SVP)
r
k kj
f
k
f
k ik
SVP
r
k kjk ik
SVDij
ba
baY
1
1
1
))((
(0 f 1)
Singular values Matrixcharacterising the rows
Matrixcharacterising the columns
SVD = PCA?
SVD:
SVP:
The rank of Y, i.e.,the minimum numberof PC required tofully represent Y
Rows scores Column scores
BiplotPlot Plot
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Biplot interpretations
Inner-product propertyInterpretations based on biplots with f = 1
approximates YY T, the distance matrix Similarity/dissimilarity among row (genotype) factors
Interpretations based on biplots with f = 0approximates Y TY, 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)
))((1
1
r
k kj f k f k ik ij baY
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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.
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Data centering prior to biplot analysis
The general linear model for a G-by-Edata 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 scaling prior to biplot analysis
Different GGE biplots Yij = ( i + ij )/s j
S j = 1 no scaling S j = (s.d.) j all environments are equally important S j = (s.e.) j heterogeneity among environments is removed
(built in GGEbiplot)
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Four questions must be askedbefore trying to interpret a biplot
1. 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)
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MEGA-ENVIRONMENT
ANALYSIS
TESTENVIRONMENTEVALUATION
GENOTYPEEVALUATION
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Sample G-by-E data(Yield data of 18 genotypes in 9 environments, 1993, Ontario, Canada)
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Before trying to interpret a 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 analysis
MEGA-ENVIRONMENT
ANALYSIS
TEST
ENVIRONMENTEVALUATION
GENOTYPEEVALUATION
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 environments The Environment - vector view
Angle vs.correlation
The anglesamong testenvironments
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)
G12
G7G18
G8G13
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Are there meaningful 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 crossover patterns*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 intomultiple mega-environments
GE CANNOT be exploited GE must be avoided by testing across locations and
years
*Not the environment-grouping patterns Mega-environment is a group of geographical locations that share the same (set of) bestgenotypes consistently across years.
Multi-year data are needed
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Classify your target environment intoone of three categories
With Crossover GE No CrossoverGE
Repeatable (2) Multiple MEsSelect for specifically adaptedgenotypes for each ME
(1) Singlesimple MEA single test location,single year suffices toselect a single bestvariety
Not repeatable (3) Single
complex MESelect for generally adaptedgenotypes across the wholeregions across multiple years
ME: mega-environment
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G By E data analysis
MEGA-ENVIRONMENT
ANALYSIS
TEST
ENVIRONMENTEVALUATION
GENOTYPEEVALUATION
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Discriminating ability and representativeness
Vector length: discriminating ability Angle to the AE: representativeness
Average-environment axis
Average environment
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Ideal test environments:discriminating and representative
Ideal test
environment
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Classify each test environment intoone of three categories
For each good or useful test environment: is it essential?
Discriminative Notdiscriminative
Representative (2) Good forselecting (moreimportant)
(1) Useless
Notrepresentative
(3) Useful forculling (less important)
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Vector length = discrimination= GE = GE1 + GE2
Contribution toProportionateGE
Contribution toNon-proportionateGE
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G By E data analysis
MEGA-ENVIRONMENT
ANALYSIS
TEST
ENVIRONMENTEVALUATION
GENOTYPEEVALUATION
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Vector length = GGE = G + GE
Contribution To GE(instability)
Contribution To G(mean performance)
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Mean vs. Stability
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Genotype ranking on both MEAN and STABILITY
The ideal genotype
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Genotype classification
MeanStability
High meanperformance
Low meanperformance
High stability Generally adapted(VERY GOOD)
Bad everywhere(VERY BAD)
Low stability Specifically Adapted(GOOD)
Bad somewhere(BAD)
Are there stability genes?!
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G x E data analysis summary
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
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Contact: [email protected]
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Inner-product property
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Ranking on a single environment
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Ranking on two environments
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Relative adaptation of a genotype
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Compare any two genotypes
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Contact: [email protected]
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Objectives of G By T data analysis
Genotype evaluation based on traitprofiles
Relationship among breeding objectives
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Data of 4 traits for 19 covered oat varieties (Ontario 2004)
(Background info: High yield, high groat, high protein, and low oil are desirable for milling oats)
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Relationships among traits
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Trait profile of each genotype
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Trait profile of a genotype
T i fil i b
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Trait profile comparison betweentwo genotypes
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Genotype ranking based on a trait
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Parent selection based on trait profiles
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Independent culling
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Contact: [email protected]
MET data are more informativethan you thought
A G E T 3 d t t t i
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A G-E-T 3-way dataset contains various 2-way tables
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
G i i b i bi l
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Genetic-covariate by environment biplot(QTL by environment biplot)
Barley GenomicsData
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Trait-association by environment biplot
OatMETData
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Four-way data analysis
Year
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Conclusion (1)
GGE biplot analysis is an effective toolfor G by E data analysis to achieveunderstandings about.
1. the target environment,2. the test environments, and3. the genotypes
4. stability analysis is useful only to a singlemega-environment
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Conclusion (2)
GGE biplot analysis is an effective toolfor G by T data analysis to achieveunderstandings about.
1. the interconnected plant system,2. positively correlated traits3. negatively correlated traits
4. the strength and weakness of thegenotypes
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Conclusion (3)
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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Conclusion (4)
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
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Contact: Weikai Yan: [email protected] web: www.ggebiplot.com
mailto:[email protected]://www.ggebiplot.com/http://www.ggebiplot.com/mailto:[email protected]