Introduction to Randomized block designs Accounting for predicted but random variance

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.

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Walter & O’Dowd (1992) • Effects of domatia (cavities on leaves) on number of mites • Two treatments (Factor A): – shaving domatia removes domatia from leaves – normal domatia as control

• Required 14 leaves for each treatment

Completely randomized design: - 28 leaves randomly allocated to each of 2 treatments

Control leaves

Shaved domatia leaves

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Completely randomized ANOVA • Factor A with p groups (p = 2 for domatia) • n replicates within each group (n = 14 pairs of leaves) Source Factor A Residual Total

general df

example df

p-1 p(n-1)

1 26

pn-1

27

Walter & O’Dowd (1992) • Required 14 leaves for each treatment • Set up as blocked design – paired leaves (14 pairs) chosen – 1 leaf in each pair shaved, 1 leaf in each pair control

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1 block

Control leaves

Shaved domatia leaves

Rationale for blocking • Micro-temperature, humidity, leaf age, etc. more similar within block than between blocks • Variation in response variable (mite number) between leaves within block (leaf pair) < variation between leaves between blocks

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Rationale for blocking • Some of unexplained (residual) variation in response variable from completely randomized design now explained by differences between blocks • More precise estimate of treatment effects than if leaves were chosen completely randomly

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

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Null hypotheses • No effect of factor B (blocks): – no difference between blocks (leaf pairs), pooling treatments

• Blocks usually random factor: – sample of blocks from population of blocks – H0: 2 = 0

Randomized blocks ANOVA • Factor A with p groups (p = 2 treatments for domatia) • Factor B with q blocks (q = 14 pairs of leaves) Source

general

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

example 1 13 13 27

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Randomized block ANOVA • Randomized 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

Example – effect of watering on plant growth • Factor 1: Watering, no watering • Factor 2: Blocks (1-4). One replicate of each treatment (watering, no watering) in each of 4 plots • Replication: 1 plant for each watering/black combination (8 total)

No water Water

7

Results 15

Treatment No Water Water No Water Water No Water Water No Water Water

Growth 6 10 4 6 11 15 5 8

10

Growth

Block 1 1 2 2 3 3 4 4

5

0

No Water

Water Treatment

Results 15.0

Treatment No Water Water No Water Water No Water Water No Water Water

Growth 6 10 4 6 11 15 5 8

10.0

Growth

Block 1 1 2 2 3 3 4 4

5.0

0.0

1.0

2.0

3.0

4.0

Block

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Results 15

Treatment No Water Water

Treatment No Water Water No Water Water No Water Water No Water Water

Growth 6 10 4 6 11 15 5 8

10

Growth

Block 1 1 2 2 3 3 4 4

5

0

1.0

2.0

3.0

4.0

Block

Expected mean squares If factor A fixed and factor B (Blocks) random: MSwatering MSBlocks MSResidual

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EMSResidual = 2 + 2 : Why is this not simply 2 15

Treatment No Water Water

Treatment No Water Water No Water Water No Water Water No Water Water

Growth 6 10 4 6 11 15 5 8

10

Growth

Block 1 1 2 2 3 3 4 4

5

0

1.0

2.0

3.0

4.0

Block

Residual • Cannot separately estimate 2 and 2: – no replicates within each block-treatment combination

• MSResidual estimates 2 + 2

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Expected mean squares If factor A fixed and factor B (Blocks) random: MSwatering MSBlocks

2 + 2

MSResidual

EMSBlocks =2 + n2 15.0

Treatment No Water Water No Water Water No Water Water No Water Water

Growth 6 10 4 6 11 15 5 8

10.0

Growth

Block 1 1 2 2 3 3 4 4

5.0

0.0

1.0

2.0

3.0

4.0

Block

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Expected mean squares If factor A fixed and factor B (Blocks) random: MSwatering MSBlocks

2 + n2

MSResidual

2 + 2

EMSwatering= 2 + 2 + n (i)2/p-1 15

Treatment No Water Water No Water Water No Water Water No Water Water

Growth 6 10 4 6 11 15 5 8

10

Growth

Block 1 1 2 2 3 3 4 4

5

0

No Water

Water Treatment

Why does the EMS for watering include 2 , which is the effect of the interaction? Block is a random effect, hence there are unsampled combinations of block and watering that could affect the estimates of EMSwatering

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Expected mean squares If factor A fixed and factor B (Blocks) random: MSwatering

2 + 2 + n (i)2/p-1

MSBlocks

2 + n2

MSResidual

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

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Walter & O’Dowd (1992) • Factor A (treatment - shaved and unshaved domatia) - fixed • Blocks (14 pairs of leaves) - random Source

df

Treatment Block Residual

1 13 13

MS

F

P

31.34 1.77 2.77

11.32 0.64

0.005 0.784 ??

Should this be reported??

