Design and Analysis of Experiments Lecture 5 · Lecture 5.1 3 © 2016 Michael Stuart How Fast...
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Lecture 5.1 1
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
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Lecture 5.1 2
© 2016 Michael Stuart
How Much
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Lecture 5.1 3
© 2016 Michael Stuart
How Fast
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5432
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Lecture 5.1 4
© 2016 Michael Stuart
Review of Lecture 4.2
• Unit structure with no treatments
– Hierarchical Nesting vs Crossing
– Fixed and Random Factor Effects
– Implications for design and analysis
• Split Units Analysis
– right and wrong models
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Lecture 5.1 5
© 2016 Michael Stuart
Values
Values
m
•
sS
sT
eP
eS
sP
eT
e = eP + eS + eT
Model for variation in moisture content
Process
variation
Sampling
variation
Testing
variation
y
•
•
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Lecture 5.1 6
© 2016 Michael Stuart
Model for variation in moisture content
Basic model:
Y = m + eP + eS + eT
Components of variance:
2
T
2
S
2
P
2
Y ssss
2
T
2
S
2
PY ssss
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Lecture 5.1 7
© 2016 Michael Stuart
Aside on Nesting and Crossing
• Two factors are crossed when
each level of one factor occurs with
every level of the other factor.
– usually concerns treatment factors
• A factor is nested in another factor (the nesting
factor) if
each level of the nested factor occurs with
exactly one level of the nesting factor.
– usually concerns unit / plot factors.
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Lecture 5.1 8
© 2016 Michael Stuart
Recall Iron Content of Food study
• 4 samples of each of 3 food types were cooked in
each of 3 pot types,
• iron content in each sample measured in
milligrams of iron per 100 grams of cooked food.
Pot Type Food 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
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Lecture 5.1 9
© 2016 Michael Stuart
Hierachical / Nested Structure
Diploma in Statistics
Design and Analysis of Experiments
Batches
Samples
Subsamples
(Tests)
1
1
1 2
2 3 4
2
5
3 4
6 7 8
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Lecture 5.1 10
© 2016 Michael Stuart
Aside on Fixed and Random Factor Effects
Experimental factor variation usually has fixed effect,
e.g., Temperature, Pressure, Pot Type, Food Type,
Experimental unit variation usually has random effect,
e.g., Batch, Sample, Board, Panel,
(with exceptions)
Implications for Analysis of Variance
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Lecture 5.1 12
© 2016 Michael Stuart
Units
Batches
Samples
Tests
Unit / Treatment Structure Diagram
Factor
Process
Sampling
method
Test method
ANOVA
MS(Process)
MS(Batch variation)
MS(Sampling method)
MS(P x S)
MS(Sampling variation)
MS(Test method)
MS(Interactions)
MS(Testing error) Postgraduate Certificate in Statistics
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Lecture 5.1 13
© 2016 Michael Stuart
Split units design:
Right and Wrong Analysis
• Testing water resistance of four wood stains
• Stains applied to four panels cut from a board
• Boards are pretreated with one of two treatments.
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Lecture 5.1 14
© 2016 Michael Stuart
Assessing variation
• Variation between boards due to
– chance
– Pretreatments?
• Variation between panels due to
– chance
– stains?
– pretreatment by stain interaction?
