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Biom-32-GxE I.ppt

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Genotype x environment interactions occur when genotypes respond differently to varying environmental conditions. Researchers conduct multi-location trials over multiple years to investigate these interactions and identify genotypes that perform well across different environments or are specifically adapted to certain environments. Analyzing the data from such trials involves testing for homogeneity of error variances between locations and years before performing analyses of variance to partition variance components and determine which genotypes interact least or perform best on average over environments.

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Genotype x Environment Interactions Analyses of Multiple Location Trials
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.
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.
What causes Genotype x Environment interactions?
B A Yield Locations No interaction
B A Yield Locations B A Yield Locations No interaction Cross-over interaction
B A Yield Locations B A Yield Locations B A Yield Locations No interaction Scalar interaction Cross-over interaction
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.
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.
Why? 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 in particular environments. Specific adaptability.
How many environments do I need? Where should they be?
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).
Location of Environments Variability of environment throughout the target region. Proximity to research base. Availability of good cooperators. $$$s.
Analyses of Multiple Experiments
Points to Consider before Analyses Normality. Homoscalestisity (homogeneity) of error variance. Additive. Randomness.
Points to Consider before Analyses Normality. Homoscalestisity (homogeneity) of error variance. Additive. Randomness.
Bartlett Test (same degrees of freedom) M = df{nLn(S) - Ln2} Where, S = ワ2/n 2 n-1 = M/C C = 1 + (n+1)/3ndf n = number of variances, df is the df of each variance
Bartlett Test (same degrees of freedom) df 2 Ln(2 ) 5 178 5.148 5 60 4.094 5 98 4.585 5 68 4.202 Total 404 18.081 S = 101.0; Ln(S) = 4.614
Bartlett Test (same degrees of freedom) df 2 Ln(2 ) 5 178 5.148 5 60 4.094 5 98 4.585 5 68 4.202 Total 404 18.081 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
Bartlett Test (same degrees of freedom) df 2 Ln(2 ) 5 178 5.148 5 60 4.094 5 98 4.585 5 68 4.202 Total 404 18.081 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
Bartlett Test (different degrees of freedom) M = ( df)nLn(S) - dfLn2 Where, S = [df.2]/(df) 2 n-1 = M/C C = 1+{(1)/[3(n-1)]}.[(1/df)-1/ (df)] n = number of variances
Bartlett Test (different degrees of freedom) df 2 Ln(2 ) 1/df 9 0.909 -0.095 0.111 7 0.497 -0.699 0.1429 9 0.076 -2.577 0.1111 7 5 0.103 0.146 -2.273 -1.942 0.1429 0.2000 37 0.7080 S = [df.2]/(df) = 13.79/37 = 0.3727 (df)Ln(S) = (37)(-0.9870) = -36.519
Bartlett Test (different degrees of freedom) df 2 Ln(2 ) 1/df 9 0.909 -0.095 0.111 7 0.497 -0.699 0.1429 9 0.076 -2.577 0.1111 7 5 0.103 0.146 -2.273 -1.942 0.1429 0.2000 37 0.7080 M = (df)Ln(S) - dfLn 2 = -36.519 -(54.472) = 17.96 C = 1+[1/(3)(4)](0.7080 - 0.0270) = 1.057
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
Heterogeneity of Error Variance 0 10 20 30 40 50 60 70 80 Mosc Gene Tens Gran Pend Colf Kalt Mocc Boze Seed Yield
Significant Bartlett Test 金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
Heterogeneity of Error Variance 0 2 4 6 8 10 Mosc Gene Tens Gran Pend Colf Kalt Mocc Boze SQRT[Seed Yield]
Significant Bartlett Test 金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 sites data to the Standardized Normal Distribution [xi-]/
Significant Bartlett Test 金What can I do where there is significant heterogeneity of error variances? Transform the raw data Use non-parametric statistics
Analyses of Variance
Model ~ Multiple sites Yijk = + gi + ej + geij + Eijk i gi = j ej = ij geij Environments and Replicate blocks are usually considered to be Random effects. Genotypes are usually considered to be Fixed effects.
Analysis of Variance over sites Source d.f. EMSq Sites (s) Rep w Sites (r) Genotypes (g) Geno x Site Replicate error
Source d.f. EMSq Sites (s) s-1 Rep w Sites (r) s(1-r) Genotypes (g) g-1 Geno x Site (g-1)(s-1) Replicate error s(r-1)(g-1) Analysis of Variance over sites
Source d.f. EMSq Sites (s) s-1 2 e + g2 rws + rg2 s Rep w Sites (r) s(1-r) 2 e + g2 rws Genotypes (g) g-1 2 e + r2 gs+ rs2 g Geno x Site (g-1)(s-1) 2 e + r2 gs Replicate error s(r-1)(g-1) 2 e Analysis of Variance over sites
Yijkl = +gi+sj+yk+gsij+gyik+syjk+gsyijk+Eijkl igi=jsj=kyk= 0 ijgsij=ikgyik=jksyij = 0 ijkgsyijk = 0 Models ~ Years and sites
Analysis of Variance Source d.f. EMSq Years (y) y-1 2 e+gy2 rwswy+rg2 swy+rgs2 y Sites w Years (s) y(s-1) 2 e + g2 rwswy + rg2 swy Rep w Sites w year (r) ys(1-r) 2 e + g2 rwswy Genotypes (g) g-1 2 e + r2 gswy + rs2 gy + rl2 g Geno x year (y-1)(g-1) 2 e + r2 gswy + rs2 gy Geno x Site w Year y(g-1)(s-1) 2 e + r2 gswy Replicate error ys(r-1)(g-1) 2 e

