Unreplicated ANOVA designs
Block and repeated measures analyses
Gerry Quinn & Mick Keough, 1998Do 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 ANOVANo. of treatments or groups for factor A = a (2 for domatia), number of replicates = n (14 pairs of leaves)
Source general df example df
Factor A a-1 1Residual a(n-1) 26
Total 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
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: 1 = 2 = 3 = ... =
– HO: 1 = 2 = 3 = ... = 0 (i = i - )
– no effect of shaving domatia, pooling blocks
• No effect of blocks (?)– no difference between blocks (leaf pairs),
pooling treatments
No. of treatments or groups for factor A = p (2 for domatia), number of blocks = q (14 pairs of leaves)
Source general example
Factor A p-1 1Blocks q-1 13Residual (p-1)(q-1) 13Total pq-1 27
Randomised blocks ANOVA
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
If factor A is fixed and Blocks (B) are random:
MSA (Treatments) 2 + 2 + (i)2/a-1
MSBlocks 2 + n2
MSResidual 2 + 2
Cannot separately estimate 2 and 2:
• no replicates within each block-treatment combination.
Expected mean squares
Null hypotheses
• If HO of no effects of factor A is true:
– all i’s = 0 and all ’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 df MS F P
Treatment 1 31.34 11.32 0.005Block 13 1.77 0.64 0.784 ??Residual 13 2.77
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”)
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 blockSource dfTreatments p-1Blocks q-1Residual (p-1)(q-1)Total pq-1
Repeated measuresSource dfBetween “subjects” q-1Within subjects
Treatments p-1Residual (p-1)(q-1)
Total pq-1
Driscoll & Roberts (1997)
Source df MS F P
Betweendrainages 5 1046.28
Withindrainages 12 443.33
Years 2 246.78 6.28 0.017Residual 10 196.56
Computer set-up - repeated measures
Subject Time 1 Time 2 Time 3 etc.1 y11 y21 y31
2 y12 y22 y32
3 y13 y23 y33
Both analyses produce identical results
Block T1 - T2 T2 - T3 T1 - T3 etc.
1 y11-y21 y21-y31 y11-y31
2 y12-y22 y22-y32 y12-y32
3 y13-y23 y23-y33 y13-y33
etc.
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 () statistic indicates degree to which sphericity is not met– further is from 1, more variances of treatment
differences are different
• Two versions of – Greenhouse-Geisser – Huyhn-Feldt
Dealing with non-sphericity
If 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
– 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
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)
1
234
5
12
345
Container 1 Container 2
Similarly for containers 3, 4, 5, 6 and 7
Source df MS F P
Genotype 4 0.125 3.81 0.016Block 6 0.118Residual 24 0.033Total 34
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 (0.5-3.5cm 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
mussel bed with added mussels
mussel bed without added mussels
+ -
- +
- ++
-
1 block (pair of mussel beds)
+
-
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