1. Orthogonal Contrasts

2. Orthogonality ci = 0 [c1i x c2i] = 0 -1 +1 -1 +1 -- ci = 0

3. Calculating Orthogonal Contrasts
d.f. (single contrast) = 1
S.Sq(contrast) = M.Sq = [ci x Yi]2/nci2]

4. More Orthogonal Contrasts …
Trend Analyses

5. Aim of Analyses of Variance
Detect significant differences between treatment means.
Determine trends that may exist as a result of varying specific factor levels.

6. Linear

7. Quadratic

8. Cubic

9. Quartic

10. Trend Analyses
The F-value associates with a trend contrast is significant.
All higher order trend contrasts are not significant.

11. Get more tang from your mustard
Ten yellow mustard (S. alba) cultivars.
Five different nitrogen application rates (50, 75, 100, 125, and 150)

12. Analysis of Variance

13. Get more tang from your mustard

14. Get more tang from your mustard

15. Get more tang from your mustard
Linear
Quadratic
Cubic

16. Get more tang from your mustard

17. Get more tang from your mustard

18. Get more tang from your mustard

19. Analysis of Variance

20. Get more tang from your mustard
There is a significant nitrogen effect.
The relationship between yield and nitrogen is significantly linear.
All genotypes follow a linear relationship.

21. When rabbits come to dinner
Two carrot cultivars (‘Orange Gold’ and ‘Bugs Delight’.
Four seeding rates (1.5, 2.0, 2.5 and 3.0 lb/acre).
Three replicates.
TQ #1 p. 155.

22. When rabbits come to dinner

23. Analysis of Variance

24. When rabbits come to dinner

25. When rabbits come to dinner

26. When rabbits come to dinner
Cultivars
Seeding rate
lb/acre
1.5
2.0
2.5
3.0
Orange Gold
4.53
4.01
5.23
4.48
Bug’s Delight
3.25
3.70
5.41
6.08
Total
23.34
23.13
31.92
31.68

27. When rabbits come to dinner
Cultivars
Seeding rate
lb/acre
1.5
2.0
2.5
3.0
Orange Gold
4.53
4.01
5.23
4.48
Bug’s Delight
3.25
3.70
5.41
6.08
Total
23.34
23.13
31.92
31.68
Linear
Quadratic
Cubic

28. When rabbits come to dinner
Cultivars
Seeding rate
lb/acre
1.5
2.0
2.5
3.0
Orange Gold
4.53
4.01
5.23
4.48
Bug’s Delight
3.25
3.70
5.41
6.08
Total
23.34
23.13
31.92
31.68
Linear -3 -1 1 3
Quadratic
Cubic

29. When rabbits come to dinner
Cultivars
Seeding rate
lb/acre
1.5
2.0
2.5
3.0
Orange Gold
4.53
4.01
5.23
4.48
Bug’s Delight
3.25
3.70
5.41
6.08
Total
23.34
23.13
31.92
31.68
SSq
Linear -3 -1 1 3
1143/120
Quadratic
0.0/24
Cubic
325/120

30. Analysis of Variance

31. When rabbits come to dinner
Cultivar
Seeding Rate
Yield
Cv’s
Linear
C x L
Orange Gold
0.5
4.53
2.0
4.01
2.5
5.23
3.0
4.48
Bug’s Delight
3.25
3.97
5.41
6.8

32. When rabbits come to dinner
Cultivar
Seeding Rate
Yield
Cv’s
Linear
C x L
Orange Gold
0.5
4.53 -1
2.0
4.01
2.5
5.23
3.0
4.48
Bug’s Delight
3.25 +1
3.97
5.41
6.8

33. When rabbits come to dinner
Cultivar
Seeding Rate
Yield
Cv’s
Linear
C x L
Orange Gold
0.5
4.53 -1 -3
2.0
4.01
2.5
5.23 +1
3.0
4.48 +3
Bug’s Delight
3.25
3.97
5.41
6.8

34. When rabbits come to dinner
Cultivar
Seeding Rate
Yield
Cv’s
Linear
C x L
Orange Gold
0.5
4.53 -1 -3 +3
2.0
4.01 +1
2.5
5.23
3.0
4.48
Bug’s Delight
3.25
3.97
5.41
6.8

35. Analysis of Variance

36. When rabbits come to dinner
Orange Gold
Bug’s Delight

37. End of Analyses of Variance Section