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Biom-32-GxE I.ppt

  • 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?
  • 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.
  • 10. Why? 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 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.
  • 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) - Ln2} Where, S = ワ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) df 2 Ln(2 ) 5 178 5.148 5 60 4.094 5 98 4.585 5 68 4.202 Total 404 18.081 S = 101.0; Ln(S) = 4.614
  • 19. Bartlett Test (same degrees of freedom) df 2 Ln(2 ) 5 178 5.148 5 60 4.094 5 98 4.585 5 68 4.202 Total 404 18.081 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) df 2 Ln(2 ) 5 178 5.148 5 60 4.094 5 98 4.585 5 68 4.202 Total 404 18.081 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) - dfLn2 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) df 2 Ln(2 ) 1/df 9 0.909 -0.095 0.111 7 0.497 -0.699 0.1429 9 0.076 -2.577 0.1111 7 5 0.103 0.146 -2.273 -1.942 0.1429 0.2000 37 0.7080 S = [df.2]/(df) = 13.79/37 = 0.3727 (df)Ln(S) = (37)(-0.9870) = -36.519
  • 23. Bartlett Test (different degrees of freedom) df 2 Ln(2 ) 1/df 9 0.909 -0.095 0.111 7 0.497 -0.699 0.1429 9 0.076 -2.577 0.1111 7 5 0.103 0.146 -2.273 -1.942 0.1429 0.2000 37 0.7080 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 0 10 20 30 40 50 60 70 80 Mosc Gene Tens Gran Pend Colf Kalt Mocc Boze Seed Yield
  • 26. Significant Bartlett Test 金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 0 2 4 6 8 10 Mosc Gene Tens Gran Pend Colf Kalt Mocc Boze SQRT[Seed Yield]
  • 28. Significant Bartlett Test 金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 sites data to the Standardized Normal Distribution [xi-]/
  • 29. Significant Bartlett Test 金What can I do where there is significant heterogeneity of error variances? Transform the raw data Use non-parametric statistics
  • 31. Model ~ Multiple sites Yijk = + gi + ej + geij + Eijk i gi = j ej = ij geij 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 Source d.f. EMSq Sites (s) Rep w Sites (r) Genotypes (g) Geno x Site Replicate error
  • 33. Source d.f. EMSq Sites (s) s-1 Rep w Sites (r) s(1-r) Genotypes (g) g-1 Geno x Site (g-1)(s-1) Replicate error s(r-1)(g-1) Analysis of Variance over sites
  • 34. Source d.f. EMSq Sites (s) s-1 2 e + g2 rws + rg2 s Rep w Sites (r) s(1-r) 2 e + g2 rws Genotypes (g) g-1 2 e + r2 gs+ rs2 g Geno x Site (g-1)(s-1) 2 e + r2 gs Replicate error s(r-1)(g-1) 2 e Analysis of Variance over sites
  • 35. Yijkl = +gi+sj+yk+gsij+gyik+syjk+gsyijk+Eijkl igi=jsj=kyk= 0 ijgsij=ikgyik=jksyij = 0 ijkgsyijk = 0 Models ~ Years and sites
  • 36. Analysis of Variance Source d.f. EMSq Years (y) y-1 2 e+gy2 rwswy+rg2 swy+rgs2 y Sites w Years (s) y(s-1) 2 e + g2 rwswy + rg2 swy Rep w Sites w year (r) ys(1-r) 2 e + g2 rwswy Genotypes (g) g-1 2 e + r2 gswy + rs2 gy + rl2 g Geno x year (y-1)(g-1) 2 e + r2 gswy + rs2 gy Geno x Site w Year y(g-1)(s-1) 2 e + r2 gswy Replicate error ys(r-1)(g-1) 2 e