Design and Analysis of Experiments
Lecture 6.1
1. Review of split unit experiments
2. Review of Laboratory 2
3. Review of special topics (part)
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Minute Test: How Much
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Minute Test: How Fast
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Split units experiments
arise when
– one set of treatment factors is applied to
experimental units,
– a second set of factors is applied to sub units
of these experimental units.
originated in agriculture where they are referred to as
split plot designs.
"Most industrial experiments are ... split plot in their
design.“ C. Daniel (1976) p. 175
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Reasons for using split units
• Adding another factor after the experiment
started
• Changing one factor is
– more difficult
– more expensive
– more time consuming
•
than changing others
• Some factors require better precision than others
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Recognising Plot and Treatment Structure
Factor
Units
Blocks
Whole unit
Treatment
Subunit
Treatment
Whole units
Subunits
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ANOVA
MS(Blocks)
MS(WTreatments)
MS(Whole units)
MS(B x WT)
MS(STreatments)
MS(Interactions)
MS(Subunits)
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© 2012 Michael Stuart
Illustration: Water resistance of wood stains
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Plot structure, Treatment structure
Factor
Units
Pretreatment
Boards
Stain
Panels
24 subunits (panels) nested in 6 whole units (boards).
2 pretreatments allocated to whole units.
4 stains allocated to subunits within whole units.
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Assessing variation
• Variation between boards due to
– chance
– Pretreatments?
• Variation between panels due to
– chance
– stains?
– pretreatment by stain interaction?
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Analysis of Variance
Factor
ANOVA
Units
Pretreatment
Boards
MS(Pretreatment)
MS(Boards)
Stain
Panels
MS(Stain)
MS(Interaction)
MS(Panels)
Minitab model
Pretreatment Board(Pretreatment)
Stain Pretreatment*Stain
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Analysis of Variance
Source
DF
SS
MS
F
P
Pretreatment
Board(Pretreatment)
1
4
782.04
775.36
782.04
193.84
4.03
15.25
0.115
0.000
Stain
Pretreatment*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
2038.72
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© 2012 Michael Stuart
Justifying the ANOVA
1
2
3
4
5
Source
Pretreat
Board(Pretreat)
Stain
Pretreat*Stain
Error
Expected Mean Square
(5) + 4.0000 (2) + Q[1]
(5) + 4.0000 (2)
(5) + Q[3]
(5) + Q[4]
(5)
Alternative notation:
Source
Expected Mean Square
1
Pretreat
2
Board(Pretreat)
3
Stain
P2 + 4  B2 + Pretreatment effect
P2 + 4  B2
P2 + Stain effect
4
Pretreat*Stain
P2 + interaction effect
5
Error
P2
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Extending the unit structure
Suppose the 6 boards were in 3 blocks of 2
e.g. 2 boards selected from 3 production runs,
e.g. 2 boards treated on 3 successive days
B
1
1
1
1
Block 1
P
S
1
1
1
2
1
3
1
4
R
43.0
51.8
40.8
45.5
4
4
4
4
2
2
2
2
46.6
53.5
35.4
32.5
1
2
3
4
B
2
2
2
2
Block 2
P
S
1
1
1
2
1
3
1
4
R
57.4
60.9
51.1
55.3
5
5
5
5
2
2
2
2
52.2
48.3
45.9
44.6
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1
2
3
4
B
3
3
3
3
Block 3
P
S
1
1
1
2
1
3
1
4
R
52.8
59.2
51.7
55.3
6
6
6
6
2
2
2
2
32.1
34.4
32.2
30.1
1
2
3
4
Lecture 6.1
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© 2012 Michael Stuart
Expanded Unit and Treatment Structure
