1. A Bestiary of ANOVA tables

2. Randomized Block

3. Null hypotheses No effects of treatment
No effects of block B. However this hypothesis is usually not relevant because we are not interested in the differences among blocks per se. Formally, you also need to assume that the interaction is not present and you should consider the added variance due to restricted error.

4. Randomized Block
Each set of treatments is physically grouped in a block, with each treatment represented exactly once in each block

5. ANOVA table for randomized block design
Source df
Sum of squares
Mean square
Expected mean square
F ratio
Among groups
a-1
Blocks
b-1
Within groups
(a-1)(b-1)
Total
ab-1
NOTE: the Expected Mean Square terms in brackets are assumed to be absent for the randomized block design

6. Tribolium castaneum
Mean dry weights (in milligrams) of 3 genotypes of beetles, reared at a density of 20 beetles per gram of flour. Four series of experiments represent blocks

7. Tribolium castaneum Blocks (B) genotypes (A) ++ +b bb 1 0.958 0.986
0.925
0.9563 2
0.971
1.051
0.952
0.9913 3
0.927
0.891
0.829
0.8823 4
1.010
0.955
0.9787
0.9568
0.9845
0.9153

8. ANOVA Table Source of variation df SS MS Fs P MSA Genotype 2 0.010
0.005
6.97
0.03
MSB
Block 3
0.021
0.007
10.23
0.009
MSE(RB)
Error 6
0.004
0.001

9. Relative efficiency
To compare two designs we compute the relative efficiency. This is a ratio of the variances resulting from the two designs
It is an estimate of the sensitivity of the original design to the one is compared
However other aspects should be considered as the relative costs of the two designs
(Sokal and Rohlf 2000)

10. In the expression in the following slide
Had we ignored differences among series and simply analyzed these data as four replicates for each genotype, what our variance would have been for a completely randomized design?
In the expression in the following slide
MSE(CR) = expected error mean square in the completely randomized design
MSE(RB) = observed error mean square in the randomized block design
MSB is the observed mean square among blocks

11. Relative efficiency

12. Nested analysis of variance

13. Nested analysis of variance
Data are organized hierarchically, with one class of objects nested within another

14. Null hypothesis No effects of treatment
No effects of B nested within A

15. Effects of Insect Pollination
Enclosures C
No enclosures PC
Enclosures with openings E PC C E E C PC PC C E
Effects of Insect Pollination PC PC C C E

16. Enclosures with openings
Data
Treatment (i) 1 2 3
Control
Enclosures with openings
Enclosures
replicate j 4 5
Subsample (k) 82 79 90 75 38 92 62 67 95 70 74 47 60 43 84
100 93 64 80 97 76 71 88 53 44 73 65 99 83 63 85 77 72 54 86 48 16 56 87 52 45 49 6 66 55
Mean (j,i)
78.3
75.2
83.5
91.5
70.3
76.2
77.3
77.7
87.8
74.5
71.2
61.8
68.5
56.3
42.3
Variance
108
264
309
181
188 98 33
129
394
217
185
Mean (i)
79.8
78.7
60.0
Gl. Mean
72.8

17. Expected mean squares for test of null hypothesis for two factor nested (A fixed, B random)
Source df
Sum of squares
Mean square
Expected mean square
F ratio
Among groups
a-1
Among replicates within groups
a(b-1)
Subsamples within replicates
ab(n-1)
Total
abn-1

18. Source df
Sum of squares
Mean square
F ratio P
Among groups 2
3694.9
8.210
0.006
Among replicates within groups 12
450.04
2.824
0.003
Subsamples within replicates 75
159.3
Total
Source df
Sum of squares
Mean square
F ratio P
Among groups 2
615.82
8.210
0.006
Error 12
900.08
75.01