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

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 1Algae 2Caging x Algae (interaction) 2Residual 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)

ANOVASource of variation dfBetween plotsCaging 1Plots within caging (Residual 1) 4

Within plotsAlgae 2Caging x Algae (interaction) 2Plots within caging x algae (Residual 2) 8

Total 17

ANOVA worked exampleSource of variation df MS F PBetween plotsCaging 1 1494.22 17.81 0.013Plots within caging 4 83.89

Within plotsAlgae 2 247.39 65.01 <0.001Caging x Algae 2 23.72 6.23 0.023Plots withincaging x algae 12 3.81

Total 17

Westley (1993)

Effects of infloresence bud removal on asexual investment in the Jeralusem artichoke:

Populations 1 2 3 4

Genotypeswithin pops 1 2 3 4 5

Treatments C IR

Genotypes = tubers from single individualsTreatments applied to different tubers from each genotype

Westley (1993)Source of variation df

Between plots (genotypes)Population 3Genotypes within population (Residual 1) 16

Within plots (genotypes)Treatment 1Population x Treatment (interaction) 3Genotypes within Population x Treatment(Residual 2) 16

Total 39

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[O2]

Breathing Toad 1 2 3 4 5 6 7 8type

Lung 1 x x x x x x x xLung 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 xBuccal 12 x x x x x x x x... ... ... ... ... ... ... ... ... ...Buccal 21 x x x x x x x x

ANOVASource of variation df

Between subjects (toads)Breathing type 1Toads within breathing type (Residual 1) 19

Within subjects (toads)[O2] 7Breathing type x [O2] 7Toads within Breathing type x [O2](Residual 2) 133

Total 167

ANOVA toad exampleSource of variation df MS F P

Between subjects (toads)Breathing type 1 39.92 5.76 0.027Toads (breathing type) 19 6.93

Within subjects (toads)[O2] 7 3.68 4.88 <0.001Breathing type x [O2] 7 8.05 10.69 <0.001Toads (Breathing type) x [O2] 133 0.75

Total 167

Partly nested ANOVAThese 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

ANOVA table

The ANOVA looks like:

Source dfA (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)

Linear model

yijkl = + i + j(i) + k + ik + j(i)k + ijkl

grand mean (constant)i effect of factor Aj(i) effect of factor B nested w/i Ak effect of factor Cik interaction b/w A and Cj(i)k interaction b/w B(A) and Cijkl 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 (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 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

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 exampleWithin subjects (toads)Source df F P GG-P

[O2] 7 4.88 <0.001 0.004Breathing type x [O2] (interaction) 7 10.69 <0.001 <0.001Toads 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

Kohout (1995)

1 2... ..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

source

sink

DV = % greeningof nodules per band

Between plotsSpecies 1Treatment 5Species x Treatment 5Plates within Species & Treatment (Residual 1) 24

Within plotsBand 9Band x Species 9Band x Treatment 45Band x Treatment x Species 45Plots within Species & Treatment x Band (Residual 2) 216

Total Lots

Source of variation df

Parkinson (1996)

Billabong type Permanent Temporary Woodland

Billabong subjectsBillabong type between subjectsMonth and Time of day within subjects

Billabong 1 2 3 4 5 6 7 8 9 1011 121314 15

Month Nov Dec Jan Feb

Time of day AM PM

Between subjects (bongs)Type 2Bongs within Type (Residual 1) 12

Within subjects (bongs)Month 3Type x Month 6Month x Bongs within Type (Residual 2) 36Time 1Type x Time 2Time x Bongs within Type (Residual 3) 12Month x Time 3Type x Month x Time 6Month x Time x Bongs within Type (Residual 4) 36

Source of variation df