Design and Analysis of Experiments Lecture 4 · Design and Analysis of Experiments Lecture 4.1 1....
Transcript of Design and Analysis of Experiments Lecture 4 · Design and Analysis of Experiments Lecture 4.1 1....
Lecture 4.1 1
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 4.1
1. Review of Lecture 3.1 / Laboratory 1
2. Introduction to
– Fractional Factorial Designs
– Blocking factorial designs
3. Introduction to Split Plot designs
– Fisher on Potatoes
– Water resistance of wood stains
– Cambridge Grassland Experiment
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Review of Laboratory 1:
Soybean seed germination rates
Table 1: Numbers of failures in 25 plots of 100 soybean seeds, arranged in blocks of 5 plots, with random allocation of seed treatments to plots within blocks.
Block Treatment I II III IV V
Check 8 10 12 13 11 Arasan 2 6 7 11 5 Spergon 4 10 9 8 10 Semesan 3 5 9 10 6 Fermate 9 7 5 5 3
.
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Block
Failu
res
54321
14
12
10
8
6
4
2
Treatment
Fermate
Semesan
Spergon
Arasan
Check
Failure Profiles for Five Treatments
Soybean seed germination rates
Graphical analysis
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Soybean seed germination rates
Graphical analysis
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Block
Failu
res
54321
14
12
10
8
6
4
2
Treatment
Fermate
Semesan
Spergon
Arasan
Check
Failure Profiles for Five Treatments
Soybean seed germination rates
Graphical analysis
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• Treatments appear almost universally better than
no treatment (Check)
• General pattern of increasing rates from Block 1 to
Block 4, reducing for Block 5
– broadly consistent with
homogeneity within blocks
and
differences between blocks,
as desired
Summary
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• Important exceptions, including
– high rates for Fermate in Blocks 1 and 2,
otherwise Fermate is best
– low rates for Spergon in Blocks 3 and 4
Best variety?
• Fermate best in Blocks 3, 4, 5
Arasan and Semesan best in Blocks 1, 2
Summary
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• Further investigation of Fermate in Blocks 1 and 2
indicated
– potential for gain in understanding
• Possibly investigate Spergon in Blocks 3 and 4
Next steps?
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Analysis of Variance for Failures
Source DF Adj SS Adj MS F P
Treatment 4 83.840 20.960 3.87 0.022
Block 4 49.840 12.460 2.30 0.103
Error 16 86.560 5.410
Total 24 220.240
Analysis of Variance for Failures
Source DF Adj SS Adj MS F P
Treatment 4 83.840 20.960 3.07 0.040
Error 20 136.400 6.820
Total 24 220.240
Was blocking effective?
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Source DF Adj SS Adj MS F P
Treatment 4 113.400 28.350 10.92 0.000
Block 4 84.650 21.162 8.15 0.001
Error 15 38.950 2.597
Total 23 217.958
S = 1.61142
Source DF Adj SS Adj MS F P
Treatment 4 94.358 23.590 3.63 0.023
Error 19 123.600 6.505
Total 23 217.958
S = 2.55054
Was blocking effective?
Exceptional case deleted:
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Was blocking effective?
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Test for interaction?
Analysis of Variance for Rate, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Block 4 49.8400 49.8400 12.4600 **
Treatment 4 83.8400 83.8400 20.9600 **
Block*Treatment 16 86.5600 86.5600 5.4100 **
Error 0 * * *
Total 24 220.2400
Compare with: Source DF Seq SS Adj SS Adj MS F P
Treatment 4 83.840 83.840 20.960 3.87 0.022
Block 4 49.840 49.840 12.460 2.30 0.103
Error 16 86.560 86.560 5.410
Total 24 220.240
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Model including interaction
Failures equals
overall mean
plus
Treatment effect
plus
Block effect
plus
Treatment by Block interaction effect
plus
chance variation
No replication implies no measure of chance variation,
same as unreplicated 24 design (Lecture 3.1, Part 3)
UNLESS no interaction effect.
