Randomized block designs

1
Randomized block designs

• Ó Environmental sampling and analysis (Quinn
  Keough, 2002)

2
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.

3
Null hypotheses

• No main effect of Factor A
• H0 m1 m2 mi ... m
• H0 a1 a2 ai ... 0 (ai mi - m)
• no effect of shaving domatia, pooling blocks
• Factor A usually fixed

4
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

5
Randomised blocks ANOVA

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

6
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

7
Expected mean squares
If factor A fixed and factor B (Blocks)
random MSA s2 sab2 n å(ai)2/p-1 MSBlocks s2
nsb2 MSResidual s2 sab2
8
Residual

• Cannot separately estimate s2 and sab2
• no replicates within each block-treatment
  combination
• MSResidual estimates s2 sab2

9
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

10
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

11
Checks for interaction

• No real test because no within-cell variation
  measured
• Tukeys test for non-additivity
• detect some forms of interaction
• Plot treatment values against block (interaction
  plot)

12
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

13
Sphericity assumption

• Applies to randomised block and repeated measures
  designs
• Epsilon (e) statistic indicates degree to which
  sphericity is not met
• further e is from 1, more variances of treatment
  differences are different
• Two versions of e
• Greenhouse-Geisser e
• Huyhn-Feldt e

14
Dealing with non-sphericity

• If e 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 e
• Multivariate ANOVA (MANOVA)
• treatments considered as multiple response
  variables in MANOVA

15
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)

16
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

17
Partly nested ANOVA

• Ó Environmental sampling and analysis (Quinn
  Keough, 2002)

18
Partly nested ANOVA

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

19
Partly nested ANOVA
Experimental 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
20
ANOVA table
Source df Fixed or random A (p-1) Either, usually
fixed B(A) p(q-1) Random C (r-1) Either, usually
fixed A C (p-1)(r-1) Usually fixed B(A)
C p(q-1)(r-1) Random Residual pqr(n-1)
21
Linear model
yijkl m ai bj(i) dk adik bdj(i)k
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 bdj(i)k interaction b/w B(A) and
C eijkl residual variation
22
Expected mean squares
Factor A (p levels, fixed), factor B(A) (q
levels, random), factor C (r levels,
fixed) Source df EMS Test A p-1 ??2 nr??2
nqr??2 MSA/MSB(A) B(A) p(q-1) ??2
nr??2 MSB/MSRES C r-1 ??2 n???2
npq??2 MSC/MSB(A)C AC (p-1)(r-1) ??2 n???2
nq???2 MSAC/MSB(A)C B(A) C p(q-1)(r-1) ??2
n???2 MSBC/MSRES Residual pqr(n-1) ??2
23
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

24
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)

25
ANOVA
Source of variation df Between plots Site 2 Plots
within site (Residual 1) 3 Within
plots Trampling 3 Site x trampling
(interaction) 6 Plots within site x trampling
(Residual 2) 9 Total 23
26
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

27
Repeated measures designs

• Factor A
• units of replication termed subjects
• Factor B (subjects) nested within A
• Factor C
• repeated recordings on each subject

28
Repeated measures design
O2 Breathing Toad 1 2 3 4 5 6 7 8 type Lun
g 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 ... ... ... ... ... ... ... ... ... ... B
uccal 21 x x x x x x x x
29
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
(Breathing type) x O2 (Residual
2) 133 Total 167
30
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

31

• 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

32
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.

33
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

34
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

35
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

36
ANOVA options

• Multivariate (MANOVA) tests for within subjects
  or plots factors
• responses from each subject used in MANOVA
• doesnt require sphericity
• sometimes more powerful than GG adjusted
  univariate, sometimes not