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Multiple Comparisons.  Multiple Range Tests  Tukey’s and Duncan’s  Orthogonal Contrasts.

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Presentation on theme: "Multiple Comparisons.  Multiple Range Tests  Tukey’s and Duncan’s  Orthogonal Contrasts."— Presentation transcript:

1 Multiple Comparisons

2  Multiple Range Tests  Tukey’s and Duncan’s  Orthogonal Contrasts

3 Orthogonal Contrasts

4 AOV Orthogonal Contrasts

5 Tukey’s Multiple Range Test

6 Consider that cultivars A and B were developed in Idaho and C and D developed in California  Do the two Idaho cultivars have the same yield potential?  Do the two California cultivars have the same yield potential?  Are Idaho cultivars higher yielding than California cultivars?

7 Analysis of Variance

8 Orthogonality  c i = 0  [c 1i x c 2i ] = 0  c i = 0 -1 -1 +1 +1 --  c i = 0  c i = 0 -1 +1 -1 +1 --  c i = 0  c i = 0 +1 -1 -1 +1 --  c i = 0

9 Calculating Orthogonal Contrasts d.f. (single contrast) = 1 S.Sq(contrast) = M.Sq = [  c i x Y i ] 2 /n  c i 2 ]

10 Orthogonal Contrasts - Example

11 S.Sq = [  c i x Y i ]/[n  c i 2 ] S.Sq(1) [(-1)64.1+(-1)76.6+(1)40.1+(1)47.8] 2 / n  c i 2 = 52.8 2 /(3 x 4) = 232.32

12 S.Sq(2) [(-1) x 64.1+(+1) x 76.6] 2 /(3x2) 26.04 S.Sq(3) [(-1) x 40.1+(+1) x 47.8] 2 /(3x2) 9.88

13 Orthogonal Contrasts

14  Five dry bean cultivars (A, B, C, D, and E).  Cultivars A and B are drought susceptible.  Cultivars C, D and E are drought resistant.  Four Replicate RCB, one location  Limited irrigation applied.

15 Analysis of Variance

16 Orthogonal Contrast Example #2 Tukey’s Multiple Range Test

17 Orthogonal Contrasts  Is there any difference in yield potential between drought resistant and susceptible cultivars?  Is there any difference in yield potential between the two drought susceptible cultivars?  Are there any differences in yield potential between the three drought resistant cultivars?

18 Orthogonal Contrasts

19 S.Sq(1)= [(-3)130+(-3)124+(2)141+(2)186+(2)119] 2 /n  c i 2 130 2 /(4 x 40) = 140.8 S.Sq(2)= [(-1)130+(+1)124] 2 /n  c i 2 6 2 /(4 x 2) = 4.5 S.Sq(Rem) = S.Sq(Cult)-S.Sq(1)-S.Sq(2) 728.2-140.8-4.5 = 582.9 (with 2 d.f.)

20 Analysis of Variance

21 Partition Contrast(rem)

22 Analysis of Variance

23 Alternative Contrasts

24 S.Sq(1)= [(-3)130+(-3)124+(2)141+(2)186+(2)119] 2 /n  c i 2 130 2 /(4 x 40) = 140.8 S.Sq(2)= [(-1)130+(-1)124+(-1)141+(4)186+(-1)119] 2 /n  c i 2 230 2 /(4 x 20) = 661.2 S.Sq(Rem) = S.Sq(Cult)-S.Sq(1)-S.Sq(2) 728.2-140.8-661.2 = -73.8 (Oops !!!) (with 2 d.f.)

25  c 1i = 0 (-3) + (-3) + (+2) + (+2) + (+2) = 0 =  c 2i = 0 (-1) + (-1) + (-1) + (+4) + (-1) = 0 =  [c 1i x c 2i ] = 0 (-3)(-1)+(-3)(-1)+2(-1)+2(4)+2(-1) =10 =  Orthogonality

26 More Appropriate Contrasts

27 Analysis of Variance

28 Conclusions  Almost all the variation between cultivars is accounted for by the difference between cv ‘D’ and the others.  The remaining 4 cultivars are not significantly different.  Orthogonal contrast result is exactly the same are the result from Tukey’s contrasts.

