Unreplicated ANOVA designs
Block and repeated measures
analyses
 Gerry Quinn & Mick Keough, 1998
Do not copy or distribute without
permission of authors.
Blocking
• Aim:
– Reduce unexplained variation, without
increasing size of experiment.
• Approach:
– Group experimental units (“replicates”) into
blocks.
– Blocks usually spatial units, 1 experimental
unit from each treatment in each block.
Walter & O’Dowd (1992)
• Effects of domatia (cavities of leaves) on
number of mites - use a single shrub in field
• Two treatments
– shaving domatia which removes domatia from
leaves
– normal domatia as control
• Required 14 leaves for each treament
• Set up as completely randomised design
– 28 leaves randomly allocated to each of 2
treatments
Completely randomised design
Control leaves
Shaved domatia leaves
Completely randomised ANOVA
No. of treatments or groups for factor A = a (2 for
domatia), number of replicates = n (14 pairs of
leaves)
Source
Factor A
Residual
Total
general df
example df
a-1
a(n-1)
1
26
an-1
27
Walter & O’Dowd (1992)
• Effects of domatia (cavities of leaves) on
number of mites - use a single shrub in field
• Two treatments
– shaving domatia which removes domatia from
leaves
– normal domatia as control
• Required 14 leaves for each treament
• Set up as blocked design
– paired leaves (14 pairs) chosen - 1 leaf in each
pair shaved, 1 leaf in each pair control
1 block
Control leaves
Shaved domatia leaves
Rationale for blocking
• Micro-temperature, humidity, leaf age,
etc. more similar within block than
between blocks
• Variation in DV (mite number) between
leaves within block (leaf pair) < variation
between leaves between blocks
Rationale for blocking
• Some of unexplained (residual)
variation in DV from completely
randomised design now explained by
differences between blocks
• More precise estimate of treatment
effects than if leaves were chosen
completely randomly from shrub
Null hypotheses
• No effect of treatment (Factor A)
– HO: m1 = m2 = m3 = ... = m
– HO: a1 = a2 = a3 = ... = 0 (ai = mi - m)
– no effect of shaving domatia, pooling
blocks
• No effect of blocks (?)
– no difference between blocks (leaf pairs),
pooling treatments
Randomised blocks ANOVA
No. of treatments or groups for factor A = p (2 for
domatia), number of blocks = q (14 pairs of
leaves)
Source
Factor A
Blocks
Residual
Total
general
example
p-1
q-1
(p-1)(q-1)
pq-1
1
13
13
27
Randomised block ANOVA
• Randomised block ANOVA is 2 factor
factorial design
– BUT no replicates within each cell
(treatment-block combination), i.e.
unreplicated 2 factor design
– No measure of within-cell variation
– No test for treatment by block interaction
Expected mean squares
If factor A is fixed and Blocks (B) are random:
MSA (Treatments)
s2 + sab2 + n (ai)2/a-1
MSBlocks
s2 + nsb2
MSResidual
s2 + sab2
Cannot separately estimate s2 and sab2:
• no replicates within each block-treatment
combination.
Null hypotheses
• If HO of no effects of factor A is true:
– all ai’s = 0 and all m’s are the same
– then F-ratio MSA / MSResidual  1.
• If HO of no effects of factor A is false:
– then F-ratio MSA / MSResidual > 1.
Walter & O’Dowd (1992)
Factor A (treatment - shaved and unshaved
domatia) - fixed, Blocks (14 pairs of leaves) random:
Source
Treatment
Block
Residual
df
1
13
13
MS
F
31.34
1.77
2.77
11.32
0.64
P
0.005
0.784 ??
Randomised block vs completely
randomised designs
• Total number of replicates is same in both
designs
– 28 leaves in total for domatia experiment
• Block designs rearrange spatial pattern of
replicates into blocks:
– “replicates” in block designs are the blocks
• Test of factor A (treatments) has fewer df in
block design:
– reduced power of test
Randomised block vs completely
randomised designs
• MSResidual smaller in block design if
blocks explain some of variation in DV:
– increased power of test
• If decrease in MSResidual (unexplained
variation) outweighs loss of df, then
block design is better:
– when blocks explain a lot of variation in DV
Assumptions
• Normality of DV
– boxplots etc.