Factor
Pretreatment
Stain
Units
ANOVA
Blocks
MS(Blocks)
Boards
Panels
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MS(Pretreatment)
MS(Boards)
MS(B x PT)
MS(Stain)
MS(Interactions)
MS(Panels)
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© 2012 Michael Stuart
Analysis of Variance
Minitab model
Block
Pretreatment Block*Pretreatment
Stain Block*Stain Pretreatment*Stain
Source
DF
Block
2
376.99
188.49
0.95
0.514
Pretreatment
Block*Pretreatment
1
2
782.04
398.38
782.04
199.19
3.93
15.67
0.186
0.000
Stain
Pretreatment*Stain
Error
3
3
12
266.01
62.79
152.52
88.67
20.93
12.71
6.98
1.65
0.006
0.231
Total
23 2038.72
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SS
MS
F
P
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© 2012 Michael Stuart
Extending the treatment structure
4 Stain levels ↔ two 2-level factors:
Stain type
1 or 2
number of Coats applied
1 or 2
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Expanded Unit and Treatment Structure
Factor
Pretreatment
Stain x Coat
Units
ANOVA
Blocks
MS(Blocks)
Boards
MS(Pretreatment)
MS(B x PT)
Panels
MS(Stain)
MS(Coat)
MS(Interactions)
MS(Panels)
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Minitab model
Block
Pretreatment
Block*Pretreatment
Stain Coat Stain*Coat
Pretreatment*Stain Pretreatment*Coat
Pretreatment*Stain*Coat
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Analysis of Variance
Source
DF
SS
MS
F
P
Block
Pretreatment
Block*Pretreatment
2
1
2
376.99
782.04
398.38
188.49
782.04
199.19
0.95
3.93
15.67
0.514
0.186
0.000
Stain
Coat
Stain*Coat
1
1
1
38.00
214.80
13.20
38.00
214.80
13.20
2.99
16.90
1.04
0.109
0.001
0.328
Pretreatment*Stain
Pretreatment*Coat
Pretreatment*Stain*Coat
Error
1
1
1
12
43.20
18.38
1.21
152.52
43.20
18.38
1.21
12.71
3.40
1.45
0.10
0.090
0.252
0.762
Total
23 2038.72
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© 2012 Michael Stuart
Laboratory 2, Exercise 1:
Soup mix packet filling machine
Questions:
What factors affect soup powder fill variation?
How can fill variation be minimised?
Potential factors
A:
B:
C:
D:
E:
Number of ports for adding oil,
1 or 3,
Mixer vessel temperature,
ambient or cooled,
Mixing time,
60 or 80 seconds,
Batch weight,
1500 or 2000 lbs,
Delay between mixing and packaging, 1 or 7 days.
Response: Spread of weights of 5 sample packets
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Minitab analysis
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Minitab analysis
Normal plot vs Pareto Principle vs Lenth?
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Minitab analysis
Estimated Effects for Y
Term
E
Effect
Alias
-0.470
E + A*B*C*D
B*E
0.405
B*E + A*C*D
D*E
-0.315
D*E + A*B*C
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Graphical and numerical summaries
Interaction Plot for Y
Interaction Plot for Y
B
+
1.7
1.6
1.5
1.6
1.5
1.4
Mean
1.4
Mean
D
+
1.7
1.3
1.2
1.3
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
-
+
-
E
B
+
E
E
E
–
–
+
1.71 0.83
–
–
+
1.31 1.17
+
1.22 1.15
+
1.60 0.82
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D
Lecture 6.1
24
© 2012 Michael Stuart
Best conditions
Interaction Plot for Y
Interaction Plot for Y
B
+
1.7
1.6
1.5
1.6
1.5
1.4
Mean
1.4
Mean
D
+
1.7
1.3
1.2
1.3
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8
-
+
-
E
+
E
Best conditions:
B Low, D High, E High.
Best conditions with E Low:
B High, D Low.
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Reduced model
Fit model using active terms:
B + D + E + BE + DE
Pareto Chart of the Standardized Effects
2.262
E
Term
BE
DE
B
D
0
1
2
3
4
5
6
7
Standardized Effect
DE confirmed as active.