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Lecture 5.1 15
© 2016 Michael Stuart
Results
Pretreatment 1 Pretreatment 2
Board 1 2 3 4 5 6
Panels 1-4 5-8 9-12 13-16 17-20 21-24
Stain 1 43.0 57.4 52.8 46.6 52.2 32.1
Stain 2 51.8 60.9 59.2 53.5 48.3 34.4
Stain 3 40.8 51.1 51.7 35.4 45.9 32.2
Stain 4 45.5 55.3 55.3 32.5 44.6 30.1
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Lecture 5.1 16
© 2016 Michael Stuart
Minitab analysis
Minitab model: Pretreat Board(Pretreat)
Stain Pretreat * Stain
Source DF SS MS F P
Pretreat 1 782.04 782.04 4.03 0.115
Board(Pretreat) 4 775.36 193.84 15.25 0.000
Stain 3 266.00 88.67 6.98 0.006
Pretreat*Stain 3 62.79 20.93 1.65 0.231
Error 12 152.52 12.71
Total 23 2038.72
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Lecture 5.1 17
© 2016 Michael Stuart
Results
Pretreatment 1 Pretreatment 2
Board 1 2 3 4 5 6
Panels 1-4 5-8 9-12 13-16 17-20 21-24
Stain 1 43.0 57.4 52.8 46.6 52.2 32.1
Stain 2 51.8 60.9 59.2 53.5 48.3 34.4
Stain 3 40.8 51.1 51.7 35.4 45.9 32.2
Stain 4 45.5 55.3 55.3 32.5 44.6 30.1
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Lecture 5.1 18
© 2016 Michael Stuart
Results
Pretreatment 1 Pretreatment 2
Panels 1-4 5-8 9-12 13-16 17-20 21-24
Stain 1 43.0 57.4 52.8 46.6 52.2 32.1
Stain 2 51.8 60.9 59.2 53.5 48.3 34.4
Stain 3 40.8 51.1 51.7 35.4 45.9 32.2
Stain 4 45.5 55.3 55.3 32.5 44.6 30.1
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Lecture 5.1 19
© 2016 Michael Stuart
The wrong analysis
Assuming each Pretreatment / Stain combination run
4 times in random order
Analysis of Variance for Resistance
Source DF SS MS F P
Pretreat 1 782.04 782.04 13.49 0.002
Stain 3 266.01 88.67 1.53 0.245
Pretreat*Stain 3 62.79 20.93 0.36 0.782
Error 16 927.88 57.99
Total 23 2038.72
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Lecture 5.1 20
© 2016 Michael Stuart
Right and Wrong
Right: Source DF SS MS F P
Pretreat 1 782.04 782.04 4.03 0.115
Board(Pretreat) 4 775.36 193.84 15.25 0.000
Stain 3 266.00 88.67 6.98 0.006
Pretreat*Stain 3 62.79 20.93 1.65 0.231
Error 12 152.52 12.71
Total 23 2038.72
Wrong: Source DF SS MS F P
Pretreat 1 782.04 782.04 13.49 0.002
Stain 3 266.01 88.67 1.53 0.245
Pretreat*Stain 3 62.79 20.93 0.36 0.782
Error 16 927.88 57.99
Total 23 2038.72
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Lecture 5.1 21
© 2016 Michael Stuart
Right and Wrong
Right: Source DF SS MS F P
Pretreat 1 782.04 782.04 4.03 0.115
Board(Pretreat) 4 775.36 193.84 15.25 0.000
Stain 3 266.00 88.67 6.98 0.006
Pretreat*Stain 3 62.79 20.93 1.65 0.231
Error 12 152.52 12.71
Total 23 2038.72
Wrong: Source DF SS MS F P
Pretreat 1 782.04 782.04 13.49 0.002
Stain 3 266.01 88.67 1.53 0.245
Pretreat*Stain 3 62.79 20.93 0.36 0.782
Error 16 927.88 57.99
Total 23 2038.72
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Lecture 5.1 22
© 2016 Michael Stuart
The wrong analysis
F(Pretreat) has smaller denominator, MS(Error);
F(Pretreat) is now highly significant.