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Review of Laboratory 1, Part 2
An unreplicated 24 experiment:
A process improvement study to reduce impurity
• Lenth's method
• Reduced model
• Design projection
– which model?
• Optimum conditions
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2-Level Factorial Experiments
are important because they
• are relatively simple to set up
• are relatively simple to analyse
• permit several factors to be investigated in relatively
few experimental runs,
• permit even more factors to be investigated by using
carefully chosen subsets of a full experiment,
• provide clues to seeking better operating conditions.
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Process improvement study to
reduce impurity
Chemical manufacturing:
impurity levels 55 - 65 gms per Kg
target ≤ 35 gms per Kg
Key input factors:
catalyst concentration (%), 5 and 7,
concentration of NaOH (%), 40 and 45,
agitation speed (rpm), 10 and 20,
temperature (°F), 150 and 180.
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Design Point
Run Order
Catalyst Concentration
Sodium Hydroxide
Concentration
Agitation Speed
Temperature Impurity
1 2 5 40 10 150 38 2 6 7 40 10 150 40 3 12 5 45 10 150 27 4 4 7 45 10 150 30 5 1 5 40 20 150 58 6 7 7 40 20 150 56 7 14 5 45 20 150 30 8 3 7 45 20 150 32 9 8 5 40 10 180 59
10 10 7 40 10 180 62 11 15 5 45 10 180 53 12 11 7 45 10 180 50 13 16 5 40 20 180 79 14 9 7 40 20 180 75 15 5 5 45 20 180 53 16 13 7 45 20 180 54
Impurity levels in gm. per Kg. resulting from
varying levels of four two level factors
in a 24 design run in completely random order
.
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Process improvement study to
reduce impurity
20
10
0
-10
-20
210-1-2
Eff
ect
Score
A CatCon
B NaOHCon
C Speed
D Temp
Factor Name
BC
D
C
B
Normal Effects Plot
Lenth's PSE = 1.125
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Process improvement study to
reduce impurity
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Process improvement study to
reduce impurity
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Apply Lenth's analysis to
soybean seed treatments?
• Effects of 2-level factors, including interactions,
summarized in a set of independent contrasts
• Main effects of 5-level factors summarised as
5 correlated deviations from mean, with 4 df,
• Interaction effects summarised as
25 correlated deviations from mean, with 16 df.
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Process improvement study
Visualising the results
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Mean
Temperature Impurity
150 38.88
180 60.63
Speed 10 20
40 49.75 67.00 NaOH
45 40.00 42.25
Lecture 4.1 23
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Process improvement study
Visualising the results
Speed 10 20
40 49.75 67.00 NaOH
45 40.00 42.25
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Process improvement study
Visualising the results
2010
70
65
60
55
50
45
40
Speed
Im
pu
rity
40
45
NaOHCon
Interaction Plot
Speed 10 20
40 49.75 67.00 NaOH
45 40.00 42.25
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Process improvement study:
Summarising results
• Reducing temperature will reduce impurities
• Increasing concentration of NaOH will reduce
impurities
• Under those conditions, changing either catalyst
concentration or agitation speed will have little
effect
– use cheapest or most convenient levels
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Full model Effect Estimate
CatCon 0.25
NaOHCon -17.25
Speed 9.75
Temp 21.75
CatCon*NaOHCon 0.50
CatCon*Speed -1.00
CatCon*Temp -1.00
NaOHCon*Speed -7.50
NaOHCon*Temp 1.00
Speed*Temp -0.50
CatCon*NaOHCon*Speed 1.75
CatCon*NaOHCon*Temp -0.75
CatCon*Speed*Temp 0.25
NaOHCon*Speed*Temp 0.25
CatCon*NaOHCon*Speed*Temp 1.00
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Reduced model
NaOHCon -17.25
Speed 9.75
Temp 21.75
NaOHCon*Speed -7.50
s = 1.74, df = 11 = 15 − 4
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Projected model
NaOHCon -17.25
Speed 9.75
Temp 21.75
NaOHCon*Speed -7.50
NaOHCon*Temp 1.00
Speed*Temp -0.50
NaOHCon*Speed*Temp 0.25
s = 1.87, df = 8 = 15 − 7
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Comparison of fits of
Full, Reduced, Projected models All effect estimates are the same; SE's vary.