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© 2012 Michael Stuart
Diagnostics
Diagnostic Plot
Deleted Residual
2
1
0
-1
-2
-3
0.8
1.0
1.2
1.4
1.6
1.8
Fitted Value
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Diagnostics
Normal Probability Plot
Deleted Residual
3
2
1
0
-1
-2
-3
-2
-1
0
1
2
Score
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Delete Design point 5, iterate analysis
• Effect estimates similar
• Interaction patterns similar
• s = 0.15,
df = 9 ( = 14 – 5 )
Least Squares Means for Y
Mean SE Mean
B*D*E
- - - 1.7000
0.1532
+ - - 1.2050
0.1083 1.205  2.26×0.15/√2 = 0.965 to 1.445
- + - 1.9750
0.1083
+ + - 1.2250
0.1083
- - + 0.9750
0.1083
+ - + 1.3600
0.1083
0.69  2.26×0.15/√2 = 0.37 to 1.01
- + + 0.6900
0.1083
+ + + 0.9400
0.1083
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Laboratory 2, Exercise 2
Cambridge Grassland Experiment
3 grassland treatments
Rejuvenator
R
Harrow
H
no treatment
C
randomly allocated to 3 neighbouring plots,
replicated in 6 neighbouring blocks
4 fertilisers
Farmyard manure
F
Straw
S
Artificial fertiliser
A
no fertiliser
C
randomly allocated to 4 sub plots within each plot.
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Cambridge Grassland Experiment
Blocks
Whole Plots
Treatments
1
H
1
2
C
1
H
2
2
R
3
R
Sub Plot 1
C
A
Sub Plot 2
A
Sub Plot 3
Sub Plot 4
1
C
3
2
H
3
C
A
C
F
S
C
A
F
C
F
S
F
S
3
R
F
A
A
S
A
C
F
C
C
S
A
S
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1
H
4
2
R
1
C
5
2
H
3
C
A
A
F
C
F
F
S
F
S
F
S
C
1
C
6
2
R
3
R
3
H
F
F
C
A
F
F
C
A
S
S
A
S
A
S
S
C
S
A
C
S
C
C
C
F
S
C
C
A
F
F
S
A
A
Lecture 6.1
31
© 2012 Michael Stuart
Randomised Blocks analysis for
Treatments
Source
DF
SS
MS
F
P
WB
5
149700
29940
15.99
0.000
WT
2
49884
24942
13.32
0.002
WB*WT
10
18725
1872
**
Error
0
Total
17
Model:
*
*
218309
WB + WT + WBxWT
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© 2012 Michael Stuart
Diagnostics
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© 2012 Michael Stuart
Diagnostics
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© 2012 Michael Stuart
WB x WT Interaction Plot
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WT x WB Interaction Plot
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© 2012 Michael Stuart
Split Plots Analysis
Model:
B+T+BxT
+F+BxF+TxF
Source DF
SS
B
5 37425.1
T
2 12471.0
B*T
10
4681.1
MS
7485.0
6235.5
468.1
F
21.37
13.32
1.94
56022.7 18674.2
1852.1
123.5
781.5
130.3
7239.6
241.3
151.24
0.51
0.54
F
B*F
T*F
Error
3
15
6
30
Total
71 120473.3
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P
0.002 x
0.002
0.079
0.000
0.914
0.774
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© 2012 Michael Stuart
Diagnostics
Residuals Versus Fitted Values
Deleted Residual
2
1
0
-1
-2
-3
100
150
200
250
Fitted Value
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Same diagnostic, Different interpretation?
