1. Fixed and Random Effects

2. 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

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

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

5. 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 # reps times the product of factors levels that do not appear in the factor name.

6. Expected Mean Squares Which interactions to include in an EMS?
All the factors appear in the interaction.
All the other factors in the interaction are Random Effects.

7. A and B Fixed Effects

8. A and B Fixed Effects

9. A and B Fixed Effects

10. A and B Random Effects

11. A and B Random Effects

12. A and B Random Effects

13. A Fixed and B Random

14. A Fixed and B Random

15. A Fixed and B Random

16. A, B, and C are Fixed

17. A, B, and C are Random

18. A, B, and C are Random

19. A, B, and C are Random

20. A, B, and C are Random

21. A Fixed, B and C are Random

22. A Fixed, B and C are Random

23. A Fixed, B and C are Random

24. A Fixed, B and C are Random

25. Pooling Sums of Squares

26. Example Ten yellow mustard lines.
Five different nitrogen levels (50, 75, 100, 125, 150 units of N).
Three replicates.
Raw data presented in Table 7 (Page 110 & 111)

27. Example Source df SS MS ‘F’ Reps Genotypes Nitrogen G x N 2 9 4 32
12875
14881
24705
32809
6438
1666
6176
911
8.94 **
1.83 ns
6.78 **
1.26 ns
G x R
N x R
G x N x R 18 8 72
7180
7612
55819
399
951
775
0.51 ns
1.23 ns
Total
149
155992
1047

28. Example Source df SS MS ‘F’ Reps Genotypes Nitrogen G x N 2 9 4 32
12875
14881
24705
32809
6438
1666
6176
911
8.94 **
1.83 ns
6.78 **
1.26 ns
G x R
N x R
G x N x R 18 8 72
7180
7612
55819
399
951
775
0.51 ns
1.23 ns
Pooled Error 98
70610
720
Total
149
155992
1047

29. Example ~ Genotype Main-plot
Source df SS MS
‘F’
Reps
Genotypes
Nitrogen
G x N 2 9 4 32
12875
14881
24705
32809
6438
1666
6176
911
8.94 **
1.83 ns
6.78 **
1.26 ns
G x R
N x R
G x N x R 18 8 72
7180
7612
55819
399
951
775
0.51 ns
1.23 ns
Total
149
155992
1047

30. Example ~ Genotype Main-plot
Source df SS MS
‘F’
Reps
Genotypes
Nitrogen
G x N 2 9 4 32
12875
14881
24705
32809
6438
1666
6176
911
8.94 **
1.83 ns
6.78 **
1.26 ns
G x R
N x R
G x N x R 18 8 72
7180
7612
55819
399
951
775
0.51 ns
1.23 ns
Total
149
155992
1047

31. Example ~ Genotypes Main-plot
Source df SS MS
‘F’
Reps
Genotypes
Error (1) 2 9 18
12875
14881
7180
6438
1666
399
16.26 ***
4.18 ** -
Nitrogen
G x N
Error (2) 4 32 80
24705
32809
63430
6176
911
793
7.78 **
1.15 ns
Total
149
155992
1047

32. Example ~ Nitrogen Main-plot
Source df SS MS
‘F’
Reps
Nitrogen
Error (1) 2 4 8
12875
24705
7612
6438
6176
951
8.94 **
6.78 ** -
Genotypes
G x N
Error (2) 9 32 90
14881
32809
70611
1666
911
720
2.31 ns
1.26 ns
Total
149
155992
1047

33. Example ~ Strip-plot Source df SS MS ‘F’ Reps Genotypes Error(Geno) 29 18
12875
14881
7180
6438
1666
399
8.94***
4.18 ** -
Nitrogen
Error(Nitro) 4 8
24705
7612
6176
951
6.49 ***
G x N
Error (GxN) 32 72
32809
55819
911
775
1.18 ns
Total
149
155992
1047

34. Another Example Four species of Brassica.
Ten lines within each species.
Three insecticide treatments (Thiodan, Furidan, none).
Three replicates.
Raw data in Table 8.

35. Example Source df SS MS Species x Rep 6 75 12.4 Genotype x Rep 18 217
12.0
Treat x Rep 4 50
12.5
S x G x R 54
698
12.9
S x T x R 12
123
10.3
G x T x R 36 53
14.8
G x T x S x R
108
1577
14.6

36. Example Source df SS MS ‘F’ Rand Fixed Reps Treat Species Genotype 2 39 31
490
1046
225
15.7
244.9
348.7
25.0
1.5 ns
17.95***
25.47 ***
1.58 ns
Spec x Geno
Spec x Treat
Geno x Treat 27 6 18
1489
587
189
55.1
97.9
10.5
2.06 *
3.65 *
0.39 ns
4.05 ***
7.20 ***
0.77 ns
S x G x T 54
1445
26.8
1.96 ns
Pooled Error
240
3273
13.6 -

37. Example Source df SS MS ‘F’ Rand Fixed Reps Treat Species Genotype 2 39 31
490
1046
225
15.7
244.9
348.7
25.0
1.15 ns
2.50 ns
3.56 ns
0.45 ns
1.5 ns
17.95***
25.47 ***
1.58 ns
Spec x Geno
Spec x Treat
Geno x Treat 27 6 18
1489
587
189
55.1
97.9
10.5
2.06 *
3.65 *
0.39 ns
4.05 ***
7.20 ***
0.77 ns
S x G x T 54
1445
26.8
1.96 ns
Pooled Error
240
3273
13.6 -

38. Cross-Classified Genotypes Nitrogen 50 units Sunrise Helios Premier

39. Cross-Classified Genotypes Species B. napus Sunrise Helios Premier
S. alba
B. juncea
B. rapa

40. Nested Genotypes Species B. napus Sunrise Helios Premier S. alba
IdaGold
Goldfield
Tilney
B. juncea
P. Gold
Kodiak
Cutlass
B. rapa
Raven
Reston
Goldrush

41. Example ~ Nesting Source df SS MS ‘F’ Fix Rand Reps 2 31 15.7 1.15 ns
Treat
490
244.9
2.50 ns
17.95***
Species 3
1046
348.7
3.56
25.47 ***
Genotype 9
225
25.0
0.45 ns
1.58 ns
Spec x Geno 27
1489
55.1
2.06 *
4.05 ***
Spec x Treat 6
587
97.9
3.65 *
7.20 ***
Geno x Treat 18
189
10.5
0.39 ns
0.77 ns
S x G x T 54
1445
26.8
1.96 ns

42. Example ~ Nested Split-plot
Source df SS MS
‘F’
Rand
Fixed
Reps 2 31
15.7
1.15 ns
1.5 ns
Treat
490
244.9
2.50 ns
17.95***
Species 3
1046
348.7
3.56 ns
25.47 ***
Geno w Spec 36
1940
53.9
1.58 ns
Spec x Treat 6
587
97.9
7.17 ***
4.31 ***
Geno w S x T 72
16334
22.7
1.66 ns
Pooled Error
240
3273
13.6 -

43. Example ~ Nest Split-plot
Source df SS MS
‘F’
Reps 2 31
15.7
1.25 ns
Treat
490
244.9
19.57***
Error (1) 4 50
12.5 -
Species 3
1046
348.7
3.36 *
Geno w Spec 36
1940
53.9
2.37 *
Spec x Treat 6
587
97.9
7.17 ***
Geno w S x T 72
16334
22.7
1.66 ns
Error (2)
236
3223
13.6

44. Multiple Comparisons