Lenth: s = 2.25, df = 3, PSE(effect) = 1.125
Reduced: s = 1.74, df = 11, SE(effect) = 0.870
Projected: s = 1.87, df = 8, SE(effect) = 0.940
"Projected" model includes 3 interactions not
included in the "Reduced" model.
Adding null effects (chance variation) to a model
may increase or decrease s, depending on chance.
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Ref: Models for Experiments (Extra Notes)
Lab 1 Feedback
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Degrees of freedom
"Error" degrees of freedom relevant for t
– Lenth's formula
– check ANOVA table
– count estimated effects (Slides 27, 28)
– use replications (Lecture 3.1,
Slides 9, 34)
t3, .05 = 2.57
t8, .05 = 2.31
t11,.05 = 2.20
s = 2.25
s = 1.87
s = 1.74 Ref: Models for Experiments (Extra Notes)
Lab 1 Feedback
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Design and Analysis of Experiments
Lecture 4.1
1. Review of Lecture 3.1 / Laboratory 1
2. Introduction to
– Fractional Factorial Designs
– Blocking factorial designs
3. Introduction to Split Plot designs
– Fisher on Potatoes
– Water resistance of wood stains
– Cambridge Grassland Experiment
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Several 2-level factorsfactors: how many design points?
Factors Design points
2 22 = 4
3 23 = 8
4 24 = 16
5 25 = 32
6 26 = 64
7 27 = 128
Part 2: Introduction to
Fractional Factorial Designs
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Problems with big experiments
Many experimental units (plots, runs)
– large area (long time)
• inhomogeneous conditions?
– high materials cost
– high labour costs
– difficult logistics
Solution:
choose an informative subset of design points
NB: design matrix columns key to this development
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A 24 with 16 design points
Design Point
A B C D
1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +
10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +
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The first 8 design points
Design Point
A B C D
1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +
10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +
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Design Point
A B C D
1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +
10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +
The middle 8 design points
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Another 8 design points
Design Point
A B C D
1 – – – – 2 + – – – 3 – + – – 4 + + – – 5 – – + – 6 + – + – 7 – + + – 8 + + + – 9 – – – +
10 + – – + 11 – + – + 12 + + – + 13 – – + + 14 + – + + 15 – + + + 16 + + + +
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Design Point
A B C D AB AC AD BC BD CD
2 + – – – – – – + + + 3 – + – – – + + – – + 5 – – + – + – + – + – 8 + + + – + + – + – – 10 + – – + – – + + – – 11 – + – + – + – – + – 13 – – + + + – – – – + 16 + + + + + + + + + +
Same 8 design points, with 2fi’s
Confounded effects:
A = BC
B = AC
C = AB
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Design Point
A B C D AB AC AD BC BD CD
2 + – – – – – – + + + 3 – + – – – + + – – + 5 – – + – + – + – + – 8 + + + – + + – + – – 10 + – – + – – + + – – 11 – + – + – + – – + – 13 – – + + + – – – – + 16 + + + + + + + + + +
Same 8 design points, with 2fi’s
Confounded effects:
A = BC
B = AC
C = AB
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Same 8 design points,
with all interactions Design Point
A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
2 + – – – – – – + + + + + + – – 3 – + – – – + + – – + + + – + – 5 – – + – + – + – + – + – + + – 8 + + + – + + – + – – + – – – –
10 + – – + – – + + – – + – – + + 11 – + – + – + – – + – + – + – + 13 – – + + + – – – – + + + – – + 16 + + + + + + + + + + + + + + +
AD = BCD
BD = ACD
CD = ABD
I = ABC
Confounded effects:
A = BC
B = AC
C = AB
D = ABCD Postgraduate Certificate in Statistics
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Same 8 design points,
with all interactions Design Point
A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
2 + – – – – – – + + + + + + – – 3 – + – – – + + – – + + + – + – 5 – – + – + – + – + – + – + + – 8 + + + – + + – + – – + – – – –
10 + – – + – – + + – – + – – + + 11 – + – + – + – – + – + – + – + 13 – – + + + – – – – + + + – – + 16 + + + + + + + + + + + + + + +
AD = BCD
BD = ACD
CD = ABD
I = ABC
Confounded effects:
A = BC
B = AC
C = AB
D = ABCD Postgraduate Certificate in Statistics
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Design Point
A B C Y
1 – – – Y1
2 + – – Y2 3 – + – Y3 4 + + – Y4 5 – – + Y5 6 + – + Y6 7 – + + Y7 8 + + + Y8
ABC
– + + – + – – +
Clever design
Each row gives design points for a 4-factor experiment
Fourth column estimates D main effect.