• No interaction between blocks and
treatments, otherwise
– MSResidual will increase proportionally more
than MSA with reduced power of F-test for
A (treatments)
– interpretation of treatment effects may be
difficult, just like replicated factorial ANOVA
Checks for interaction
• No real test because no within-cell
variation measured
• Tukey’s test for non-additivity:
– detect some forms of interaction
• Plot treatment values against block
(“interaction plot”)
Interaction plots
DV
No interaction
DV
Interaction
Block
Repeated measures designs
• A common experimental design in
biology (and psychology)
• Different treatments applied to whole
experimental units (called “subjects”)
or
• Experimental units recorded through
time
Repeated measures designs
• The effect of four experimental drugs on
heart rate of rats:
– five rats used
– each rat receives all four drugs in random
order
• Time as treatment factor is most
common use of repeated measures
designs in biology
Driscoll & Roberts (1997)
• Effect of fuel-reduction burning on frogs
• Six drainages:
– blocks or subjects
• Three treatments (times):
– pre-burn, post-burn 1, post-burn 2
• DV:
– difference between no. calling males on
paired burnt-unburnt sites at each drainage
Repeated measures cf.
randomised block
• Simple repeated measures designs are
analysed as unreplicated two factor
ANOVAs
• Like randomised block designs
– experimental units or “subjects” are blocks
– treatments comprise factor A
Randomised block
Source
Treatments
Blocks
Residual
Total
Repeated measures
Source
Between “subjects”
Within subjects
Treatments
Residual
Total
df
p-1
q-1
(p-1)(q-1)
pq-1
df
q-1
p-1
(p-1)(q-1)
pq-1
Driscoll & Roberts (1997)
Source
df
MS
Between
drainages
5
1046.28
12
2
10
443.33
246.78
196.56
Within
drainages
Years
Residual
F
6.28
P
0.017
Computer set-up - randomised
block
Treatment
1
2
3
1
2
etc.
Block
1
1
1
2
2
DV
y11
y21
y31
y12
y22
Computer set-up - repeated
measures
Subject Time 1 Time 2 Time 3
1
y11
y21
y31
2
y12
y22
y32
3
y13
y23
y33
etc.
Both analyses produce identical results
Sphericity assumption
Block
Treat 1
Treat 2
1
2
3
etc.
y11
y12
y13
y21
y22
y23
Treat 3 etc.
y31
y32
y33
Block
T1 - T2
T2 - T3
T1 - T3 etc.
1
2
3
etc.
y11-y21
y12-y22
y13-y23
y21-y31
y22-y32
y23-y33
y11-y31
y12-y32
y13-y33
Sphericity assumption
• Pattern of variances and covariances
within and between “times”:
– sphericity of variance-covariance matrix
• Variances of differences between all
pairs of treatments are equal:
– variance of (T1 - T2)’s = variance of (T2 T3)’s = variance of (T1 - T3)’s etc.
• If assumption not met:
– F-test produces too many Type I errors
Sphericity assumption
• Applies to randomised block and
repeated measures designs
• Epsilon (e) statistic indicates degree to
which sphericity is not met
– further e is from 1, more variances of
treatment differences are different
• Two versions of e
– Greenhouse-Geisser e
– Huyhn-Feldt e
Dealing with non-sphericity
If e not close to 1 and sphericity not met,
there are 2 approaches:
– Adjusted ANOVA F-tests
• df for F-tests from ANOVA adjusted
downwards (made more conservative)
depending on value e
– Multivariate ANOVA (MANOVA)
• treatments considered as multiple DVs
in MANOVA
Sphericity assumption
• Assumption of sphericity probably OK
for randomised block designs:
– treatments randomly applied to
experimental units within blocks
• Assumption of sphericity probably also
OK for repeated measures designs:
– if order each “subject” receives each
treatment is randomised (eg. rats and
drugs)
Sphericity assumption
• Assumption of sphericity probably not
OK for repeated measures designs
involving time:
– because DV for times closer together more
correlated than for times further apart
– sphericity unlikely to be met
– use Greenhouse-Geisser adjusted tests or
MANOVA
Examples from literature
Poorter et al. (1990)
• Growth of five genotypes (3 fast and 2
slow) of Plantago major (a dicot plant
called ribwort)
• One replicate seedling of each
genotype was placed in each of 7
plastic containers in growth chamber
• Genotypes (1, 2, 3, 4, 5) are treatments,
containers are blocks, DV is total plant
weight (g) after 12 days
Poorter et al. (1990)
3
4
1
2
5
Container 1
1
2
5
4
Container 2
Similarly for containers 3, 4, 5, 6 and 7
3
Source
Genotype
Block
Residual
Total
df
4
6
24
34
MS
F
0.125
0.118
0.033
3.81
P
0.016
Conclusions:
• Large variation between containers (= blocks) so
block design probably better than completely
randomised design
• Significant difference in growth between
genotypes
Robles et al. (1995)
• Effect of increased mussel (Mytilus spp.)
recruitment on seastar numbers
• Two treatments: 30-40L of Mytilus (0.53.5cm long) added, no Mytilus added
• Four matched pairs of mussel beds chosen,
each pair = block
• Treatments randomly assigned to mussel
beds within a pair
• DV is % change in seastar numbers
+
+
+
-
-
-
+
1 block (pair of mussel beds)
+
-
mussel bed with added mussels
mussel bed without added mussels
Source
df
Blocks
Treatment
Residual
3
1
3
MS
62.82
5237.21
115.09
F
45.50
P
0.007
Conclusions:
• Relatively little variation between blocks so a
completely randomised design probably better
because treatments would have 1,6 df
• Significant treatment effect - more seastars
where mussels added