Fixed and Random Effects

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

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

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.

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.

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.

A and B Fixed Effects

A and B Fixed Effects

A and B Fixed Effects

A and B Random Effects

A and B Random Effects

A and B Random Effects

A Fixed and B Random

A Fixed and B Random

A Fixed and B Random

A, B, and C are Fixed

A, B, and C are Random

A, B, and C are Random

A, B, and C are Random

A, B, and C are Random

A Fixed, B and C are Random

A Fixed, B and C are Random

A Fixed, B and C are Random

A Fixed, B and C are Random

Pooling Sums of Squares

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)

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

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

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

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

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

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

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

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

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

Example Source df SS MS ‘F’ Rand Fixed Reps Treat Species Genotype 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 Pooled Error 240 3273 13.6 -

Example Source df SS MS ‘F’ Rand Fixed Reps Treat Species Genotype 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 Pooled Error 240 3273 13.6 -

Cross-Classified Genotypes Nitrogen 50 units Sunrise Helios Premier

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

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

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

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 -

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

Multiple Comparisons