Fourth column also estimates ABC interaction effect.
In fact, fourth column estimates D + ABC in 24-1.
D=
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Classwork Exercise: Confirm confounding patterns
Design Point
A= BCD
B= ACD
C= ABD
D= ABC
Y
1 – – – – Y1
2 + – – + Y2
3 – + – + Y3
4 + + – – Y4
5 – – + + Y5
6 + – + – Y6
7 – + + – Y7
8 + + + + Y8
Confirm "confounding" or "aliasing" patterns shown.
Also, confirm AB = CD.
What other effects are aliased?
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Fractional factorial designs First half fraction
Design Point
A B C D Y
1 – – – – 70
2 + – – + 62
3 – + – + 88
4 + + – – 81
5 – – + + 60
6 + – + – 49
7 – + + – 88
8 + + + + 79
Full factorial design
Design Point
A B C D Y
1 – – – – 70 2 + – – – 60 3 – + – – 89 4 + + – – 81 5 – – + – 60 6 + – + – 49 7 – + + – 88 8 + + + – 82 9 – – – + 69
10 + – – + 62 11 – + – + 88 12 + + – + 81 13 – – + + 60 14 + – + + 52 15 – + + + 86 16 + + + + 79
Identify corresponding
design points
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Fractional factorial designs First half fraction
Design Point
A B C D Y
1 – – – – 70
2 + – – + 62
3 – + – + 88
4 + + – – 81
5 – – + + 60
6 + – + – 49
7 – + + – 88
8 + + + + 79
Full factorial design
Design Point
A B C D Y
1 – – – – 70 2 + – – – 60 3 – + – – 89 4 + + – – 81 5 – – + – 60 6 + – + – 49 7 – + + – 88 8 + + + – 82 9 – – – + 69
10 + – – + 62 11 – + – + 88 12 + + – + 81 13 – – + + 60 14 + – + + 52 15 – + + + 86 16 + + + + 79
Identify corresponding
design points
1
2 3
4
5
6 7
8 Postgraduate Certificate in Statistics
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Fractional factorial designs
Column A estimates A + BCD Column A estimates A – BCD
Full 24 design: Column A estimates ½[(A + BCD) + (A – BCD)] = A
Second half fraction
Design Point
A B C D Y
9 – – – + 69 1 + – – – 60 2 – + – – 89
12 + + – + 81 5 – – + – 60
14 + – + + 52 15 – + + + 86 8 + + + – 82
First half fraction
Design Point
A B C D Y
1 – – – – 70 10 + – – + 62 11 – + – + 88 4 + + – – 81
13 – – + + 60 6 + – + – 49 7 – + + – 88
16 + + + + 79
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Fractional factorial designs
With bigger designs (more factors) use smaller
fractions, e.g.
25 = 32 design points;
identify 4 ¼ fractions of 8 design points each.
Choose fractions to alias
main effects with 4-factor interactions,
2-factor interaction with 3-factor interactions.
Run one fraction.
If doubtful about a 2fi, run another appropriate
fraction to resolve the alias.
