A Bestiary of ANOVA tables

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A Bestiary of ANOVA tables
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Randomized Block
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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.

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Randomized Block

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

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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
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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
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Tribolium castaneum
Blocks (B) genotypes (A) genotypes (A) genotypes (A) 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 0.971 1.010 0.955 0.9787
0.9568 0.9845 0.9153
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ANOVA Table
Source of variation 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
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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)
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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

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Relative efficiency
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Nested analysis of variance
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Nested analysis of variance

• Data are organized hierarchically, with one class
  of objects nested within another

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Null hypothesis

• No effects of treatment
• No effects of B nested within A

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E
Enclosures
C
No enclosures
PC
Enclosures with openings
E
PC
E
C
E
PC
C
PC
C
E
Effects of Insect Pollination
PC
PC
C
C
E
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Data
Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i) Treatment (i)
1 1 1 1 1 2 2 2 2 2 3 3 3 3 3
Control Control Control Control Control Enclosures with openings Enclosures with openings Enclosures with openings Enclosures with openings Enclosures with openings Enclosures Enclosures Enclosures Enclosures Enclosures
replicate j 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
Subsample (k)
1 82 79 90 75 38 92 62 67 95 70 74 47 60 43 47
2 67 84 100 93 64 80 97 64 93 62 76 71 88 53 44
3 73 70 65 99 80 83 63 85 100 77 72 54 86 48 16
4 70 71 99 95 74 77 77 83 80 80 71 56 84 79 43
5 83 67 84 92 87 52 88 79 83 71 60 77 45 70 49
6 95 80 63 95 79 73 77 88 76 87 74 66 48 45 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 45 264 71 309 181 188 98 90 76 33 129 394 217 185
Mean (i) 79.8 78.7 60.0
Gl. Mean 72.8
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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
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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