1. Copyright, Gerry Quinn & Mick Keough, 1998 Please do not copy or distribute this file without the authors’ permission Experimental design and analysis Partly nested designs

2. Designs with 3 or more factors Factor A and C crossed Factor B nested within A, crossed with C

3. 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

4. Colonisation by stream insects Colonisation of stream insects to stones Effects of algal cover: –No algae, half algae, full algae 3 replicates for each algal treatment Design options: –completely randomised –randomised block

5. Completely randomised Rock with no algae Rock with half algae Rock with full algae

6. Randomised block Rock with no algae Rock with half algae Rock with full algae

7. Colonisation of stream insects Colonisation of stream insects to rocks Effects of algal cover –No algae, half algae, full algae 3 replicates for each algal treatment Effects of predation by fish –Caged vs cage controls 3 replicates for each predation

8. Completely randomised Uncaged Caged Rock with no algae Rock with half algae Rock with full algae

9. ANOVA Source of variationdf Caging1 Algae2 Caging x Algae (interaction) 2 Residual12 (stones within caging & algae) Total17

10. Split-plot design Factor A is caging: –fish excluded vs controls –applied to blocks = plots Factor B is plots nested within A Factor C is algal treatment –no algae, half algae, full algae –applied to stones = subplots within each plot

11. Split plot Uncaged Caged Rock with no algae Rock with half algae Rock with full algae

12. Advantages Uses randomised block (= plot) design for factor C (algal treatment): –better if blocks (plots) explain variation in DV More efficient: –only need cages over blocks (plots), not over individual stones

13. 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)

14. ANOVA Source of variationdf Between plots Caging1 Plots within caging (Residual 1)4 Within plots Algae2 Caging x Algae (interaction)2 Plots within caging x algae (Residual 2)8 Total17

15. ANOVA worked example Source of variationdfMSFP Between plots Caging11494.2217.810.013 Plots within caging483.89 Within plots Algae2247.3965.01<0.001 Caging x Algae223.726.230.023 Plots within caging x algae123.81 Total17

16. Westley (1993) Effects of infloresence bud removal on asexual investment in the Jeralusem artichoke: Populations1234 Genotypes within pops12345 TreatmentsCIR Genotypes = tubers from single individuals Treatments applied to different tubers from each genotype

17. Westley (1993) Source of variationdf Between plots (genotypes) Population3 Genotypes within population (Residual 1)16 Within plots (genotypes) Treatment1 Population x Treatment (interaction)3 Genotypes within Population x Treatment (Residual 2)16 Total39

18. Repeated measures designs Each whole plot is measured repeatedly under different treatments and/or times Within plots factor is often time, or at least treatments applied through time Plots termed “subjects” in repeated measures terminology Groups x trials designs –Groups are between subjects factor –Trials are within subjects factor

19. Cane toads and hypoxia How do cane toads respond to conditions of hypoxia? Two factors: –Breathing type buccal vs lung breathers –O 2 concentration 8 different [O 2 ] 10 replicates per breathing type and [O 2 ] combination

20. Completely randomised design 2 factor design (2 x 8) with 10 replicates –total number of toads = 160 Toads are expensive –reduce number of toads? Lots of variation between individual toads –reduce between toad variation?

21. Repeated measures design [O 2 ] BreathingToad12345678 type Lung1xxxxxxxx Lung2xxxxxxxx.............................. Lung9xxxxxxxx Buccal10xxxxxxxx Buccal12xxxxxxxx.............................. Buccal21xxxxxxxx

22. 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 within Breathing type x [O 2 ] (Residual 2)133 Total167

23. ANOVA toad example Source of variationdfMSFP Between subjects (toads) Breathing type139.925.760.027 Toads (breathing type)196.93 Within subjects (toads) [O 2 ]73.684.88<0.001 Breathing type x [O 2 ] 78.0510.69<0.001 Toads (Breathing type) x [O 2 ]1330.75 Total167

24. Partly nested ANOVA These are experimental designs where a factor is crossed with one factor but nested within another. A123etc. B(A)123456789 C123etc. Reps123n

25. ANOVA table The ANOVA looks like: Sourcedf A(p-1) B(A)p(q-1) C(r-1) A * C(p-1)(r-1) B(A) * Cp(q-1)(r-1) Residualpqr(n-1)

26. 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

27. Assumptions Normality of DV & 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

28. 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”

29. Sphericity of variances- covariances Sphericity of variance-covariance matrix –variances of paired differences between levels of within-plots (or subjects) factor must be same and consistent 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. –important if MS B(A) x C is used as error terms for tests of C and A x C

30. Sphericity (compound symmetry) More likely to be met for split-plot designs –within plot treatment levels randomly allocated to subplots More likely to be met for repeated measures designs –if order of within subjects treatments is randomised Unlikely to be met for repeated measures designs when within subjects factor is time –order of time cannot be randomised

31. 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 Adjusted univariate F tests for within-subjects factors and their interactions –conservative tests when sphericity is not met –Greenhouse-Geisser better than Huyhn-Feldt

32. ANOVA options Multivariate (MANOVA) tests for within subjects factors –treats responses from each subject as multiple DV’s in MANOVA –uses differences between successive responses –doesn’t require sphericity –sometimes more powerful than GG adjusted univariate, sometimes not –SYSTAT & SPSS automatically produce both

33. Toad example Within subjects (toads) Sourcedf F P GG-P [O 2 ]74.88<0.0010.004 Breathing type x [O 2 ] (interaction)710.69<0.001<0.001 Toads within Breathing type x [O 2 ]133 Greenhouse-Geisser Epsilon: 0.4282 Multivariate tests: Breathing type: PILLAI TRACE: df = 7,13, F = 14.277, p < 0.001 Breathing type x [O 2 ] PILLAI TRACE: df = 7,13, F = 3.853, p = 0.017

34. Kohout (1995) 12.....10 Between plates: 2 species= Trifolium alexandrinum = T. resupinatum 6 treatments= PIBT - sink = PIT - BAP = etc. 3 replicate plates per species/treatment combination Within plates: 10 bands sourcesource sinksink DV = % greening of nodules per band

35. Between plots Species1 Treatment5 Species x Treatment5 Plates within Species & Treatment (Residual 1)24 Within plots Band9 Band x Species9 Band x Treatment45 Band x Treatment x Species45 Plots within Species & Treatment x Band (Residual 2)216 TotalLots Source of variationdf

36. Parkinson (1996) Billabong typePermanentTemporaryWoodland Billabongsubjects Billabong typebetween subjects Month and Time of daywithin subjects Billabong123456789101112131415 MonthNovDecJanFeb Time of dayAMPM

37. Between subjects (bongs) Type2 Bongs within Type (Residual 1)12 Within subjects (bongs) Month3 Type x Month6 Month x Bongs within Type (Residual 2)36 Time1 Type x Time2 Time x Bongs within Type (Residual 3)12 Month x Time3 Type x Month x Time6 Month x Time x Bongs within Type (Residual 4)36 Source of variationdf