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Design and Analysis of Experiments
Lecture 4.1
1. Review of Lecture 3.1 / Laboratory 1
2. Introduction to
– Fractional Factorial Designs
– Blocking factorial designs
3. Introduction to Split Plot designs
– Fisher on Potatoes
– Water resistance of wood stains
– Cambridge Grassland Experiment
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In multifactor experiments
requiring
several runs in inhomogeneous conditions,
fractions may be used as blocks.
Block effects are aliased with suitable high level
interactions.
Blocking Factorials Designs
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Blocking a 24 experiment
Block 1: ABCD = + Block 2: ABCD = –
Second half fraction
Design Point
A B C D Y
9 – – – + 69 1 + – – – 60 2 – + – – 89
12 + + – + 81 5 – – + – 60
14 + – + + 52 15 – + + + 86 8 + + + – 82
First half fraction
Design Point
A B C D Y
1 – – – – 70 10 + – – + 62 11 – + – + 88 4 + + – – 81
13 – – + + 60 6 + – + – 49 7 – + + – 88
16 + + + + 79
ABCD effect confounded with block difference
All other effects unconfounded, estimated separately
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Design and Analysis of Experiments
Lecture 4.1
1. Review of Lecture 3.1 / Laboratory 1
2. Introduction to
– Fractional Factorial Designs
– Blocking factorial designs
3. Introduction to Split Plot designs
– Fisher on Potatoes
– Water resistance of wood stains
– Cambridge Grassland Experiment
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Recall Randomised Blocks Case Study
Reducing yield loss in a chemical process
• Process: chemicals blended, filtered and dried
• Problem: yield loss at filtration stage
• Proposal: adjust initial blend to reduce yield loss
• Plan:
– prepare five different blends
– use each blend in successive process runs, in
random order
– repeat at later times (blocks)
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Unit Structure
Block 1 Block 2 Block 3
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Unit 1_1
Unit 1_2
Unit 1_3
Unit 1_4
Unit 1_5
Unit 2_1
Unit 2_2
Unit 2_3
Unit 2_4
Unit 2_5
Unit 3_1
Unit 3_2
Unit 3_3
Unit 3_4
Unit 3_5
Blocks
Units
Units nested in Blocks
15 experimental units
grouped in 3 blocks
less variation
more variation
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Introduction to Split Plots designs
• The first ever split plots design? (Fisher, 1925)
• Think of Broadbalk (Lecture 1.2, slide 71, Notes p.15)
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© Rothamsted Research
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Potatoes
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Potatoes
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Experimental factors
Varieties:
Ajax
Arran Comrade
British Queen
Duke of York
Epicure
Great Scott
Iron Duke
King of Kings
Kerr's Pink
Nithsdale
Tinwald Perfection
Up-to-Date
Fertilisers:
Basal manure dressing
Manure with added
Potassium Sulphate
Manure with added
Potassium Chloride.
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The first ever split plots design?
• Twelve varieties of potatoes planted in 36 plots
– each variety planted in three plots "scattered
over the area"
• Each plot divided into three subplots,
– each subplot fertilised with one of three
fertilisers.
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Field layout
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Field layout
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Whole Plots Numbered
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1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24
25 26 27 28
29 30 31 32
33 34 35 36
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Assignment of Varieties to Whole Plots
Varieties:
