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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
1
© 2015 Michael Stuart
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.
Treatment
Check
Arasan
Spergon
Semesan
Fermate
I
8
2
4
3
9
II
10
6
10
5
7
Block
III
12
7
9
9
5
IV
13
11
8
10
5
V
11
5
10
6
3
.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
2
© 2015 Michael Stuart
Soybean seed germination rates
Graphical analysis
Failure Profiles for Five Treatments
14
Treatment
Arasan
Check
Fermate
Semesan
Spergon
12
Failures
10
8
6
4
2
1
2
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
3
Block
4
5
Lecture 4.1
3
© 2015 Michael Stuart
Soybean seed germination rates
Graphical analysis
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
4
© 2015 Michael Stuart
Soybean seed germination rates
Graphical analysis
Failure Profiles for Five Treatments
14
Treatment
Arasan
Check
Fermate
Semesan
Spergon
12
Failures
10
8
6
4
2
1
2
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
3
Block
4
5
Lecture 4.1
5
© 2015 Michael Stuart
Summary
• 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
• 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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
6
© 2015 Michael Stuart
Summary
Best variety?
• Fermate best in Blocks 3, 4, 5
Arasan and Semesan best in Blocks 1, 2
Next steps?
• Further investigation of Fermate in Blocks 1 and 2
indicated
– potential for gain in understanding
• Possibly investigate Spergon in Blocks 3 and 4
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
7
© 2015 Michael Stuart
Was blocking effective?
Analysis of Variance for Failures
Source
Treatment
Block
Error
Total
DF
4
4
16
24
Adj SS
83.840
49.840
86.560
220.240
Adj MS
20.960
12.460
5.410
F
3.87
2.30
P
0.022
0.103
Analysis of Variance for Failures
Source
Treatment
Error
Total
DF
4
20
24
Adj SS
83.840
136.400
220.240
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Adj MS
20.960
6.820
F
3.07
P
0.040
Lecture 4.1
8
© 2015 Michael Stuart
Was blocking effective?
Exceptional case deleted:
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
P
S = 1.61142
Source
DF
Adj SS
Adj MS
F
4
94.358
23.590
3.63
Error
19
123.600
6.505
Total
23
217.958
Treatment
0.023
S = 2.55054
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
9
© 2015 Michael Stuart
Was blocking effective?
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
10
© 2015 Michael Stuart
Test for interaction?
Analysis of Variance for Rate, using Adjusted SS for Tests
Source
Block
Treatment
Block*Treatment
Error
Total
DF
4
4
16
0
24
Seq SS
49.8400
83.8400
86.5600
*
220.2400
DF
4
4
16
24
Seq SS
83.840
49.840
86.560
220.240
Adj SS
49.8400
83.8400
86.5600
*
Adj MS
12.4600
20.9600
5.4100
*
F
**
**
**
P
Compare with:
Source
Treatment
Block
Error
Total
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Adj SS
83.840
49.840
86.560
Adj MS
20.960
12.460
5.410
F
3.87
2.30
P
0.022
0.103
Lecture 4.1
11
© 2015 Michael Stuart
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.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
12
© 2015 Michael Stuart
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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
13
© 2015 Michael Stuart
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.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
14
© 2015 Michael Stuart
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.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
15
© 2015 Michael Stuart
Impurity levels in gm. per Kg. resulting from
varying levels of four two level factors
in a 24 design run in completely random order
Design Run
Catalyst
.