Residuals Versus Fitted Values
Deleted Residual
2
1
0
-1
-2
-3
100
150
200
250
Fitted Value
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Treatment comparisons
• First step: summary statistics
Variable
Y
s2
=
Treatment
C
H
R
Count
24
24
24
MS(B*T) = 468.1;
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Mean
181.46
150.96
157.17
df = 10
Lecture 6.1
40
© 2012 Michael Stuart
Compare Harrow with Control
YH  YC  150.96  181.46  30.5
s2 s2
SE 


nH nC
t
468.1 468.1

 6.25
24
24
 30.5
 4.88
6.25
t 10 ,.05  2.23
difference is statistica lly significan t
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© 2012 Michael Stuart
Compare Harrow with Control
YH  YC  150.96  181.46  30.5
s2 s2
SE 


nH nC
t
468.1 468.1

 6.25
24
24
 30.5
 4.88
6.25
t 10 ,.05  2.23
difference is statistica lly significan t
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© 2012 Michael Stuart
Compare Harrow with Control
YH  YC  150.96  181.46  30.5
s2 s2
SE 


nH nC
468.1 468.1

 6.25
24
24
 30.5
t
 4.88
6.25
t 10 ,.05  2.23
difference is statistica lly significan t
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© 2012 Michael Stuart
Compare Harrow with Control
YH  YC  150.96  181.46  30.5
s2 s2


SE 
nH nC
468.1 468.1
 6.25

24
24
 30.5
 4.88
t
6.25
t 10 ,.05  2.23
difference is statistica lly significan t
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© 2012 Michael Stuart
Compare Harrow with Control
YH  YC  150.96  181.46  30.5
s2 s2
SE 


nH nC
468.1 468.1

 6.25
24
24
 30.5
t
 4.88
6.25
t 10 ,.05  2.23
difference is statistica lly significan t
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© 2012 Michael Stuart
Laboratory 2, Exercise 3
Fertiliser experiment
Fertilisers potentially affecting bean yield
Dung (D):
Nitrochalk (N):
SuperPhosphate (P):
Muriate of Potash (K):
Low
none
none
none
none
High
10 tons per acre
0.4 cwt per acre
0.6 cwt per acre
1.0 cwt per acre
Questions:
What factors affect bean yield?
How can bean yield be maximised?
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© 2012 Michael Stuart
Exercise 2,
Minitab analysis
Pareto Chart of the Effects
Normal Plot of the Effects
Factor
A
B
C
D
B
BCD
AC
D
BC
ACD
ABC
ABD
BD
AB
A
CD
C
AD
Name
D
N
P
K
Effect Type
Not Significant
Significant
5
Effect
Term
7.880
0
-5
B
0
2
4
6
8
Effect
Lenth's PSE = 3
-10
-2
-1
0
1
2
Score
N is marginally significant.
Pareto Principle suggests adding NPK and DP.
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© 2012 Michael Stuart
Term
Constant
Block
D
N
P
K
D*N
D*P
D*K
N*P
N*K
P*K
D*N*P
D*N*K
D*P*K
N*P*K
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Effect
-0.75
-0.750
-8.000
0.250
-2.250
1.250
4.500
0.000
2.250
-1.750
0.500
-2.000
2.000
2.250
-5.500
Coef
47.250
-0.375
-0.375
-4.000
0.125
-1.125
0.625
2.250
0.000
1.125
-0.875
0.250
-1.000
1.000
1.125
-2.750
Lecture 6.1
48
© 2012 Michael Stuart
Reduced model
Fit model using active terms:
D, N, P, K, DP, NP, NK, PK, NPK
Pareto Chart of the Standardized Effects
Term
2.447
Factor
A
B
C
D
B
BCD
AC
D
BC
BD
A
CD
C
0
1
2
3
4
Name
D
N
P
K
5
Standardized Effect
Active effects confirmed.
Diagnostics unremarkable
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© 2012 Michael Stuart
3-factor interaction
Interaction Plot for Y
N Low
56
56
P
+
54
52
50
48
46
52
50
48
46
44
44
42
42
40
40
-
P
+
54
Mean
Mean
Interaction Plot for Y
N High
+
K
-
+
K
Adding N reduces yield overall, but has a small positive effect at
high P and low K.
At low N, the P effect is negative at low K and positive at high K.
At high N, this interaction is reversed.
The best combination is no fertiliser.
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© 2012 Michael Stuart
Least Squares Means for Y
N*P*K
- - + - - + + + - - +
+ - +
- + +
+ + +
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Mean
SE Mean
55.5
41.5
47.5
49.0
49.0
42.5
53.0
40.0
2.419
2.419
2.419
2.419
2.419
2.419
2.419
2.419
Lecture 6.1
51
© 2012 Michael Stuart
Best combination
N*P*K
Best:
- - Next best: - + +
Mean
55.5
53.0
SE
2.419
2.419
Difference:
2.5
3.42
t:
2.5/3.42 = 0.73,
not statistically significantly different.
