A Bestiary of ANOVA tables

Randomized Block

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.

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

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

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

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

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

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)

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

Relative efficiency

Nested analysis of variance

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

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

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

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

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

Source df Sum of squares Mean square F ratio P Among groups 2 7389.87 3694.9 8.210 0.006 Among replicates within groups 12 5400.47 450.04 2.824 0.003 Subsamples within replicates 75 11950.17 159.3 Total 24740.5 Source df Sum of squares Mean square F ratio P Among groups 2 1231.64 615.82 8.210 0.006 Error 12 900.08 75.01