Point Order Concentration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2
6
12
4
1
7
14
3
8
10
15
11
16
9
5
13
5
7
5
7
5
7
5
7
5
7
5
7
5
7
5
7
Sodium
Agitation
Hydroxide
Temperature Impurity
Speed
Concentration
40
10
150
38
40
10
150
40
45
10
150
27
45
10
150
30
40
20
150
58
40
20
150
56
45
20
150
30
45
20
150
32
40
10
180
59
40
10
180
62
45
10
180
53
45
10
180
50
40
20
180
79
40
20
180
75
45
20
180
53
45
20
180
54
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
16
© 2015 Michael Stuart
Process improvement study to
reduce impurity
Normal Effects Plot
D
Factor
A
B
C
D
20
C
Effect
10
0
Name
CatCon
NaOHCon
Speed
Temp
BC
-10
B
-20
-2
-1
0
1
2
Score
Lenth's PSE = 1.125
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
17
© 2015 Michael Stuart
Process improvement study to
reduce impurity
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
18
© 2015 Michael Stuart
Process improvement study to
reduce impurity
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
19
© 2015 Michael Stuart
Lenth's analysis
Lenth's PSE, or pseudo standard error:
Given a sample of Normal values from N(0,s),
ŝ = 1.5×median(absolute values).
Given null effect estimates,
SE0 = 1.5×median(absolute values).
Given some non-null effects, > 2.5xSE0
PSE = 1.5×median(absolute values of the rest).
Effect critical value = tm/3,0.05 x PSE,
where m = number of effects.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
20
© 2015 Michael Stuart
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.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
21
© 2015 Michael Stuart
Process improvement study
Visualising the results
Main Effects Plot for Impurity
60
Mean
55
50
45
40
150
180
Temperature
Temperature
150
180
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Mean
Impurity
38.88
60.63
Lecture 4.1
22
© 2015 Michael Stuart
Process improvement study
Visualising the results
Interaction Plot
70
NaOHCon
40
45
Impurity
65
60
55
50
Speed
45
NaOH
40
10
40
45
10
49.75
40.00
20
67.00
42.25
20
Speed
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
23
© 2015 Michael Stuart
Process improvement study
Visualising the results
Interaction Plot
70
NaOHCon
40
45
Impurity
65
60
55
50
Speed
45
NaOH
40
10
40
45
10
49.75
40.00
20
67.00
42.25
20
Speed
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
24
© 2015 Michael Stuart
Process improvement study
Visualising the results
Interaction Plot
70
NaOHCon
40
45
Impurity
65
60
55
50
Speed
45
NaOH
40
10
40
45
10
49.75
40.00
20
67.00
42.25
20
Speed
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
25
© 2015 Michael Stuart
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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
26
© 2015 Michael Stuart
Full model
Effect
CatCon
NaOHCon
Speed
Temp
CatCon*NaOHCon
CatCon*Speed
CatCon*Temp
NaOHCon*Speed
NaOHCon*Temp
Speed*Temp
CatCon*NaOHCon*Speed
CatCon*NaOHCon*Temp
CatCon*Speed*Temp
NaOHCon*Speed*Temp
CatCon*NaOHCon*Speed*Temp
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Estimate
0.25
-17.25
9.75
21.75
0.50
-1.00
-1.00
-7.50
1.00
-0.50
1.75
-0.75
0.25
0.25
1.00
Lecture 4.1
27
© 2015 Michael Stuart
Reduced model
NaOHCon
Speed
Temp
NaOHCon*Speed
s = 1.74,
-17.25
9.75
21.75
-7.50
df = 11 = 15 − 4
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
28
© 2015 Michael Stuart
Projected model
NaOHCon
Speed
Temp
NaOHCon*Speed
NaOHCon*Temp
Speed*Temp
NaOHCon*Speed*Temp
s = 1.87,
-17.25
9.75
21.75
-7.50
1.00
-0.50
0.25
df = 8 = 15 − 7
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
29
© 2015 Michael Stuart
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.
