1. BIPLOT ANALYSIS OF AUTOMOBILE EVALUATION DATA Weikai Yan, Ph. D Email: wyan@ggebiplot.comwyan@ggebiplot.com Web: www.ggebiplot.comwww.ggebiplot.com

2. Weikai_Yan 2005 Why biplot? One picture is worth of 10,000 words. Biplot is a very informative picture of research data

3. www.ggebiplot.com Weikai_Yan 2005 Three types of biplot will be used in this study Automobile by parameter biplot –Genotype by trait biplot in terms of agricultural studies (Yan and Rajcan 2002, Crop Science ) Automobile by judge biplot –Genotype by environment biplot in terms of agricultural studies (Yan 2001, Agronomy Journal) parameter by judge biplot –Genetic covariate by environment biplot (Yan and Tinker 2005, Crop Science)

4. www.ggebiplot.com Weikai_Yan 2005 Car by parameter table 'Preference Ratings for Automobiles Manufactured in 1980, obtained from: http://ftp.sas.com/techsup/download/sample/samp_lib/statsampPrincipal_Components_Analysis_of.html Rating for 10 parameters

5. www.ggebiplot.com Weikai_Yan 2005 Car by parameter biplot Biplot PC1 vs. PC2 (Primary biplot) Cars: blue parameters: red Four questions to ask before trying to interpret a biplot Mathematical model? –Model =1 (parameter- centered data = GGE biplot) Goodness of fit? –64% S.V.P.? –SVP = 1 ( = 1), car- metric preserving Axes drawn to scale? –Always Yes by GGEbiplot

6. www.ggebiplot.com Weikai_Yan 2005 Relationships among parameters Cosine of an angle between two parameters –Correlation between two parameters Acute angles: Positive correlations Obtuse angles: Negative correlations Right angles: no correlation Vector length –Discriminating ability of the parameter –A short vector: Not related to any other parameters Lack of variation or not well represented in the biplot

7. www.ggebiplot.com Weikai_Yan 2005 Biplot of PC3 vs. PC4 Display variations that are not displayed by the primary biplot of PC1 vs. PC2 To check if the primary biplot is adequate

8. www.ggebiplot.com Weikai_Yan 2005 Rank cars based on any parameter MPG High: –Civic_Honda –Chvette_GMC –… Low –Firebird_GMC

9. www.ggebiplot.com Weikai_Yan 2005 Rank cars based on any parameter Ride Best: –Continental –Granfury –DL Poorest –Pinto –Chevette –Mustang

10. www.ggebiplot.com Weikai_Yan 2005 Rank cars based on any two parameters MPG and RIDE Best –DL_Volvo Poorest –Firebird

11. www.ggebiplot.com Weikai_Yan 2005 Rank cars on all parameters Best –DL_Volvo Poorest –Firebird_GMC The average position of all parameters

12. www.ggebiplot.com Weikai_Yan 2005 The parameter profile of any car: Ford Continental Best in –Ride –Comfort Poorest in –MPG

13. www.ggebiplot.com Weikai_Yan 2005 The parameter profile of any car: Volvo DL Best in –Cargo –Comfort –Reliability Better than average for everything except Acceleration

14. www.ggebiplot.com Weikai_Yan 2005 Compare any two cars: Volvo DL vs. Continental Continental is better in –Ride Both are similar in –Comfort –Quiet –Accel DL is better in everything else Equality line

15. www.ggebiplot.com Weikai_Yan 2005 Which car gets the highest scores for what? Vertices –Continental –DL –Civic –Chevette –Pinto –Firebird

16. www.ggebiplot.com Weikai_Yan 2005 Which car gets the lowest scores for what? Vertices –Pinto –Firebird –DL

17. www.ggebiplot.com Weikai_Yan 2005 Car by judge data (personal preference) Preference of 25 judges 'Preference Ratings for Automobiles Manufactured in 1980, obtained from: http://ftp.sas.com/techsup/download/sample/samp_lib/statsampPrincipal_Components_Analysis_of.html

18. www.ggebiplot.com Weikai_Yan 2005 Car by judge biplot

19. www.ggebiplot.com Weikai_Yan 2005 Similarity among judges in terms of car preferences Angles –Similarity among judges in preference Vector length –Discriminativeness of the judges –J8 and J22?

20. www.ggebiplot.com Weikai_Yan 2005 Biplot of PC3 vs. PC4 Little variation is left for PC3 and PC4 The main biplot is adequate

21. www.ggebiplot.com Weikai_Yan 2005 Similarity among cars from the eyes of the judges Similarity among cars in the eye of the judges

22. www.ggebiplot.com Weikai_Yan 2005 Genotype evaluation: who favors what most? DL and imported car lovers –14 judges Continental and Eldorado lovers –7 judges Pinto and Chevette Lover –J24 Why?

23. www.ggebiplot.com Weikai_Yan 2005 Joint two-way table of car by parameter + car by judge What are the bases of the preference of the judges? Explanatoryvariables Responsevariables

24. www.ggebiplot.com Weikai_Yan 2005 Response variable by explanatory variable table (correlation coefficients)

25. www.ggebiplot.com Weikai_Yan 2005 Parameter by judge biplot The angle between a judge and a parameter: –Positive attitude: acute angles –Negative attitude: obtuse angles –Indifference: a right angle

26. www.ggebiplot.com Weikai_Yan 2005 parameter by judge biplot Who values what most? The most important thing for different judges –Braking J24 –MPG 6 –Reliability 8 –Quietness 6 –Ride 4

27. www.ggebiplot.com Weikai_Yan 2005 A rotating 3D-biplot 3D-biplot In case the primary biplot is not adequate…

28. www.ggebiplot.com Weikai_Yan 2005 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) Or 3D-biplots for rank-3 matrix!

29. Limitations of Biplot Analysis Biplot analysis is a very powerful tool, but…

30. www.ggebiplot.com Weikai_Yan 2005 What can biplots do? Revealing linear patterns, generating hypotheses –Patterns among rows –Patterns among columns –Interactions between rows and columns

31. www.ggebiplot.com Weikai_Yan 2005 What biplots cannot do? Revealing non-linear relationships among variables Hypothesis test (Hypothesis test is NOT always necessary)

32. www.ggebiplot.com Weikai_Yan 2005 Biplot Analysis & Statistical test are complementary Biplot Analysis Statistical Tests Decisions Hypothesis Testing Pattern discovery Hypothesis generating Data inspection & visualization Research data

33. www.ggebiplot.com Weikai_Yan 2005 Conclusions Biplot analysis has evolved into an elegant, powerful, generic tool for research data exploration Using user-friendly software GGEbiplot, biplot analysis is easy and fun. GGEbiplot beta is freely available at www.ggebiplot.com. Visit www.ggebiplot.com for more about biplot analysis.www.ggebiplot.com Dont be discouraged by the math; you dont have to know how a car is made to drive it