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© 2012 Michael Stuart
Review topics (part)
• Time related issues
– repeated measures
– cross-over designs
• Complex block structures
• Analysis of Covariance
• Robustness studies
• Response surface designs
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© 2012 Michael Stuart
Time related issues
1.
Repeated measures
Example:
Calves fed diet supplements to improve growth
Initial weight recorded,
Y0
blocks formed based on initial weight,
weights recorded at
4 weeks,
8 weeks
12 weeks
16 weeks
Y1
Y2
Y3
Y4
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© 2012 Michael Stuart
Split plots analysis?
• Calves are whole units
• Time periods are sub units
Problems:
correlation structure
varying standard deviation
Solution:
Multivariate analysis
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© 2012 Michael Stuart
Time related issues
2.
Crossover designs
• Repeated measures designs compares diets on
different calves,
• reduce variation by comparing diets on same
calves,
• e.g.
diet A for weeks 1 to 4
diet B for weeks 5 to 8
diet C for weeks 9 to 12
diet D for weeks 13 to 16
• requires attention to order of diets
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
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© 2012 Michael Stuart
Crossover design
Calf
1
2
3
4
1-4
A
B
C
D
• Every diet occurs
Time Period
5 – 8 9 – 12
B
C
D
A
A
D
C
B
13 - 16
D
C
B
A
? correlation structure
– once for each calf,
? carry over
– once in each time
period
? experimental set up
versus actual use
– Latin square
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
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© 2012 Michael Stuart
Complex blocking
• 2 blocking factors
– calves and time periods
– Latin square
– Latin rectangle
• Incomplete blocks
– more treatments than plots in a block
– balanced incomplete blocks
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
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© 2012 Michael Stuart
Analysis of Covariance
Objective: take account of variation in uncontrolled
environmental variables.
Solution: measure the environmental variables at
each design point and incorporate in the
analysis through regression methods
(Analysis of Covariance)
Effects:
reduces "error" variation, makes factor
effects more significant
adjusts factor effect estimates to take
account of extra variation source.
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
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© 2012 Michael Stuart
Analysis of Covariance; Illustration
Breaking strength of monofilament fibre (Y)
produced by three different machines (1, 2, 3)
allowing for variation in fibre thickness (X)
Machine 1
Machine 2
Machine 3
Y
36
41
39
42
49
Y
40
48
39
45
44
Y
35
37
42
34
32
X
20
25
24
25
32
Diploma in Statistics
Design and Analysis of Experiments
X
22
28
22
30
28
X
21
23
26
21
15
Lecture 6.1
60
© 2012 Michael Stuart
Analysis of Covariance; Minitab
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
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© 2012 Michael Stuart
Analysis of Covariance; Minitab
General Linear Model: Y versus Machine
Source
X
Machine
Error
Total
DF
1
2
11
14
Seq SS
305.13
13.28
27.99
346.40
Adj SS
178.01
13.28
27.99
Adj MS
178.01
6.64
2.54
F
69.97
2.61
P
0.000
0.118
S = 1.59505
One-way ANOVA: Y versus Machine
Source
Machine
Error
Total
DF
2
12
14
SS
140.4
206.0
346.4
MS
70.2
17.2
F
4.09
P
0.044
S = 4.143
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
62
© 2012 Michael Stuart
Analysis of Covariance; Minitab
Scatterplot of Y vs X
50
Machine
1
2
3
Y
45
40
35
30
15.0
17.5
20.0
Diploma in Statistics
Design and Analysis of Experiments
22.5 25.0
X
27.5
30.0
32.5
Lecture 6.1
63
© 2012 Michael Stuart
Covariance vs Blocking
Chance causes and assignable causes of variation
(W. Shewhart, 1931)
Chance causes of variation are the
many individually negligible and unpredictable
but
collectively influential
factors that affect a process or system.
Assignable causes of variation are the
few individually influential and predictable effect
factors that affect a process or system.