Ref: Models for Experiments (Extra Notes)
Lab 1 Feedback
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
30
© 2015 Michael Stuart
Degrees of freedom
"Error" degrees of freedom relevant for t
– Lenth's formula
(Slide 20)
– check ANOVA table
– count estimated effects
– use replication structure
t3, .05 = 2.57
s = 2.25
t8, .05 = 2.31
s = 1.87
t11,.05 = 2.20
s = 1.74
(Slides 29, 30)
(Lecture 3.1,
Slide 11)
Ref: Models for Experiments (Extra Notes)
Lab 1 Feedback
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
31
© 2015 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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
32
© 2015 Michael Stuart
Part 2:
Introduction to
Fractional Factorial Designs
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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
33
© 2015 Michael Stuart
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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
34
© 2015 Michael Stuart
A 24 with 16 design points
Design
Point
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
A
B
C
D
–
+
–
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Lecture 4.1
35
© 2015 Michael Stuart
The first 8 design points
Design
Point
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
A
B
C
D
–
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Lecture 4.1
36
© 2015 Michael Stuart
The middle 8 design points
Design
Point
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
A
B
C
D
–
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Lecture 4.1
37
© 2015 Michael Stuart
Another 8 design points
Design
Point
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
A
B
C
D
–
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Lecture 4.1
38
© 2015 Michael Stuart
Same 8 design points, with 2fi’s
Design
A
Point
2
+
3
–
5
–
8
+
10
+
11
–
13
–
16
+
B
C
D AB AC AD BC BD CD
–
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–
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–
–
–
+
+
Confounded effects:
A = BC
B = AC
C = AB
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
39
© 2015 Michael Stuart
Same 8 design points, with 2fi’s
Design
A
Point
2
+
3
–
5
–
8
+
10
+
11
–
13
–
16
+
B
C
D AB AC AD BC BD CD
–
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–
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–
+
+
+
–
–
–
–
+
+
Confounded effects:
A = BC
B = AC
C = AB
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
40
© 2015 Michael Stuart
Design
Point
2
3
5
8
10
11
13
16
Same 8 design points,
with all interactions
A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
+
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–
–
+
–
+
Confounded effects:
AD
A = BC
BD
B = AC
CD
C = AB
I
D = ABCD
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
+
+
–
–
–
–
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=
=
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BCD
ACD
ABD
ABC
Lecture 4.1
41
© 2015 Michael Stuart
Design
Point
2
3
5
8
10
11
13
16
Same 8 design points,
with all interactions
A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
+
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–
+
–
+
Confounded effects:
AD
A = BC
BD
B = AC
CD
C = AB
I
D = ABCD
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
+
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=
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BCD
ACD
ABD
ABC
Lecture 4.1
42
© 2015 Michael Stuart
Clever design
Design
Point
1
2
3
4
5
6
7
8
A
B
C
–
+
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+
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D=
ABC
–
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Y
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
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.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
43
© 2015 Michael Stuart
Classwork
Exercise:
Confirm confounding patterns
Design A= B= C= D=
Y
Point BCD ACD ABD ABC
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?
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
44
© 2015 Michael Stuart
Fractional factorial designs
Full factorial design
First half fraction
Design
Point
1
2
3
4
5
6
7
8
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
+
+
–
+
–
–
+
70
62
88
81
60
49
88
79
Identify corresponding
design points
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Design
Point
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
–
–
–
+
+
+
+
–
–
–
–
–
–
–
–
+
+
+
+
+
+
+
+
70
60
89
81
60
49
88
82
69
62
88
81
60
52
86
79
Lecture 4.1
45
© 2015 Michael Stuart
Fractional factorial designs
Full factorial design
First half fraction
Design
Point
1
2
3
4
5
6
7
8
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
+
+
–
+
–
–
+
70
62
88
81
60
49
88
79
Identify corresponding
design points
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Design
Point
1 1
2
3
4 4
5
6 6
7 7
8
9
2 10
3 11
12
5 13
14
15
8 16
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
–
–
–
+
+
+
+
–
–
–
–
–
–
–
–
+
+
+
+
+
+
+
+
70
60
89
81
60
49
88
82
69
62
88
81
60
52
86
79
Lecture 4.1
46
© 2015 Michael Stuart
Fractional factorial designs
First half fraction
Design
Point
1
10
11
4
13
6
7
16
Second half fraction
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
+
+
–
+
–
–
+
70
62
88
81
60
49
88
79
Column A estimates A + BCD
Design
Point
9
1
2
12
5
14
15
8
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
+
–
–
+
–
+
+
–
69
60
89
81
60
52
86
82
Column A estimates A – BCD
Full 24 design: Column A estimates ½[(A + BCD) + (A – BCD)] = A
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
47
© 2015 Michael Stuart
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.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
48
© 2015 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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
49
© 2015 Michael Stuart
Blocking Factorials Designs
In multifactor experiments
requiring
several runs in inhomogeneous conditions,
fractions may be used as blocks.
