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

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– H0: 1 = 2 = … = i = ... = – H0: 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

– H0: 2 = 0

• Factor A with p groups (p = 2 treatments for domatia)

• Factor B with q blocks (q = 14 pairs of leaves)

Source general example

Factor A p-1 1Factor B (blocks) q-1 13Residual (p-1)(q-1) 13Total pq-1 27

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:

MSA 2 + 2 + n(i)2/p-1

MSBlocks 2 + n2

MSResidual 2 + 2

Expected mean squares

Residual

• Cannot separately estimate 2 and 2:

– no replicates within each block-treatment combination

• MSResidual estimates 2 + 2

Testing null hypotheses

• Factor A fixed and blocks random

• If H0 no effects of factor A is true:

– then F-ratio MSA / MSResidual 1

• If H0 no variance among blocks is true:

– no F-ratio for test unless no interaction assumed

– if blocks fixed, then F-ratio MSB / MSResidual 1

Assumptions

• Normality of response variable– boxplots etc.

• No interaction between blocks and factor A, otherwise– MSResidual increase proportionally more than

MSA 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 ANOVAExperimental designs where a factor (B) is crossed with one factor (C) but nested within another (A).

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

Source df Fixed or randomA (p-1) Either, usually fixedB(A) p(q-1) RandomC (r-1) Either, usually fixedA * C (p-1)(r-1) Usually fixedB(A) * C p(q-1)(r-1) Random

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

Expected mean squaresFactor A (p levels, fixed), factor B(A) (q levels, random), factor C (r levels, fixed)

Source df EMS TestA p-1

2 + nr2 + nqr

2 MSA/MSB(A)

B(A) p(q-1) 2 + nr

2 MSB/MSRES

C r-1 2 + n

2 + npq2 MSC/MSB(A)C

AC (p-1)(r-1)2 + n

2 + nq2MSAC/MSB(A)C

B(A) C p(q-1)(r-1) 2 + n

2 MSBC/MSRES

Residual pqr(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)

ANOVASource of variation dfBetween plotsSite 2Plots within site (Residual 1) 3

Within plotsTrampling 3Site x trampling (interaction) 6Plots within site x trampling (Residual 2) 9

Total 23

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[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 (Breathing type) x [O2](Residual 2) 133

Total 167

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 [O2] 1 and [O2] 2 = variance of differences between [O2] 2 and [O2] 2 = variance of differences between [O2] 1 and [O2] 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