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Randomized block designs
Environmental sampling and analysis (Quinn & Keough, 2002)
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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, one
experimental unit from each treatment in each block.
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Null hypotheses
• No main effect of Factor A– H0: 1 = 2 = … = i = ... = – H0: 1 = 2 = … = i = ... = 0 (i = i - )
– no effect of shaving domatia, pooling blocks
• Factor A usually fixed
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Null hypotheses
• No effect of factor B (blocks):– no difference between blocks (leaf pairs),
pooling treatments
• Blocks usually random factor:– sample of blocks from populations of
blocks
– H0: 2 = 0
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• Factor A with p groups (p = 2 treatments for domatia)
• Factor B with q blocks (q = 14 pairs of leaves)
Source general example
Factor A p-1 1Factor B (blocks) q-1 13Residual (p-1)(q-1) 13Total pq-1 27
Randomised blocks ANOVA
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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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If factor A fixed and factor B (Blocks) random:
MSA 2 + 2 + n(i)2/p-1
MSBlocks 2 + n2
MSResidual 2 + 2
Expected mean squares
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Residual
• Cannot separately estimate 2 and 2:
– no replicates within each block-treatment combination
• MSResidual estimates 2 + 2
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Testing null hypotheses
• Factor A fixed and blocks random
• If H0 no effects of factor A is true:
– then F-ratio MSA / MSResidual 1
• If H0 no variance among blocks is true:
– no F-ratio for test unless no interaction assumed
– if blocks fixed, then F-ratio MSB / MSResidual 1
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Assumptions
• Normality of response variable– boxplots etc.
• No interaction between blocks and factor A, otherwise– MSResidual increase proportionally more than
MSA with reduced power of F-ratio test for A (treatments)
– interpretation of main 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
• Tukey’s test for non-additivity:– detect some forms of interaction
• Plot treatment values against block (“interaction plot”)
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Sphericity assumption
• Pattern of variances and covariances within and between “times”:– sphericity of variance-covariance matrix
• Equal variances of differences between all pairs of treatments : – variance of (T1 - T2)’s = variance of (T2 - T3)’s =
variance of (T1 - T3)’s etc.
• If assumption not met:– F-ratio test produces too many Type I errors
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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
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Dealing with non-sphericity
If not close to 1 and sphericity not met, there are 2 approaches:– Adjusted ANOVA F-tests
• df for F-ratio tests from ANOVA adjusted downwards (made more conservative) depending on value
– Multivariate ANOVA (MANOVA)• treatments considered as multiple
response variables 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 response variable 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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Partly nested ANOVA
Environmental sampling and analysis (Quinn & Keough, 2002)
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Partly nested ANOVA
• Designs with 3 or more factors
• Factor A and C crossed
• Factor B nested within A, crossed with C
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Partly nested ANOVAExperimental designs where a factor (B) is crossed with one factor (C) but nested within another (A).
A 1 2 3 etc.
B(A) 1 2 3 4 5 6 7 8 9
C 1 2 3 etc.
Reps 1 2 3 n
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ANOVA table
Source df Fixed or randomA (p-1) Either, usually fixedB(A) p(q-1) RandomC (r-1) Either, usually fixedA * C (p-1)(r-1) Usually fixedB(A) * C p(q-1)(r-1) Random
Residual pqr(n-1)
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Linear model
yijkl = + i + j(i) + k + ik + j(i)k + ijkl
grand mean (constant)i effect of factor Aj(i) effect of factor B nested w/i Ak effect of factor Cik interaction b/w A and Cj(i)k interaction b/w B(A) and Cijkl residual variation
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Expected mean squaresFactor A (p levels, fixed), factor B(A) (q levels, random), factor C (r levels, fixed)
Source df EMS TestA p-1
2 + nr2 + nqr
2 MSA/MSB(A)
B(A) p(q-1) 2 + nr
2 MSB/MSRES
C r-1 2 + n
2 + npq2 MSC/MSB(A)C
AC (p-1)(r-1)2 + n
2 + nq2MSAC/MSB(A)C
B(A) C p(q-1)(r-1) 2 + n
2 MSBC/MSRES
Residual pqr(n-1) 2
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Split-plot designs
• Units of replication different for different factors
• Factor A:– units of replication termed “plots”
• Factor B nested within A• Factor C:
– units of replication termed subplots within each plot
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Analysis of variance
• Between plots variation:– Factor A fixed - one factor ANOVA using plot
means– Factor B (plots) random - nested within A
(Residual 1)
• Within plots variation:– Factor C fixed– Interaction A * C fixed– Interaction B(A) * C (Residual 2)
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ANOVASource of variation dfBetween plotsSite 2Plots within site (Residual 1) 3
Within plotsTrampling 3Site x trampling (interaction) 6Plots within site x trampling (Residual 2) 9
Total 23
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Repeated measures designs
• Each whole plot measured repeatedly under different treatments and/or times
• Within plots factor often time, or at least treatments applied through time
• Plots termed “subjects” in repeated measures terminology
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Repeated measures designs
• Factor A:– units of replication termed “subjects”
• Factor B (subjects) nested within A
• Factor C:– repeated recordings on each subject
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Repeated measures design[O2]
Breathing Toad 1 2 3 4 5 6 7 8type
Lung 1 x x x x x x x xLung 2 x x x x x x x x... ... ... ... ... ... ... ... ... ...Lung 9 x x x x x x x x
Buccal 10 x x x x x x x xBuccal 12 x x x x x x x x... ... ... ... ... ... ... ... ... ...Buccal 21 x x x x x x x x
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ANOVASource of variation df
Between subjects (toads)Breathing type 1Toads within breathing type (Residual 1) 19
Within subjects (toads)[O2] 7Breathing type x [O2] 7Toads (Breathing type) x [O2](Residual 2) 133
Total 167
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Assumptions
• Normality & homogeneity of variance:– affects between-plots (between-subjects)
tests– boxplots, residual plots, variance vs mean
plots etc. for average of within-plot (within-subjects) levels
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• No “carryover” effects:– results on one subplot do not influence
results one another subplot.– time gap between successive repeated
measurements long enough to allow recovery of “subject”
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Sphericity
• Sphericity of variance-covariance matrix– variances of paired differences between
levels of within-plots (or subjects) factor equal within and between levels of between-plots (or subjects) factor
– variance of differences between [O2] 1 and [O2] 2 = variance of differences between [O2] 2 and [O2] 2 = variance of differences between [O2] 1 and [O2] 3 etc.
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Sphericity (compound symmetry)• OK for split-plot designs
– within plot treatment levels randomly allocated to subplots
• OK for repeated measures designs– if order of within subjects factor levels randomised
• Not OK for repeated measures designs when within subjects factor is time– order of time cannot be randomised
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ANOVA options
• Standard univariate partly nested analysis– only valid if sphericity assumption is met– OK for most split-plot designs and some
repeated measures designs
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ANOVA options
• Adjusted univariate F-tests for within-subjects factors and their interactions– conservative tests when sphericity is not
met– Greenhouse-Geisser better than Huyhn-
Feldt
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ANOVA options
• Multivariate (MANOVA) tests for within subjects or plots factors– responses from each subject used in
MANOVA– doesn’t require sphericity– sometimes more powerful than GG
adjusted univariate, sometimes not