Block effects are aliased with suitable high level
interactions.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
50
© 2015 Michael Stuart
Blocking a 24 experiment
Second half fraction
First half fraction
Design
Point
1
10
11
4
13
6
7
16
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
–
+
+
–
+
–
–
+
70
62
88
81
60
49
88
79
Block 1: ABCD = +
Design
Point
9
1
2
12
5
14
15
8
A
B
C
D
Y
–
+
–
+
–
+
–
+
–
–
+
+
–
–
+
+
–
–
–
–
+
+
+
+
+
–
–
+
–
+
+
–
69
60
89
81
60
52
86
82
Block 2: ABCD = –
ABCD effect confounded with block difference
All other effects unconfounded, estimated separately
within blocks
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
51
© 2015 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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
52
© 2015 Michael Stuart
Introduction to Split Plots designs
• The first ever split plots design? (Fisher, 1925)
• Think of Broadbalk (Lecture 1.2, slide 60, Notes p.14)
© Rothamsted Research
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
53
© 2015 Michael Stuart
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.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
54
© 2015 Michael Stuart
Introduction to Split Plots designs
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
55
© 2015 Michael Stuart
Introduction to Split Plots designs
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
56
© 2015 Michael Stuart
Introduction to Split Plots designs
Varieties:
Treatments:
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
Basal manure dressing,
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Manure with added
Potassium Sulphate
Manure with added
Potassium Chloride.
Lecture 4.1
57
© 2015 Michael Stuart
Field layout
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
58
© 2015 Michael Stuart
Whole Plots Numbered
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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
59
© 2015 Michael Stuart
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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Whole Plots