29 Conclusions  Important to make the “correct” orthogonal contrasts.  Important to make contrasts which have “biological sense”.  Orthogonal contrasts should be decided prior to analyses and not dependant on the data.

30 Orthogonal Contrasts  Four Brassica species (B. napus, B. rapa, B. juncea, and S. alba).  Ten cultivars ‘nested’ within each species.  Three insecticide treatments (Thiodan, Furidan, no insecticide).  Three replicate split-plot design.

31 Analysis of Variance

32 Species and Treatment Means

33 Orthogonal Contrasts

34

35 Analysis of Variance

36 Species x Treatment Interaction

37 Species x Contrast (1)

38 Species x Contrast (2)

39 Orthogonal Contrasts and Interactions  Consider a cross classified factorial design with 4 replicates.  Four cultivars; 2 from Idaho and 2 from California (again).  3 herbicide treatments; No-treatment control, Killall, and Onllik.

40 Cultivar ControlKillallOnllikTotal IdaBest 90168147405 IdaCream 75141135351 Yuppy 456475184 Total 210373357 Orthogonal Contrasts and Interactions

41  Contrasts for cultivars?  Idaho v California (-1 -1 +2),  SS(Id v CA) = 2,787;  Contrast for herbicides?  Herbicide v No-treatment control (-2 +1 +1),  SS(Herb v Not) = 1,779;  Contrast for the interaction between the first two contrasts?

42 GenotypeHerb YieldID v CA Herb v Not Interaction IdaBestCont 90 IdaBestKillall 168 IdaBestOnllik 147 IdaCreamCont 75 IdaCreamKillall 141 IdaCreamOnllik 135 YuppyCont 45 YuppyKillall 64 YuppyOnllik 75 Orthogonal Contrasts and Interactions

43 GenotypeHerb YieldID v CA Herb v Not Interaction IdaBestCont 90 IdaBestKillall 168 IdaBestOnllik 147 IdaCreamCont 75 IdaCreamKillall 141 IdaCreamOnllik 135 YuppyCont 45+2 YuppyKillall 64+2 YuppyOnllik 75+2 Orthogonal Contrasts and Interactions

44 GenotypeHerb YieldID v CA Herb v Not Interaction IdaBestCont 90-2 IdaBestKillall 168+1 IdaBestOnllik 147+1 IdaCreamCont 75-2 IdaCreamKillall 141+1 IdaCreamOnllik 135+1 YuppyCont 45+2-2 YuppyKillall 64+2+1 YuppyOnllik 75+2+1 Orthogonal Contrasts and Interactions

45 GenotypeHerb YieldID v CA Herb v Not Interaction IdaBestCont 90-2+2 IdaBestKillall 168+1 IdaBestOnllik 147+1 IdaCreamCont 75-2+2 IdaCreamKillall 141+1 IdaCreamOnllik 135+1 YuppyCont 45+2-2-4 YuppyKillall 64+2+1+2 YuppyOnllik 75+2+1+2 Orthogonal Contrasts and Interactions

46  Contrasts for cultivars?  Idaho v California (-1 -1 +2),  SS(Id v CA) = 2,787;  Contrast for herbicides?  Herbicide v No-treatment control (-2 +1 +1),  SS(Herb v Not) = 1,779;  Contrast for the interaction between the first two contrasts?  SS (Interaction) = 246.

47 Orthogonal Contrasts and Interactions

48 More Orthogonal Contrasts … Trend Analyses

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

50 Example #4  Ten yellow mustard (S. alba) cultivars.  Five different nitrogen application rates (50, 75, 100, 125, and 150)

51 Analysis of Variance

52 Orthogonal Contrasts

53 Example #4

54

55

56 Analysis of Variance

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

58 Example #4

59 Linear

60 Quadratic

61 Cubic

62 Quartic

63 Example #5  Two carrot cultivars (‘Orange Gold’ and ‘Bugs Delight’.  Four seeding rates (1.5, 2.0, 2.5 and 3.0 lb/acre).  Three replicates.

64 Example #5

65 Analysis of Variance

66

67 Orange Gold Bug’s Delight

68 End of Analyses of Variance Section

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