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Experimental design and analysis

Experimental design and analysis. Partly nested designs. Partly nested designs. Designs with 3 or more factors Factor A and C crossed Factor B nested within A, crossed with C. Split-plot designs. Units of replication different for different factors Factor A:

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Experimental design and analysis

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  1. Experimental design and analysis Partly nested designs

  2. Partly nested designs • 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 variation df Caging 1 Algae 2 Caging x Algae (interaction) 2 Residual 12 (stones within caging & algae) Total 17

  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 variation df Between plots Caging 1 Plots within caging (Residual 1) 4 Within plots Algae 2 Caging x Algae (interaction) 2 Plots within caging x algae (Residual 2) 8 Total 17

  15. ANOVA worked example Source of variation df MS FP Between plots Caging 1 1494.22 17.81 0.013 Plots within caging 4 83.89 Within plots Algae 2 247.39 65.01 <0.001 Caging x Algae 2 23.72 6.23 0.023 Plots withincaging x algae 12 3.81 Total 17

  16. Westley (1993) Effects of infloresence bud removal on asexual investment in the Jeralusem artichoke: Populations 1 2 3 4 Genotypes within pops 1 2 3 4 5 Treatments C IR Genotypes = tubers from single individuals Treatments applied to different tubers from each genotype

  17. Westley (1993) Source of variation df Between plots (genotypes) Population 3 Genotypes within population (Residual 1) 16 Within plots (genotypes) Treatment 1 Population x Treatment (interaction) 3 Genotypes within Population x Treatment (Residual 2) 16 Total 39

  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 • O2 concentration • 8 different [O2] • 10 replicates per breathing type and [O2] 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 [O2] Breathing Toad 1 2 3 4 5 6 7 8 type Lung 1 x x x x x x x x Lung 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 x Buccal 12 x x x x x x x x ... ... ... ... ... ... ... ... ... ... Buccal 21 x x x x x x x x

  22. ANOVA Source of variation df Between subjects (toads) Breathing type 1 Toads within breathing type (Residual 1) 19 Within subjects (toads) [O2] 7 Breathing type x [O2] 7 Toads within Breathing type x [O2] (Residual 2) 133 Total 167

  23. ANOVA toad example Source of variation df MS FP Between subjects (toads) Breathing type 1 39.92 5.76 0.027 Toads (breathing type) 19 6.93 Within subjects (toads) [O2] 7 3.68 4.88 <0.001 Breathing type x [O2] 7 8.05 10.69 <0.001 Toads (Breathing type) x [O2] 133 0.75 Total 167

  24. Partly nested ANOVA These are experimental designs where a factor is crossed with one factor but nested within another. 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

  25. ANOVA table The ANOVA looks like: Source df A (p-1) B(A) p(q-1) C (r-1) A * C (p-1)(r-1) B(A) * C p(q-1)(r-1) Residual pqr(n-1)

  26. Linear model yijkl = m + ai + bj(i) + dk + adik + bj(i)dk + eijkl m grand mean (constant) ai effect of factor A bj(i) effect of factor B nested w/i A dk effect of factor C adik interaction b/w A and C bj(i)dk interaction b/w B(A) and C eijkl 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) Source df FPGG-P [O2] 7 4.88 <0.001 0.004 Breathing type x [O2] (interaction) 7 10.69 <0.001 <0.001 Toads within Breathing type x [O2] 133 Greenhouse-Geisser Epsilon: 0.4282 Multivariate tests: Breathing type: PILLAI TRACE: df = 7,13, F = 14.277, p < 0.001 Breathing type x [O2] PILLAI TRACE: df = 7,13, F = 3.853, p = 0.017

  34. Kohout (1995) 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 s o u r c e s i n k 1 2 ... ..10 DV = % greening of nodules per band

  35. Source of variation df Between plots Species 1 Treatment 5 Species x Treatment 5 Plates within Species & Treatment (Residual 1) 24 Within plots Band 9 Band x Species 9 Band x Treatment 45 Band x Treatment x Species 45 Plots within Species & Treatment x Band (Residual 2) 216 Total Lots

  36. Parkinson (1996) Billabong type Permanent Temporary Woodland Billabong 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Month Nov Dec Jan Feb Time of day AM PM Billabong subjects Billabong type between subjects Month and Time of day within subjects

  37. Source of variation df Between subjects (bongs) Type 2 Bongs within Type (Residual 1) 12 Within subjects (bongs) Month 3 Type x Month 6 Month x Bongs within Type (Residual 2) 36 Time 1 Type x Time 2 Time x Bongs within Type (Residual 3) 12 Month x Time 3 Type x Month x Time 6 Month x Time x Bongs within Type (Residual 4) 36

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