Randomized block designs  Environmental sampling and analysis (Quinn & Keough, 2002)

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.

Null hypotheses No main effect of Factor A –H 0 :  1 =  2 = … =  i =... =  –H 0 :  1 =  2 = … =  i =... = 0 (  i =  i -  ) –no effect of shaving domatia, pooling blocks Factor A usually fixed

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 –H 0 :   2 = 0

Factor A with p groups (p = 2 treatments for domatia) Factor B with q blocks (q = 14 pairs of leaves) Sourcegeneral example Factor Ap-11 Factor B (blocks)q-113 Residual(p-1)(q-1)13 Totalpq-127 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 fixed and factor B (Blocks) random: MS A  2 +   2 + n  (  i ) 2 /p-1 MS Blocks  2 + n   2 MS Residual  2 +   2 Expected mean squares

Residual Cannot separately estimate  2 and   2 : –no replicates within each block-treatment combination MS Residual estimates  2 +   2

Testing null hypotheses Factor A fixed and blocks random If H 0 no effects of factor A is true: –then F-ratio MS A / MS Residual  1 If H 0 no variance among blocks is true: –no F-ratio for test unless no interaction assumed –if blocks fixed, then F-ratio MS B / MS Residual  1

Assumptions Normality of response variable –boxplots etc. No interaction between blocks and factor A, otherwise –MS Residual increase proportionally more than MS A with reduced power of F-ratio test for A (treatments) –interpretation of main 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”)

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

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-ratio tests from ANOVA adjusted downwards (made more conservative) depending on value  –Multivariate ANOVA (MANOVA) treatments considered as multiple response variables 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 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

Partly nested ANOVA  Environmental sampling and analysis (Quinn & Keough, 2002)

Partly nested ANOVA Designs with 3 or more factors Factor A and C crossed Factor B nested within A, crossed with C

Partly nested ANOVA Experimental designs where a factor (B) is crossed with one factor (C) but nested within another (A). A123etc. B(A) C123etc. Reps123n

ANOVA table SourcedfFixed or random A(p-1)Either, usually fixed B(A)p(q-1)Random C(r-1)Either, usually fixed A * C(p-1)(r-1)Usually fixed B(A) * Cp(q-1)(r-1)Random Residualpqr(n-1)

Linear model y ijkl =  +  i +  j(i) +  k +  ik +  j(i)k +  ijkl  grand mean (constant)  i effect of factor A  j(i) effect of factor B nested w/i A  k effect of factor C  ik interaction b/w A and C  j(i)k interaction b/w B(A) and C  ijkl residual variation

Expected mean squares Factor A (p levels, fixed), factor B(A) (q levels, random), factor C (r levels, fixed) Source dfEMS Test A p-1   2 + nr   2 + nqr   2 MS A /MS B(A) B(A) p(q-1)   2 + nr   2 MS B /MS RES C r-1   2 + n   2 + npq   2 MS C /MS B(A)C AC (p-1)(r-1)   2 + n   2 + nq   2 MS AC /MS B(A)C B(A) C p(q-1)(r-1)   2 + n   2 MS BC /MS RES Residualpqr(n-1)   2

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

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)

ANOVA Source of variationdf Between plots Site2 Plots within site (Residual 1)3 Within plots Trampling3 Site x trampling (interaction)6 Plots within site x trampling (Residual 2)9 Total23

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

Repeated measures designs Factor A: –units of replication termed “subjects” Factor B (subjects) nested within A Factor C: –repeated recordings on each subject

Repeated measures design [O 2 ] BreathingToad type Lung1xxxxxxxx Lung2xxxxxxxx Lung9xxxxxxxx Buccal10xxxxxxxx Buccal12xxxxxxxx Buccal21xxxxxxxx

ANOVA Source of variationdf Between subjects (toads) Breathing type1 Toads within breathing type (Residual 1)19 Within subjects (toads) [O 2 ]7 Breathing type x [O 2 ]7 Toads (Breathing type) x [O 2 ] (Residual 2)133 Total167

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

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”

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 [O 2 ] 1 and [O 2 ] 2 = variance of differences between [O 2 ] 2 and [O 2 ] 2 = variance of differences between [O 2 ] 1 and [O 2 ] 3 etc.

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

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

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

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