Unreplicated ANOVA designs

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Unreplicated ANOVA designs

• Block and repeated measures analyses

Ó Gerry Quinn Mick Keough, 1998 Do not copy or
distribute without permission of authors.
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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.

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Walter ODowd (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

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Completely randomised design
Control leaves
Shaved domatia leaves
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Completely randomised ANOVA
No. 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 1 Residual a(n-1) 26 Total an-
1 27
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Walter ODowd (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

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1 block
Control leaves
Shaved domatia leaves
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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) lt variation between
  leaves between blocks

9
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

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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

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Randomised blocks ANOVA
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 1 Blocks q-1 13 Residual (p-1)(q-1) 13 Total
pq-1 27
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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

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Expected mean squares

• If factor A is fixed and Blocks (B) are random
• MSA (Treatments) s2 sab2 n å(ai)2/a-1
• MSBlocks s2 nsb2
• MSResidual s2 sab2
• Cannot separately estimate s2 and sab2
• no replicates within each block-treatment
  combination.

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Null hypotheses

• If HO of no effects of factor A is true
• all ais 0 and all ms are the same
• then F-ratio MSA / MSResidual ? 1.
• If HO of no effects of factor A is false
• then F-ratio MSA / MSResidual gt 1.

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Walter ODowd (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 0.005 Block 13 1.77 0.64 0
.784 ?? Residual 13 2.77
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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

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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

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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

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Checks for interaction

• No real test because no within-cell variation
  measured
• Tukeys test for non-additivity
• detect some forms of interaction
• Plot treatment values against block (interaction
  plot)

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Interaction plots
DV
No interaction
Interaction
DV
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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

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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

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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

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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

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Randomised block Source df Treatments p-1 Blocks
q-1 Residual (p-1)(q-1) Total pq-1 Repeated
measures Source df Between subjects q-1 Within
subjects Treatments p-1 Residual (p-1)(q-1) Tota
l pq-1
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Driscoll Roberts (1997)
Source df MS F P
Between drainages 5 1046.28 Within drainages 12
443.33 Years 2 246.78 6.28 0.017 Residual 10 196
.56
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Computer set-up - randomised block
Treatment Block DV 1 1 y11 2 1 y21 3 1 y31 1
2 y12 2 2 y22 etc.
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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
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Sphericity assumption
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Block Treat 1 Treat 2 Treat 3 etc. 1 y11 y21 y31
2 y12 y22 y32 3 y13 y23 y33 etc.
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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.
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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

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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

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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

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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)

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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

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Examples from literature
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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

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Poorter et al. (1990)
3
1
2
2
4
5
3
4
1
5
Container 1
Container 2
Similarly for containers 3, 4, 5, 6 and 7
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Source df MS F P

• Genotype 4 0.125 3.81 0.016
• Block 6 0.118
• Residual 24 0.033
• Total 34
• Conclusions
• Large variation between containers ( blocks) so
  block design probably better than completely
  randomised design
• Significant difference in growth between genotypes

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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

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-

-
-

-

1 block (pair of mussel beds)

mussel bed with added mussels mussel bed without
added mussels
-
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Source df MS F P

• Blocks 3 62.82
• Treatment 1 5237.21 45.50 0.007
• Residual 3 115.09
• 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