Diploma i Statistics
Design and Analysis of Experiments
Lecture 6.1
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© 2012 Michael Stuart
Covariance vs Blocking
Blocking
Covariance
Diploma i Statistics
Design and Analysis of Experiments
Chance causes
Assignable causes
Lecture 6.1
65
© 2012 Michael Stuart
Robustness Studies
Seek optimal settings of experimental factors
that remain optimal,
irrespective of uncontrolled environmental factors.
Run the experimental design, the inner array,
at fixed settings of the environmental variables, the
outer array.
Popularised by Taguchi.
Improved by Box et al
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
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© 2012 Michael Stuart
Study of Detergent Robustness
Product
1
2
3
4
5
6
7
8
Design factors
A
B
C
D
–
–
–
–
+
–
–
+
–
+
–
+
+
+
–
–
–
–
+
+
+
–
+
–
–
+
+
–
+
+
+
+
Diploma in Statistics
Design and Analysis of Experiments
Environmental factors
T
–
+
–
+
H
–
–
+
+
R
+
–
–
+
i
ii
iii
iv
88 85 88 85
80 77 80 76
90 84 91 86
95 87 93 88
84 82 83 84
85 84 82 82
91 93 92 92
89 88 89 87
Mean
86.50
78.25
87.75
90.75
83.25
83.25
92.00
88.25
Range
3
4
7
8
2
3
2
2
Lecture 6.1
67
© 2012 Michael Stuart
Study of Detergent Robustness
Product
1
2
3
4
5
6
7
8
Design factors
A
B
C
D
–
–
–
–
+
–
–
+
–
+
–
+
+
+
–
–
–
–
+
+
+
–
+
–
–
+
+
–
+
+
+
+
Diploma in Statistics
Design and Analysis of Experiments
Environmental factors
T
–
+
–
+
H
–
–
+
+
R
+
–
–
+
i
ii
iii
iv
88 85 88 85
80 77 80 76
90 84 91 86
95 87 93 88
84 82 83 84
85 84 82 82
91 93 92 92
89 88 89 87
Mean
86.50
78.25
87.75
90.75
83.25
83.25
92.00
88.25
Range
3
4
7
8
2
3
2
2
Lecture 6.1
68
© 2012 Michael Stuart
Study of Detergent Robustness
Product
1
2
3
4
5
6
7
8
Design factors
A
B
C
D
–
–
–
–
+
–
–
+
–
+
–
+
+
+
–
–
–
–
+
+
+
–
+
–
–
+
+
–
+
+
+
+
Diploma in Statistics
Design and Analysis of Experiments
Environmental factors
T
–
+
–
+
H
–
–
+
+
R
+
–
–
+
i
ii
iii
iv
88 85 88 85
80 77 80 76
90 84 91 86
95 87 93 88
84 82 83 84
85 84 82 82
91 93 92 92
89 88 89 87
Mean
86.50
78.25
87.75
90.75
83.25
83.25
92.00
88.25
Range
3
4
7
8
2
3
2
2
Lecture 6.1
69
© 2012 Michael Stuart
Split plots model analysis
B significant, positive,
T and TC interaction
set at high (+) level
significant
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
70
© 2012 Michael Stuart
Split plots model analysis
At low C, whiteness is highly sensitive to T.
At high C, whiteness is relatively insensitive to T.
Diploma in Statistics
Design and Analysis of Experiments
Lecture 6.1
71
© 2012 Michael Stuart
Conclusion
• Set B and C to high levels, A and D as convenient
Product
1
2
3
4
5
6
7
8
Design factors
A
B
C
D
–
–
–
–
+
–
–
+
–
+
–
+
+
+
–
–
–
–
+
+
+
–
+
–
–
+
+
–
+
+
+
+
Diploma in Statistics
Design and Analysis of Experiments
Environmental factors
T
–
+
–
+
H
–
–
+
+
R
+
–
–
+
i
ii
iii
iv
88 85 88 85
80 77 80 76
90 84 91 86
95 87 93 88
84 82 83 84
85 84 82 82
91 93 92 92
89 88 89 87
Mean
86.50
78.25
87.75
90.75
83.25
83.25
92.00
88.25
Range
3
4
7
8
2
3
2
2
Lecture 6.1
72
© 2012 Michael Stuart