1
8
24
5
10
4
9
2
22
3
26
21
13
20
25
7
12
6
11
14
28
15
33
23
32
34
35
19
27
18
31
16
29
17
36
30
Lecture 4.1
60
© 2015 Michael Stuart
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
Plot 1 total = 8.57
Plot 13 total = 8.79
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Plot 32 total = 12.70
Lecture 4.1
61
© 2015 Michael Stuart
Analysis of Split Plots design
• Varieties vary between whole plots,
– variety effects evaluated with reference to
chance variation between whole plots
• Treatments vary between subplots
– treatment effects evaluated with reference to
chance variation between subplots
• Implications for Analysis of Variance
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
62
© 2015 Michael Stuart
Analysis of Varieties in Whole Plots
Variety
Plot
1
8
24
5
10
4
9
2
22
3
26
21
Ajax
Arran Comrade
British Queen
Duke of York
Epicure
Great Scot
Iron Duke
K. of K.
Kerr's Pink
Nithsdale
Tinwald Perfection
Up-to-Date
Whole Plot Yields
Yield
Plot
Yield
8.57
13
8.79
6.63
20
6.88
8.67
25
8.25
4.29
7
5.25
5.90
12
5.82
9.24
6
9.86
10.26
11
8.40
9.99
14
12.90
8.00
28
9.75
6.57
15
8.00
8.84
33
9.46
11.89
23
10.25
Plot
32
34
35
19
27
18
31
16
29
17
36
30
Yield
12.70
6.85
11.31
6.46
7.72
11.53
10.57
12.74
12.02
10.85
9.07
12.43
One-way ANOVA: Yield versus Variety
Source
Variety
Error
Total
DF
11
24
35
SS
130.915
52.320
183.2355
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
MS
11.901
2.180
F
5.46
P
0.000
Lecture 4.1
63
© 2015 Michael Stuart
Model for ANOVA
Yield
NB:
equals
Overall Mean
plus
Variety effect
plus
Whole-plot effect
(i.e., chance variation)
Whole-plot is a nested factor, that is
each of the 36 levels of Whole-plot occurs
with
just one level of Variety.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
64
© 2015 Michael Stuart
Whole Plots Nested
Varieties:
Whole Plot Numbers
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
1
8
24
5
10
4
9
2
22
3
26
21
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
13
20
25
7
12
6
11
14
28
15
33
23
32
34
35
19
27
18
31
16
29
17
36
30
Lecture 4.1
65
© 2015 Michael Stuart
A crossed design
Food Type
Pot Type
Meat
Legumes
Vegetables
Aluminium 1.77 2.36 1.96 2.14 2.40 2.17 2.41 2.34 1.03 1.53 1.07 1.30
Clay
2.27 1.28 2.48 2.68 2.41 2.43 2.57 2.48 1.55 0.79 1.68 1.82
Iron
5.27 5.17 4.06 4.22 3.69 3.43 3.84 3.72 2.45 2.99 2.80 2.92
With crossed factors, every level of each factor
occurs with every level of every other factor
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
66
© 2015 Michael Stuart
Minitab models
Model: Variety
Source
Variety
Error
Total
Model: Variety
DF
11
24
35
SS
130.915
52.320
183.2355
MS
11.901
2.180
F
5.46
P
0.000
Whole Plot(Variety)
Source
Variety
Whole Plot(Variety)
Error
Total
DF
11
24
0
35
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
SS
130.9152
52.3203
*
183.2355
MS
11.9014
2.1800
*
F
5.46
**
*
P
0.000
Lecture 4.1
67
© 2015 Michael Stuart
Full analysis
Model:
Variety 'Whole Plot' (Variety)
Fertiliser Variety* Fertiliser
Source
DF
SS
MS
F
P
Variety
11
43.6384
3.9671
5.46
0.000
Whole Plot(Variety)
24
17.4401
0.7267
4.32
0.000
2
0.3495
0.1748
1.04
0.362
Variety*Fertiliser
22
2.1911
0.0996
0.59
0.909
Error
48
8.0798
0.1683
Total
107
Fertiliser
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
71.6989
Lecture 4.1
68
© 2015 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
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
69
© 2015 Michael Stuart
Another illustration
• Testing water resistance of four wood stains
• Stains applied to four panels cut from a board
• Boards are pretreated with one of two treatments.
• Ideal:
• crossed two-factor design
– eight Stain / Pretreatment combinations
– one applied to each of 8 panels
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
70
© 2015 Michael Stuart
Another illustration
• Problem:
– Pretreatments can only be applied to whole board
• 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
71
© 2015 Michael Stuart
Another illustration
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
72
© 2015 Michael Stuart
Another illustration
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
73
© 2015 Michael Stuart
Results
Stain
1
Stain
2
Stain
3
Stain
4
Pretreatment 1
43.0
57.4
52.8
51.8
60.9
59.2
40.8
51.1
51.7
45.5
55.3
55.3
Pretreatment 2
46.6
52.2
32.1
53.5
48.3
34.4
35.4
45.9
32.2
32.5
44.6
30.1
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
74
© 2015 Michael Stuart
Minitab Analysis
Model:
Pretreat Board(Pretreat)
Stain Pretreat * Stain
Source
DF
SS
MS
F
P
Pretreat
Board(Pretreat)
1
4
782.04
775.36
782.04
193.84
4.03
15.25
0.115
0.000
Stain
Pretreat*Stain
Error
3
3
12
266.00
62.79
152.52
88.67
20.93
12.71
6.98
1.65
0.006
0.231
Total
23
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
2038.72
Lecture 4.1
75
© 2015 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
76
© 2015 Michael Stuart
Reading
EM §5.7, §7.4 for fractional factorial designs and
blocking
Lecture Notes:
Introduction to Split Units Design
and Analysis, pages 1-7, 8-11
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
Lecture 4.1
77
© 2015 Michael Stuart