Ajax
Arran Comrade
British Queen
Duke of York
Epicure
Great Scott
Iron Duke
K. of K.
Kerr's Pink
Nithsdale
Tinwald Perfection
Up-to-Date
Whole Plots
1 13 32
8 20 34
24 25 35
5 7 19
10 12 27
4 6 18
9 11 31
2 14 16
22 28 29
3 15 17
26 33 36
21 23 30 Postgraduate Certificate in Statistics
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Results: Yield (lbs per plant) Variety Sulphate Chloride Basal
Ajax 3.20 4.00 3.86 2.55 3.04 4.13 2.82 1.75 4.71
Arran Comrade 2.25 2.56 2.58 1.96 2.15 2.10 2.42 2.17 2.17
British Queen 3.21 2.82 3.82 2.71 2.68 4.17 2.75 2.75 3.32
Duke of York 1.11 1.25 2.25 1.57 2.00 1.75 1.61 2.00 2.46
Epicure 2.36 1.64 2.29 2.11 1.93 2.64 1.43 2.25 2.79
Great Scot 3.38 3.07 3.89 2.79 3.54 4.14 3.07 3.25 3.50
Iron Duke 3.43 3.00 3.96 3.33 3.08 3.32 3.50 2.32 3.29
K. of K. 3.71 4.07 4.21 3.39 4.63 4.21 2.89 4.20 4.32
Kerr's Pink 3.04 3.57 3.82 2.96 3.18 4.32 2.00 3.00 3.88
Nithsdale 2.57 2.21 3.58 2.04 2.93 3.71 1.96 2.86 3.56
Tinwald Perfection 3.46 3.11 2.50 2.83 2.96 3.21 2.55 3.39 3.36
Up-to-Date 4.29 2.93 4.25 3.39 3.68 4.07 4.21 3.64 4.11
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Plot 1 mean = 2.86 Plot 13 mean= 2.93 Plot 32 mean=4.23
Lecture 4.1 64
© 2016 Michael Stuart
Plot structure
108 subplots
grouped in 36 plots
Two layers of plot structure:
subplots
in
whole plots
Treatment Structure 12 varieties assigned haphazardly to whole plots
3 fertilisers assigned systematically to subplots
within whole plots
less variation
more variation
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 65
© 2016 Michael Stuart
Analysis of Split Plots design
• Varieties vary between whole plots,
– variety effects evaluated with reference to
chance variation between whole plots
• Fertilisers vary between subplots
– fertiliser effects evaluated with reference to
chance variation between subplots
• Two sources of chance variation
• Implications for Analysis of Variance
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 66
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 4.1
1. Review of Lecture 3.1 / Laboratory 1
2. Introduction to
– Fractional Factorial Designs
– Blocking factorial designs
3. Introduction to Split Plot designs
– Fisher on Potatoes
– Water resistance of wood stains
– Cambridge Grassland Experiment
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 67
© 2016 Michael Stuart
Another illustration
• Testing water resistance of four wood stains
• Stains applied to boards
• Boards are pretreated with one of two treatments.
• Ideal:
• standard two-factor design
– eight Stain / Pretreatment combinations
– one applied to each of 8 boards
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 68
© 2016 Michael Stuart
Another illustration
• Problem:
– Pretreatments can only be applied to whole board
– single replication requires 8 boards
• Solution:
– Apply Pretreatments to whole boards
– Cut pretreated boards into 4 panels
– Apply stains to panels
– Replicate 3 times
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 69
© 2016 Michael Stuart
Another illustration
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 70
© 2016 Michael Stuart
Another illustration
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 71
© 2016 Michael Stuart
Results
Stain 1
Stain 2
Stain 3
Stain 4
43.0 51.8 40.8 45.5 Pretreatment 1 57.4 60.9 51.1 55.3
52.8 59.2 51.7 55.3
46.6 53.5 35.4 32.5 Pretreatment 2 52.2 48.3 45.9 44.6
32.1 34.4 32.2 30.1
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 72
© 2016 Michael Stuart
Unit structure
24 panels (subunits)
grouped in 6 boards
(whole units)
Treatment structure
2 pretreatments assigned at random to boards
4 stains assigned at random to panels within boards
less variation
more variation
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 73
© 2016 Michael Stuart
Analysis of Split Plots design
• Pretreatments vary between boards,
– pretreatment effects evaluated with reference
to chance variation between boards
• Stains vary between panels
– stain effects evaluated with reference to
chance variation between panels
• Two sources of chance variation
• Implications for Analysis of Variance
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 74
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 4.1
1. Review of Lecture 3.1 / Laboratory 1
2. Introduction to
– Fractional Factorial Designs
– Blocking factorial designs
3. Introduction to Split Plot designs
– Fisher on Potatoes
– Water resistance of wood stains
