Fixed and Random Effects

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
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) 2 9 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 F Fixed
Reps Treat Species Genotype 2 2 3 9 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 1.96 ns
Pooled Error 240 3273 13.6 - -
37
Example
Source df SS MS F Rand F Fixed
Reps Treat Species Genotype 2 2 3 9 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 1.96 ns
Pooled Error 240 3273 13.6 - -
38
Cross-Classified
Genotypes Genotypes Genotypes
Nitrogen Genotypes Genotypes Genotypes
50 units Sunrise Helios Premier
75 units Sunrise Helios Premier
100 units Sunrise Helios Premier
125 units Sunrise Helios Premier
39
Cross-Classified
Genotypes Genotypes Genotypes
Species Genotypes Genotypes Genotypes
B. napus Sunrise Helios Premier
S. alba Sunrise Helios Premier
B. juncea Sunrise Helios Premier
B. rapa Sunrise Helios Premier
40
Nested
Genotypes Genotypes Genotypes
Species Genotypes Genotypes Genotypes
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 F Rand
Reps 2 31 15.7 1.15 ns 1.5 ns
Treat 2 490 244.9 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 1.96 ns
42
Example Nested Split-plot
Source df SS MS F Rand F Fixed
Reps 2 31 15.7 1.15 ns 1.5 ns
Treat 2 490 244.9 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 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 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 2 490 244.9 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