1. Genotype x Environment Interactions Analyses of Multiple Location Trials

2. Previous Class  How many locations are sufficient.  Where should sites be located.  Assumptions of over site analyses.  Homoscalestisity of error variance.  Bartlett Test – Same d.f., Different d.f.  Transforming the data.  Analyses of variance table and EMS.

3. Interpretation

4. Interpretation  Look at data: diagrams and graphs  Joint regression analysis  Variance comparison analyze  Probability analysis  Multivariate transformation of residuals: Additive Main Effects and Multiplicative Interactions (AMMI)

5. Multiple Experiment Interpretation Visual Inspection  Inter-plant competition study  Four crop species: Pea, Lentil, Canola, Mustard  Record plant height (cm) every week after planting  Significant species x time interaction

6. Plant Biomas x Time after Planting

7. PeaLentil Mustard Canola

8. Legume Brassica

9. Joint Regression

10. Regression Revision  Glasshouse study, relationship between time and plant biomass.  Two species: B. napus and S. alba.  Distructive sampled each week up to 14 weeks.  Dry weight recorded.

11. Dry Weight Above Ground Biomass

12. Biomass Study S. alba B. napus

13. Biomass Study (Ln Transformation) S. alba B. napus

14. Mean x = 7.5; Mean y = 0.936 SS(x)=227.5; SS(y)=61.66; SP(x,y)=114.30 Ln(Growth) = 0.5024 x Weeks - 2.8328 se(b)= 0.039361

15. B. napus Mean x = 7.5; Mean y = 0.936 SS(x)=227.5; SS(y)=61.66; SP(x,y)=114.30 Ln(Growth) = 0.5024 x Weeks - 2.8328 se(b)= 0.039361 Source df SS MS Regression 1 57.43 57.43 *** Residual 12 4.23 0.35

16. S. alba Mean x = 7.5; Mean y = 1.435 SS(x)=227.5; SS(y)=61.03; SP(x,y)=112.10 Ln(Growth) = 0.4828 x Weeks - 2.2608 se(b)= 0.046068

17. S. alba Mean x = 7.5; Mean y = 1.435 SS(x)=227.5; SS(y)=61.03; SP(x,y)=112.10 Ln(Growth) = 0.4828 x Weeks - 2.2608 se(b)= 0.046068 Source df SS MS Regression 1 55.24 55.24 *** Residual 12 5.79 0.48

18. Comparison of Regression Slopes t - Test [b 1 - b 2 ] [se(b 1 ) + se(b 2 )/2] 0.4928 - 0.5024 [(0.0460 + 0.0394)/2] 0.00960.0427145 = 0.22 ns

19. Joint Regression Analyses

20. Y ijk =  + g i + e j + ge ij + E ijk ge ij =  i e j +  ij Y ijk =  + g i + (1+  i )e j +  ij + E ijk

21. Yield Environments a b c d

22. Joint Regression Example  Class notes, Table15, Page 229.  20 canola (Brassica napus) cultivars.  Nine locations, Seed yield.

23. Joint Regression Example

24. Westar = 0.94 x Mean + 0.58 Source df SS Regression 1 b 1 sp(x,y)/ss(x) = [sp(x,y)] 2 /ss(x) Residual 12 Difference = SS(Res) Total 13 ss(y)

25. Joint Regression Example Source df SSqMSq Regression 118991899 *** Residual 7 22 3.2 Westar = 0.94 x Mean + 0.58

26. Joint Regression Example Bounty = 1.12 x Mean + 1.12 Source df SS Regression 1 b 1 sp(x,y)/ss(x) = [sp(x,y)] 2 /ss(x) Residual 12 Difference = SS(Res) Total 13 ss(y)

27. Joint Regression Example Source df SSqMSq Regression 122472247 *** Residual 7 31 4.0 Bounty = 1.12 x Mean + 1.12

28. Joint Regression Example Source SS Heter. Reg ∑[SP(x,y i )] 2 /SS(x)-[∑SP(x,y i )] 2 /[SS(x)] 2 Residual ∑SS(Res i ) G x E 459.4

29. Joint Regression Example

30. Joint Regression ~ Example #2

31. Joint Regression C A B

32. Problems with Joint Regression  Non-independence - regression of genotype values onto site means, which are derived including the site values.  The x-axis values (site means) are subject to errors, against the basic regression assumption.  Sensitivity (  -values) correlated with genotype mean.