– Cambridge Grassland Experiment
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 75
© 2016 Michael Stuart
Cambridge Grassland Experiment
(1931)
Original plan:
Investigate two new grassland cultivation treatments:
grassland “Rejuvenator” R
conventional Harrow H
by comparison with
no treatment (Control) C
in 6 independently randomised blocks of 3 adjacent
plots each.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 76
© 2016 Michael Stuart
Recall typical agricultural experimental
layouts
Broadbalk, Rothamsted
Rothamsted
Research
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 77
© 2016 Michael Stuart
Cambridge Grassland Experiment
layout Blocks 1 2 3 4 5 6
Plots 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 78
© 2016 Michael Stuart
Cambridge Grassland Experiment
Blocks 1 2 3 4 5 6
Plots 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Treatments H C R H R C C H R H R C C H R C R H
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 79
© 2016 Michael Stuart
Cambridge Grassland Experiment
Subsequent addition:
investigate 3 fertilisers
Farmyard manure F
Straw S
Artificial fertiliser A
by comparison with
no fertiliser (Control) C
allocated at random to 4 sub plots within each plot.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 80
© 2016 Michael Stuart
Cambridge Grassland Experiment
Blocks 1 2 3 4 5 6
Whole Plots 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Treatments H C R H R C C H R H R C C H R C R H
Sub Plot 1
Sub Plot 2
Sub Plot 3
Sub Plot 4
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 81
© 2016 Michael Stuart
Cambridge Grassland Experiment
Blocks 1 2 3 4 5 6
Whole Plots 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Treatments H C R H R C C H R H R C C H R C R H
Sub Plot 1 C A A
Sub Plot 2 A S C
Sub Plot 3 F C F
Sub Plot 4 S F S
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 82
© 2016 Michael Stuart
Cambridge Grassland Experiment
Blocks 1 2 3 4 5 6
Whole Plots 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Treatments H C R H R C C H R H R C C H R C R H
Sub Plot 1 C A A C F F A A A A F F F C A F F C
Sub Plot 2 A S C A S A C C F F A S S A S A S S
Sub Plot 3 F C F F C C S F S C S A C S C C C F
Sub Plot 4 S F S S A S F S C S C C A F F S A A
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 83
© 2016 Michael Stuart
Results
Yield was recorded in pounds (lbs) of green
produce from a single cut of each subplot made on
June 31, 1931 and are shown in the table below.
Block 1 Block 2 Block 3
C H R C H R C H R
A 266 213 208 210 222 266 220 184 184 C 165 127 155 150 167 163 155 118 153 F 198 180 200 247 203 228 190 168 174 S 184 127 150 188 167 157 140 128 141
Block 4 Block 5 Block 6
C H R C H R C H R
A 216 178 207 202 175 184 169 142 151 C 159 125 135 147 118 98 132 104 69 F 225 149 162 184 175 144 164 145 116 S 174 107 113 154 112 113 116 89 101
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 84
© 2016 Michael Stuart
Plot structure
72 subplots
grouped in 18 whole plots
grouped in 6 blocks
Three layers of plot structure:
subplots
in
whole plots
in
blocks
least variation
in between variation
most variation
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 85
© 2016 Michael Stuart
Treatment structure
3 grassland treatments randomly assigned
to whole plots within blocks
4 fertilisers randomly assigned
to subplots within whole plots
Treatments referred to
chance variation between whole plots
Fertilisers referred to
chance variation between subplots
What about Blocks?
Implications for Analysis of Variance
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 86
© 2016 Michael Stuart
Reasons for using split units
• Adding another factor after the experiment
started
• Some factors require better precision than others
• Changing one factor is
– more difficult
– more expensive
– more time consuming
than changing others
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 87
© 2016 Michael Stuart
Minute test
– How much did you get out of today's class?
– How did you find the pace of today's class?
– What single point caused you the most
difficulty?
– What single change by the lecturer would have
most improved this class?
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1 88
© 2016 Michael Stuart
Reading
EM §5.7, §7.4 for fractional factorial designs and
blocking
Lecture Notes: Split Units Design and Analysis,
pages 1-6, 10-11
Postgraduate Certificate in Statistics
Design and Analysis of Experiments