Explanation Blocks Treatment

1

2

3

4

5

6

7

8

9

10

11

12

13

Shaved

1

1

1

1

1

1

1

1

1

1

1

1

1

Control

1

1

1

1

1

1

1

1

1

1

1

1

1

Cells represent the possible effect of the Block by Treatment interaction but: 1) There is only one replicate per cell, therefore 2) No way to estimate variance term for each cell, therefore 3) No way to estimate the variance associated with the interaction, therefore 4) The residual term estimates 2 + 2

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Randomized Block vs Completely Randomized designs • Total number of experimental units same in both designs – 28 leaves in total for domatia experiment

• Test of factor A (treatments) has fewer df in block design: – reduced power of test

RCB vs CR designs • MSResidual smaller in block design if blocks explain some of variation in Y: – increased power of test

• If decrease in MSResidual (unexplained variation) outweighs loss of df, then block design is better: – when blocks explain much of variation in Y

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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”)

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Interaction plots Y

No interaction

Y

Interaction

Block

Growth of Plantago

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Growth of Plantago • Growth of five genotypes (3 fast, 2 slow) of Plantago major (ribwort) • Poorter et al. (1990) • One replicate seedling of each genotype placed in each of 7 plastic containers in growth chamber – Genotypes (1, 2, 3, 4, 5) are factor A – Containers (1 to 7) are blocks – Response variable is total plant weight (g) after 12 days

Poorter et al. (1990) 3

4 1

2 5

Container 1

1

2 5

4

3

Container 2

Similarly for containers 3, 4, 5, 6 and 7

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Source Genotype Block Residual Total

df 4 6 24 34

MS

F

0.125 0.118 0.033

3.81

P 0.016

Conclusions: • Large variation between containers (= blocks) so block design probably better than completely randomized design • Significant difference in growth between genotypes

Mussel recruitment and seastars

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Mussel recruitment and seastars • Effect of increased mussel (Mytilus spp.) recruitment on seastar numbers • Robles et al. (1995) – Two treatments: 30-40L of Mytilus (0.5-3.5cm long) added, no Mytilus added – Four matched pairs (blocks) of mussel beds chosen – Treatments randomly assigned to mussel beds within pair – Response variable % change in seastar numbers

+

+

+

-

-

-

+

1 block (pair of mussel beds)

+

-

mussel bed with added mussels mussel bed without added mussels

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Source

df

Blocks Treatment Residual

3 1 3

MS 62.82 5237.21 115.09

F 45.50

P 0.007

Conclusions: • Relatively little variation between blocks so a completely randomized design probably better because treatments would have 1,6 df • Significant treatment effect - more seastars where mussels added

Worked Example – seastar colors • Comparison of numbers of purple vs orange seastars along the CA coast • Data number of purple and orange seastars collected at 7 random locations • Compare models (block vs completely random vs paired t test)

sea star colors all sites two sample

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Number of Seastars as a function of color and site 600

1000

500

800

400 600 300 400

200

200

100 0

Orange

0

Purple Color

Each error bar is constructed using 1 standard error from the mean.

Site Each error bar is constructed using 1 standard error from the mean.

Any obvious problem with the data??

Normal Probability

0.91 0.88 0.84 0.8

Diagnostics: Log transform helps normality and homogeneity of variance assumptions

0.7 0.6 0.5 0.4 0.3 Log Number

0.2 0.16 0.12 0.09 0.07 0

200

400

600

800

1000

Number

Number

Normal Probability

0.91 0.88 0.84 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.16 0.12 0.09 0.07 1

1.5

2

2.5

3

Log Number

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Model 1: One factor ANOVA Why the difference? SE=0.181 2.5

SE=0.184

2

1.5

1 0.5

0

Orange

Purple Color

Each error bar is constructed using 1 standard error from the mean.

Model 2: Paired t test • Accounts for site specific (block) differences • But no way to assess site (block) differences

3.5

Value

3.0 2.5 2.0 1.5 1.0 LORANGE

LPURPLE Index of Case

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Model 3: Randomized Block Design using least squares • Accounts for and assesses (with a caveat) site specific effects

Stair

PSN

Shell Beach

Govpt

Hazards

Compare to paired t (same p value for Color) but no Site effect Compare to single factor ANOVA (look at p-value for Color). Here tradeoff between df and partitioning of variance makes for a more powerful test

Boat

1) 2)

Cayucos

Mean(Log Number)

Be careful

Any hint of Interaction (site*color)? If not then how does this change our interpretation of results? If factor A fixed and factor B (Blocks) random: MSA

2 + 2 + n (i)2/p-1

MSBlocks

2 + n2

MSResidual

2 + 2

3.5

LNUMBER

3.0 2.5 2.0 1.5 1.0

COLOR

t t r vp Boa Stai ards each ucos PSN z y Go B Ha ell Ca Sh

Orange Purple

SITE

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Model 3: Randomized Block Design - using (restricted) Maximum Likelihood Estimation • Accounts for site specific effects

Identical to least squares solution

1) 2)

Variance component used to calculate percent of variance associated with the random effect P-value for Color is identical to that from the Least Squares Estimation (this will always be true for balanced designs)

Model 3: Mixed Model Solution • Also accounts for site specific effects

Identical to least squares solution and REML

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Sphericity assumption This is for reference – much more important for repeated measures

Block

Treat 1

Treat 2

1 2 3 etc.

y11 y12 y13

y21 y22 y23

Treat 3 etc. y31 y32 y33

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Block

T1 - T2

T2 - T3

T1 - T3 etc.

1 2 3 etc.

y11-y21 y12-y22 y13-y23

y21-y31 y22-y32 y23-y33

y11-y31 y12-y32 y13-y33

Sphericity assumption • Pattern of variances and covariances within and between treatments: – 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

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Sphericity assumption • Applies to randomized block – also 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

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Sphericity assumption • Assumption of sphericity probably OK for randomized 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 randomized (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

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