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
Yij    Ai  B j   ij
ANOVA table for randomized block design
Source
df
Among
groups
a-1
Sum of
squares
a
Mean
square
b
 (Y  Y )
2
i
( a  1)
i 1 j 1
Blocks
b-1
a
b
 (Y
j
Y )
i 1 j 1
Within
groups
(a-1)(b-1)
a
b
 (Y
ij
 Yi )
2
i 1 j 1
Total
ab-1
a
b
 (Y
ij
i 1 j 1
Y )
2
SSag
2
SSblocks
(b  1)
SSwg
( a  1)( b  1)
SStotal
( ab  1)
Expected
F ratio
mean square
 2  b A2
MS ag
MS wg
 2  a B2
MS blocks
[ 2  a B2  a R2 ] MS wg
2
[ 2  a 2 AB ]
 Y2
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
YB
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
YA
0.9568
0.9845
0.9153
ANOVA Table
Source of variation df
SS
MS
Fs
MSA
Genotype
2
0.010
0.005 6.97
MSB
Block
3
0.021
0.007 10.23 0.009
MSE(RB)
Error
6
0.004
0.001
P
0.03
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)
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
MS E (CR ) 
RE 
b(a  1) MS E ( RB)  (b  1) MS B
ab  1
100  MS E (CR )
MS E ( RB)
 351.6%
Nested analysis of variance
Nested analysis of variance
• Data are organized hierarchically, with one
class of objects nested within another
Yijk    Ai  B j (i )   ijk
Null hypothesis
• No effects of treatment
• No effects of B nested within A
E
Enclosures
C
No enclosures
PC
Enclosures with
openings
E
PC
E
C
E
C
PC
PC
C
E
PC
PC
C
C
E
Effects of
Insect
Pollination
Data
Treatment (i)
replicate j
1
2
3
Control
Enclosures with openings
Enclosures
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
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
Subsample
(k)
Mean (i)
Gl. Mean
79.8
78.7
72.8
60.0
Expected mean squares for test of null hypothesis for two
factor nested (A fixed, B random)
Source
df
Among
groups
a-1
Among
replicates
within groups
a(b-1)
Subsamples
within
replicates
ab(n-1)
Total
abn-1
Sum of squares
a
b
Mean
square
n

SSag
(Yi  Y ) 2
( a  1)
i 1 j 1 k 1
a
b
n

(Y j (i )  Yi )
2
i 1 j 1 k 1
a
b
n

(Yijk  Y j (i ) ) 2
i 1 j 1 k 1
a
b
n

i 1 j 1 k 1
(Yijk  Y ) 2
SSrep( gr )
a(b  1)
SS subsamples
Expected mean
square
F ratio
 2  bn A2  n B2 ( A) MS ag
MS r ( g )
 2  n B2 ( A)
MS r ( g )
MS subsam
ab( n  1)
2
SStotal
( abn  1)
 Y2
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