1. Genotype x Environment Interactions Analyses of Multiple Location Trials

2. Why do researchers conduct multiple experiments?  Effects of factors under study vary from location to location or from year to year. To obtain an unbias estimate.  Interest in determining the effect of factors over time.  To investigate genotype (or treatment) x environment interactions.

3. What are Genotype x Environment Interactions?  Differential response of genotypes to varying environmental conditions.  Delight for statisticians who love to investigate them.  The biggest nightmare for plant breeders (and some other agricultural researchers) who try to avoid them like the plague.

4. What causes Genotype x Environment interactions?

5. B A Yield Locations No interaction

6. B A Yield Locations B A Yield Locations No interaction Cross-over interaction

7. B A Yield Locations B A Yield Locations B A Yield Locations No interaction Scalar interaction Cross-over interaction

8. Examples of Multiple Experiments  Plant breeder grows advanced breeding selections at multiple locations to determine those with general or specific adaptability ability.  A pathologist is interested in tracking the development of disease in a crop and records disease at different time intervals.  Forage agronomist is interested in forage harvest at different stages of development over time.

9. Types of Environment  Researcher controlled environments, where the researcher manipulates the environment. For example, variable nitrogen.  Semi-controlled environments, where there is an opportunity to predict conditions from year to year. For example, soil type.  Uncontrolled environments, where there is little chance of predicting environment. For example, rainfall, temperature, high winds.

10. Why? To investigate relationships between genotypes and different environmental (and other) changes. To investigate relationships between genotypes and different environmental (and other) changes. To identify genotypes which perform well over a wide range of environments. General adaptability. To identify genotypes which perform well over a wide range of environments. General adaptability. To identify genotypes which perform well in particular environments. Specific adaptability. To identify genotypes which perform well in particular environments. Specific adaptability.

11. How many environments do I need? Where should they be?

12. Number of Environments  Availability of planting material.  Diversity of environmental conditions.  Magnitude of error variances and genetic variances in any one year or location.  Availability of suitable cooperators  Cost of each trial ($’s and time).

13. Location of Environments  Variability of environment throughout the target region.  Proximity to research base.  Availability of good cooperators.  $$$’s.

14. Analyses of Multiple Experiments

15. Points to Consider before Analyses  Normality.  Homoscalestisity (homogeneity) of error variance.  Additive.  Randomness.

16. Points to Consider before Analyses  Normality.  Homoscalestisity (homogeneity) of error variance.  Additive.  Randomness.

17. Bartlett Test (same degrees of freedom) M = df{nLn(S) -  Ln  2 } Where, S =  2 /n  2 n-1 = M/C C = 1 + (n+1)/3ndf n = number of variances, df is the df of each variance

18. Bartlett Test (same degrees of freedom) S = 101.0; Ln(S) = 4.614

19. Bartlett Test (same degrees of freedom) S = 100.0; Ln(S) = 4.614 M = (5)[(4)(4.614)-18.081] = 1.880, 3df C = 1 + (5)/[(3)(4)(5)] = 1.083

20. Bartlett Test (same degrees of freedom) S = 100.0; Ln(S) = 4.614 M = (5)[(4)(4.614)-18.081] = 1.880, 3df C = 1 + (5)/[(3)(4)(5)] = 1.083  2 3df = 1.880/1.083 = 1.74 ns

21. Bartlett Test (different degrees of freedom) M = (  df)nLn(S) -  dfLn  2 Where, S = [  df.  2 ]/(  df)  2 n-1 = M/C C = 1+{(1)/[3(n-1)]}.[  (1/df)-1/ (  df)] n = number of variances

22. Bartlett Test (different degrees of freedom) S = [  df.  2 ]/(  df) = 13.79/37 = 0.3727 (  df)Ln(S) = (37)(-0.9870) = -36.519

23. Bartlett Test (different degrees of freedom) M = (  df)Ln(S) -  dfLn  2 = -36.519 -(54.472) = 17.96 C = 1+[1/(3)(4)](0.7080 - 0.0270) = 1.057

24. Bartlett Test (different degrees of freedom) S = [  df.  2 ]/(  df) = 13.79/37 = 0.3727 (  df)Ln(S) = (37)(=0.9870) = -36.519 M = (  df)Ln(S) -  dfLn  2 = -36.519 -(54.472) = 17.96 C = 1+[1/(3)(4)](0.7080 - 0.0270) = 1.057  2 3df = 17.96/1.057 = 16.99 **, 3df

25. Heterogeneity of Error Variance

26. Significant Bartlett Test  “ What can I do where there is significant heterogeneity of error variances?”  Transform the raw data: Often  ~  cw Binomial Distribution where  = np and  = npq Transform to square roots

27. Heterogeneity of Error Variance

28. Significant Bartlett Test  “What else can I do where there is significant heterogeneity of error variances?”  Transform the raw data: Homogeneity of error variance can always be achieved by transforming each site’s data to the Standardized Normal Distribution [x i -  ]/ 

29. Significant Bartlett Test  “What can I do where there is significant heterogeneity of error variances?”  Transform the raw data  Use non-parametric statistics

30. Analyses of Variance

31. Model ~ Multiple sites Y ijk =  + g i + e j + ge ij + E ijk  i g i =  j e j =  ij ge ij Environments and Replicate blocks are usually considered to be Random effects. Genotypes are usually considered to be Fixed effects.

32. Analysis of Variance over sites

35. Y ijkl =  +g i +s j +y k +gs ij +gy ik +sy jk +gsy ijk +E ijkl  i g i =  j s j =  k y k = 0  ij gs ij =  ik gy ik =  jk sy ij = 0  ijk gsy ijk = 0 Models ~ Years and sites

36. Analysis of Variance