1. Analyses of Variance Review

3. Simple Situation
Genotype A
Genotype B
135 34

4. Simple Situation Genotype A Genotype B 135 34 115 76 102 83 110 64
115.5
64.2

5. t-test
|x1-x2| 2[(12+22)/(n1+n2)]
t =

6. More than two treatments
Rep.
Genotype
Brundage
Lambert
Croft
Stephens 1 64 78 75 55 2 72 91 93 66 3 68 97 49 4 77 82 71 5 56 85 63 70 6 95 76

7. Multiple t-tests
Brundage v Lambert; Brundage v Croft; Brundage v Stephens; Lambert v Croft; Lambert v Stephens; Croft v Stephens.
Problems?
If all tests were done at 95% significance level, and one difference was significant, we have done 6 tests and would expect 1/20 to be significant, at random.

8. Analysis of Variance
Is an elegant and quicker way to calculate a pooled error term.
Analysis is simple in simple designs but can be complicated and lengthy in some designs (i.e. rectangular lattices).
In some experimental designs the ANOVA is the only method to estimate a pooled error term.

9. Analysis of Variance
It can provide an F-test to tests specific hypotheses. (i.e. to test general differences between different treatments).
Can be an invaluable initial contribution to interpretation of experiments.

10. Theory of Analysis of Variance
ij(xij-x..)2 = ij[(xij-xi.) + (xi.-x..)]2
ij[(xij-xi.)2+2(xij-xi.)(xi.-x..)+(xi.-x..)2]
ij(xij-x..)2 = ij(xij-xi.)2+ki(xi.-x..)2]
ki(xi.-x..)2 = Between Treatment SS
ij(xij-xi.)2 = Within Treatment SS

11. Theory of Analysis of Variance
BTMS ~ 2n-1 df : WTMS ~ 2nk-n df
2n-1 df 2nk-n df ~ F Dist n-1,nk-n df

12. Theory of Analysis of Variance
Source of variation df
EMS
Between treatments
n-1
e2 + kt2
Within treatments
nk-n
e2
Total
nk-1
[e2 + kt2]/e2 = 1, if kt2 = 0

13. Assumptions behind the ANOVA
Assumption of data being normally distributed.
Homogeneity of error variance.
Additivity of variance effects.
Data collected from a properly randomized experiment.

14. Analyses of CRB Designs
Yij =  + ti + eij

15. Analysis of Variance of CRB
Source df SS
Between treatments
k-1
[G12/n1 + G22/n2 … Gk2/nk] - CF
Within treatments
jk-k
By difference
Total
jk-1
[x112 + x … + xjk2] - CF
CF = [xij]2/jk

16. Analyses of RCB Designs
Yij =  + bi + tj + eij

17. Analysis of Variance of RCB
Source df SS
Blocks
r-1
[B12 + B22 + … + Br2]/t – CF
Treatments
t-1
[T12 + T22 + … + Tt2]/r – CF
Error
(r-1)(t-1)
By difference
Total
rt-1
[x112 + x … + xrt2] – CF
CF = [xij]2/rt

18. Analyses of Latin Designs
Yijk =  + ri + cj + tk(ij) + eijk

19. Analysis of Variance of Latin
Source df SS
Rows
t-1
[R12 + R22 + … + Rt2]/t – CF
Columns
[C12 + C22 + … + Ct2]/t – CF
Treatments
[T12 + T22 + … + Tt2]/t – CF
Error
(t-1)(t-2)
By difference
Total
t2-1
[x112 + x … + xtt2] – CF
CF = [xij]2/t2

20. Efficiency of Latin Squares
cw CRB Design
[MSr + MSc + (t-1)EMS]/(t+1)EMS
If value response is 325, then latin square in will increase precision by 225% over CRB
and CRD would have need 2.25 x 4 = 9 replicates to be as accurate.

21. Efficiency of Latin Squares
cw RCB Design
Row (RCB)
= [MSr + (t-1)EMS]/(t+1)EMS
Col(RCB)
= [MSc + (t-1)EMS]/(t+1)EMS

22. +266%
-19% ☺

23. -19%
+226% ☺

24. Analyses of Lattice Squares
Yijk =  + ri + baj + tak + eijk

25. Lattice Square ANOVA Source df SS MS F Reps 4 5,946 1,486 4.03 *
Blk(adj) 15
11,382
759
2.35 ns
Intra error 45
14,533
323 -
T(adj)
24,030
1,602
4.34 **
Eff. Error
16,605
369

26. Efficiency of Lattice Design
100 x [Blk(adj)SS+Intra error SS]/k(k2-1)EMS
100 [11, ,533]/4(16)369
117%
I II III IV V
I II III
IV V

27. Dealing with Wrongful Data
It is usually assumed that the data collected is correct!.
Why would data not be correct?
Mis-recording, mis-classification, transcription errors, errors in data entry.
Outliers.

