#### Transcript Partly nested ANOVA

Experimental design and analysis Partly nested designs Copyright, Gerry Quinn & Mick Keough, 1998 Please do not copy or distribute this file without the authors’ permission 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: – units of replication termed “plots” – factor B nested within A • Factor C: – units of replication termed subplots within each plot 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 Completely randomised Rock with no algae Rock with half algae Rock with full algae Randomised block Rock with no algae Rock with half algae Rock with full algae 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 Completely randomised Uncaged Caged Rock with no algae Rock with half algae Rock with full algae ANOVA Source of variation df Caging Algae Caging x Algae (interaction) Residual 1 2 2 12 (stones within caging & algae) Total 17 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 Split plot Uncaged Caged Rock with no algae Rock with half algae Rock with full algae 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 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 variation Between plots Caging Plots within caging (Residual 1) Within plots Algae Caging x Algae (interaction) Plots within caging x algae (Residual 2) Total df 1 4 2 2 8 17 ANOVA worked example Source of variation Between plots Caging Plots within caging df MS F P 1 4 1494.22 83.89 17.81 0.013 65.01 6.23 <0.001 0.023 Within plots Algae Caging x Algae Plots within caging x algae 2 2 247.39 23.72 12 3.81 Total 17 Westley (1993) Effects of infloresence bud removal on asexual investment in the Jeralusem artichoke: Populations 1 Genotypes within pops Treatments 2 3 4 1 2 3 4 5 C IR Genotypes = tubers from single individuals Treatments applied to different tubers from each genotype Westley (1993) Source of variation df Between plots (genotypes) Population Genotypes within population (Residual 1) 3 16 Within plots (genotypes) Treatment Population x Treatment (interaction) Genotypes within Population x Treatment (Residual 2) 16 Total 39 1 3 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 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 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? Repeated measures design Breathing Toad type 1 2 [O2] 3 4 5 Lung Lung ... Lung 1 2 ... 9 x x ... x x x ... x x x ... x x x ... x x x ... x x x ... x x x ... x x x ... x Buccal Buccal ... Buccal 10 12 ... 21 x x ... x x x ... x x x ... x x x ... x x x ... x x x ... x x x ... x x x ... x 6 7 8 ANOVA Source of variation df Between subjects (toads) Breathing type Toads within breathing type (Residual 1) 1 19 Within subjects (toads) [O2] Breathing type x [O2] Toads within Breathing type x [O2] (Residual 2) 133 Total 167 7 7 ANOVA toad example Source of variation df Between subjects (toads) Breathing type Toads (breathing type) 1 19 Within subjects (toads) [O2] Breathing type x [O2] Toads (Breathing type) x [O2] 7 7 133 Total 167 MS 39.92 6.93 F P 5.76 0.027 3.68 4.88 <0.001 8.05 10.69 <0.001 0.75 Partly nested ANOVA These are experimental designs where a factor is crossed with one factor but nested within another. A B(A) C Reps 1 2 3 1 2 3 4 5 6 7 8 9 1 2 3 1 2 3 n etc. etc. ANOVA table The ANOVA looks like: Source A B(A) C A*C B(A) * C df (p-1) p(q-1) (r-1) (p-1)(r-1) p(q-1)(r-1) Residual pqr(n-1) Linear model yijkl = m + ai + bj(i) + dk + adik + bj(i)dk + eijkl m ai bj(i) dk adik bj(i)dk eijkl grand mean (constant) effect of factor A effect of factor B nested w/i A effect of factor C interaction b/w A and C interaction b/w B(A) and C residual variation 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 (withinsubjects) 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 of variancescovariances • 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 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 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 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 Toad example Within subjects (toads) Source df [O2] 7 Breathing type x [O2] (interaction) 7 Toads within Breathing type x [O2] 133 Greenhouse-Geisser Epsilon: F 4.88 10.69 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 P <0.001 <0.001 GG-P 0.004 <0.001 Kohout (1995) s o u r c e 1 2... s i n k ..10 DV = % greening of nodules per band 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 Source of variation df Between plots Species Treatment Species x Treatment Plates within Species & Treatment (Residual 1) 1 5 5 24 Within plots Band Band x Species Band x Treatment Band x Treatment x Species Plots within Species & Treatment x Band (Residual 2) 9 9 45 45 216 Total Lots Parkinson (1996) Billabong type Billabong Month Time of day Permanent Temporary Woodland 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Nov Dec Jan Feb AM Billabong Billabong type Month and Time of day PM subjects between subjects within subjects Source of variation df Between subjects (bongs) Type Bongs within Type (Residual 1) 2 12 Within subjects (bongs) Month Type x Month Month x Bongs within Type (Residual 2) Time Type x Time Time x Bongs within Type (Residual 3) Month x Time Type x Month x Time Month x Time x Bongs within Type (Residual 4) 3 6 36 1 2 12 3 6 36