33. Problems with Joint Regression  Non-independence - regression of genotype values onto site means, which are derived including the site values.  Do not include genotype value in mean for that regression.  Do regression onto other values other than site means (i.e. control values).

34. Joint Regression ~ Example #2

36. Problems with Joint Regression  The x-axis values (site means) are subject to errors, against the basic regression assumption.  Sensitivity (  -values) correlated with genotype mean.

37. Addressing the Problems  Use genotype variance over sites to indicate sensitivity rather than regression coefficients.

38. Genotype Yield over Sites ‘Ark Royal’

39. Genotype Yield over Sites ‘Golden Promise’

40. Over Site Variance

41. Univariate Probability Prediction

42. Over Site Variance

43. Univariate Probability Prediction ƒ(µ¸A) T  A.

44.   TT  TT ƒ (  A  d  dA T  A. Univariate Probability Prediction

45. Environmental Variation  1 1 1 1  2 2 2 2 T

46. Use of Normal Distribution Function Tables |T – m|  g to predict values greater than the target (T) |m – T|  g to predict values less than the target (T)

47. The mean (m) and environmental variance (  g 2 ) of a genotype is 12.0 t/ha and 16.0 2, respectively (so  = 4). What is the probability that the yield of that given genotype will exceed 14 t/ha when grown at any site in the region chosen at random from all possible sites. Use of Normal Distribution Function Tables

48. T – m  g  = = = = 14 – 12 4 = Use of Normal Distribution Function Tables = 0.5 Using normal dist. tables we have the probability from -  to T is 0.6915. Actual answer is 1 – 0.6916 = 30.85 (or 38.85% of all sites in the region).

49. Use of Normal Distribution Function Tables The mean (m) and environmental variance (  g 2 ) of a genotype is 12.0 t/ha and 16.0 2, respectively (so  = 4). What is the probability that the yield of that given genotype will exceed 11 t/ha when grown at any site in the region chosen at random from all possible sites.

50. T – m  g  = = = = 11 – 12 4 = Use of Normal Distribution Function Tables = -0.25 Using normal dist. tables we have  (0.25) = 0.5987, but because  is negative our answer is 1 – (1 – 0.5987) = 0.5987 or 60% of all sites in the region.

51.  Exceed the target; and (T-m)/  positive, then probability = 1 – table value.  Exceed the target; and (T-m)/  negative, then probability = table value.  Less than the target; and (m-T)/  positive, then probability = table value.  Less than target; and (m-T)/  negative, then probability = 1 – table value. Use of Normal Distribution Function Tables

52. Univariate Probability

55. Multivariate Probability Prediction T1T1--T1T1-- T2T2--T2T2-- TnTn--TnTn-- …. f (x 1,x 2,..., x n ) dx 1, dx 2,..., dx n

56. Problems with Probability Technique  Setting suitable/appropriate target values:  Control performance  Industry (or other) standard  Past experience  Experimental averages

57.  Complexity of analytical estimations where number of variables are high:  Use of rank sums Problems with Probability Technique

58. Additive Main Effects and Multiplicative Interactions AMMI  AMMI analysis partitions the residual interaction effects using principal components.  Inspection of scatter plot of first two eigen values (PC1 and PC2) or first eigen value onto the mean.

59. AMMI Analyses Y ijk =  + g i + e j + ge ij + E ijk

60. AMMI Analyses Y ijk -  - g i - e j - E ijk = ge ij

61. AMMI Analyses Y ijk -  - g i - e j - E ijk = ge ij ge 11 ge 12 ge 13 ….. ge 1n ge 21 ge 22 ge 23 ….. ge 2n... …...... …... ge i1 ge i2 ge i3 ….. ge in... …...... …... ge k1 ge k2 ge k3 ….. ge kn

62. AMMI Analysis Seed Yield G1 S4 S7 S3 S5 G4 S1 S6 G2 G3 S2

63. G1 S4 S7 S3 S5 G4 S1 S6 G2 G3 S2 AMMI Analysis Seed Yield

64. G1 S4 S7 S3 S5 G4 S1 S6 G2 G3 S2 AMMI Analysis Seed Yield

65. Time Square Chi-Square