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Unreplicated ANOVA designs Block and repeated measures analyses 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 & 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

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Completely randomised design Control leavesShaved 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) Sourcegeneral dfexample df Factor Aa-11 Residuala(n-1)26 Totalan-127

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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

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1 block Control leavesShaved 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) < variation between leaves between blocks

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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) –H O : 1 = 2 = 3 =... = –H O : 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

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No. of treatments or groups for factor A = p (2 for domatia), number of blocks = q (14 pairs of leaves) Sourcegeneralexample Factor Ap-11 Blocksq-113 Residual(p-1)(q-1)13 Totalpq-127 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 is fixed and Blocks (B) are random: MS A (Treatments) 2 + 2 + ( i ) 2 /a-1 MS Blocks 2 + n 2 MS Residual 2 + 2 Cannot separately estimate 2 and 2 : no replicates within each block-treatment combination. Expected mean squares

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Null hypotheses If H O of no effects of factor A is true: –all i ’s = 0 and all ’s are the same –then F-ratio MS A / MS Residual 1. If H O of no effects of factor A is false: –then F-ratio MS A / MS Residual > 1.

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Walter & O’Dowd (1992) Factor A (treatment - shaved and unshaved domatia) - fixed, Blocks (14 pairs of leaves) - random: SourcedfMSFP Treatment Block ?? Residual132.77

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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 MS Residual smaller in block design if blocks explain some of variation in DV: –increased power of test If decrease in MS Residual (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 –MS Residual will increase proportionally more than MS A 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 Tukey’s test for non-additivity: –detect some forms of interaction Plot treatment values against block (“interaction plot”)

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Interaction plots DV Block DV No interaction Interaction

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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 Sourcedf Treatmentsp-1 Blocksq-1 Residual(p-1)(q-1) Totalpq-1 Repeated measures Sourcedf Between “subjects”q-1 Within subjects Treatmentsp-1 Residual(p-1)(q-1) Totalpq-1

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Driscoll & Roberts (1997) SourcedfMSFP Between drainages Within drainages Years Residual

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TreatmentBlockDV 11y 11 21y 21 31y 31 12y 12 22y 22 etc. Computer set-up - randomised block

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Computer set-up - repeated measures SubjectTime 1Time 2Time 3etc. 1y 11 y 21 y 31 2y 12 y 22 y 32 3y 13 y 23 y 33 Both analyses produce identical results

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Sphericity assumption

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BlockTreat 1Treat 2Treat 3 etc. 1y 11 y 21 y 31 2y 12 y 22 y 32 3y 13 y 23 y 33 etc.

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BlockT1 - T2T2 - T3T1 - T3etc. 1y 11 -y 21 y 21 -y 31 y 11 -y 31 2y 12 -y 22 y 22 -y 32 y 12 -y 32 3y 13 -y 23 y 23 -y 33 y 13 -y 33 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 ( ) 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-tests from ANOVA adjusted downwards (made more conservative) depending on value –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) Container 1 Container 2 Similarly for containers 3, 4, 5, 6 and 7

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SourcedfMSFP Genotype Block Residual Total34 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 ( cm 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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mussel bed with added mussels mussel bed without added mussels block (pair of mussel beds) + -

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SourcedfMSFP Blocks Treatment Residual 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

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