28. Dealing with Wrongful Data
What things can help?
Keep detailed records, on each experimental unit.
Decide beforehand what values would arouse suspision.

29. Dealing with Wrongful Data
What do you do with suspicios data?
If correct, and it is discarded, then valuable information is lost. This will bias the results.
If wrong and included, will bias results and may have extreme consequences.

30. Checking ANOVA Accurucy
Coefficient of variation: [e/]x100.
CV=(√100.9/73.75)*100=13.6%
R2 value = {[TSS-ESS]/TSS}x100.
R2 = (1654/3654)*100 = 44.7%.
Compare the effect of blocking or sub-blocking (discussed later).

31. Marvelous Marvin father of the Groom
Alaskan Wedding Feast

32. ANOVA of Factorial Designs

33. Factorial AOV Example Source df SS MS F Reps 2 0.01 0.005 ns
Seed Density
2.75
1.375
33.9 ***
Nitrogen 5
81.56
16.312
401.9***
S x N 10
1.33
0.133
3.28***
Error 34
1.38
0.041
Total 53
87.03

34. Factorial AOV Example Source df SS MS F Reps x Seed Rate 4 0.2268
0.0567
1.63 ns
Rep x N rate 10
0.4528
0.0453
1.30 ns
Rep x Seed x N 20
0.6936
0.0347

35. Split-plot AOV Source df SS MS F Reps 2 0.01 0.005 ns Seed Density
2.75
1.375
24.2 ***
Error (1) 4
0.2268
0.057 -
Nitrogen 5
81.56
16.312
426.9***
S x N 10
1.33
0.133
3.5***
Error (2) 30
1.1464
0.038
Total 53
87.03

36. Strip-plot AOV Source df SS MS F Reps 2 0.01 0.005 ns Seed Density
2.75
1.375
24.2 ***
Error 1 (Seed) 4
0.2268
0.0567 -
Nitrogen 5
81.56
16.312
360.1***
Error 2 (N) 10
0.4528
0.0453
S x N
1.33
0.133
3.83***
Error 3 (SxN) 20
0.6936
0.0347
Total 53
87.03

37. Fixed and Random Effects

38. Expected Mean Squares
Dependant on whether factor effects are Fixed or Random.
Necessary to determine which F-tests are appropriate and which are not.

39. Setting Expected Mean Squares
The expected mean square for a source of variation (say X) contains.
the error term.
a term in 2x. (or S2x )
a variance term for other selected interactions involving the factor X.

40. Coefficient for error mean square is always 1
Coefficients for EMS
Coefficient for error mean square is always 1
Coefficient of other expected mean squares is n times the product of factors levels that do not appear in the factor name.

41. Expected Mean Squares Which interactions to include in an EMS?
All the letter (i.e. A, B, C, …) appear in X.
All the other letters in the interaction (except those in X) are Random Effects.

42. A and B Fixed Effects
Model yield=A B A*B;

43. A and B Random Effects
Model yield=A B A*B;
Test h = A B e=A*B;

44. A Fixed and B Random
Model yield=A B A*B;
Test h = A e=A*B;

46. Multiple Comparisons Multiple Range Tests: Orthogonal Contrasts.
Tukey’s and Duncan’s.
Orthogonal Contrasts.

47. Tukey’s Multiple Range Test
W = q(p,f) x se[x]
se[x] = (2/n)
(94,773/4) = 153.9
W = 4.64 x = 714.1

48. Tukey’s Multiple Range Test

49. Duncan’s Multiple Range Test

50. Duncan’s Multiple Range Test

51. Multiple Comparisons Genotype Tukey Duncan A 2678 a B 2552 ab C
2127 abcd
2127 bcd E
1796 cde F
1681 cdef G
1316 ef

52. Orthogonal Contrasts

53. Orthogonal Contrasts
Maximum number of orthogonal contrasts is df for treatment.
SS of all contrasts must equal SS of treatment effect.
Rem SS is difference beyween treatment SS and sum of contrast SS.
Contrasts can help understand main effects and interactions.

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

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

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

57. Trend Analyses
Linear
Quadratic
Cubic
Quartic

58. 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.

59. Analysis of Variance

60. When rabbits come to dinner

61. 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

62. Analysis of Variance

63. 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

64. Analysis of Variance

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

66. End of Analyses of Variance Section