MS(Error) is increased, F(Stain) is reduced;
F(Stain) is now not statistically significant
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Lecture 5.1 23
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
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Lecture 5.1 24
© 2016 Michael Stuart
Fisher's Potato Study
• 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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Lecture 5.1 25
© 2016 Michael Stuart
Fisher's Potato Study
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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Lecture 5.1 26
© 2016 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
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Lecture 5.1 27
© 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
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Lecture 5.1 28
© 2016 Michael Stuart
Unit / Treatment Structure Diagram
Factors Units
Variety Whole Plots
↓
Fertiliser Subplots
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Lecture 5.1 29
© 2016 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
• Two sources of chance variation
• Implications for Analysis of Variance
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Lecture 5.1 30
© 2016 Michael Stuart
Minitab analysis
Model: Variety WholePlot(Variety)
Fertiliser Variety* Fertiliser
Fixed: Variety, Fertiliser
Random: WholePlot
Source DF SS MS F P
Variety 11 43.64 3.97 5.46 0.000
WholePlot(Variety) 24 17.44 0.73 4.32 0.000
Fertiliser 2 0.35 0.17 1.04 0.362
Variety*Fertiliser 22 2.19 0.10 0.59 0.909
Error 48 8.10 0.17
Total 107 71.70
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Lecture 5.1 31
© 2016 Michael Stuart
Minitab analysis
Model: Variety WholePlot(Variety)
Fertiliser Variety* Fertiliser
Fixed: Variety, Fertiliser
Random: WholePlot
Source DF SS MS F P
Variety 11 43.64 3.97 5.46 0.000
WholePlot(Variety) 24 17.44 0.73 4.32 0.000
Fertiliser 2 0.35 0.17 1.04 0.362
Variety*Fertiliser 22 2.19 0.10 0.59 0.909
Error 48 8.10 0.17
Total 107 71.70
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Lecture 5.1 32
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
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Lecture 5.1 33
© 2016 Michael Stuart
Decomposing the Split Plot Analysis
• Analysis of Varieties in Whole Plots
– Fertilisers irrelevant,
– analyse averages in whole plots
• Analysis of Fertilisers in subplots
– allowing for differences between whole plots
– randomised blocks structure
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Lecture 5.1 34
© 2016 Michael Stuart
Analysis of Varieties in Whole Plots
Postgraduate Certificate in Statistics
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Variety Whole Plot Yields
Ajax 2.86 2.93 4.23
Arran Comrade 2.21 2.29 2.28
British Queen 2.89 2.75 3.77
Duke of York 1.43 1.75 2.15
Epicure 1.97 1.94 2.57
Great Scot 3.08 3.29 3.84
Iron Duke 3.42 2.80 3.52
King of King 3.33 4.30 4.25
Kerr's Pink 2.67 3.25 4.01
Nithsdale 2.19 2.67 3.62
Tinwald Perfection 2.95 3.15 3.02
Up-to-Date 3.96 3.42 4.14
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Lecture 5.1 35
© 2016 Michael Stuart
"One Way"Analysis of
Varieties in Whole Plots
Minitab model: Variety
Source DF SS MS F P
Variety 11 14.55 1.32 5.46 0.000
Error 24 5.81 0.24
Total 35 20.36
Split Plot Analysis
Source DF SS MS F P
Variety 11 43.64 3.97 5.46 0.000
WholePlot(Variety) 24 17.44 0.73 4.32 0.000
Fertiliser 2 0.35 0.17 1.04 0.362
Variety*Fertiliser 22 2.19 0.10 0.59 0.909
Error 48 8.10 0.17
Total 107 71.70
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Lecture 5.1 36
© 2016 Michael Stuart
"One Way"Analysis of
Varieties in Whole Plots
Minitab model: Variety
Source DF SS MS F P
Variety 11 14.55 1.32 5.46 0.000
Error 24 5.81 0.24
Total 35 20.36
Split Plot Analysis
Source DF SS MS F P
Variety 11 43.64 3.97 5.46 0.000
WholePlot(Variety) 24 17.44 0.73 4.32 0.000
Fertiliser 2 0.35 0.17 1.04 0.362
Variety*Fertiliser 22 2.19 0.10 0.59 0.909
Error 48 8.10 0.17
Total 107 71.70
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Lecture 5.1 37
© 2016 Michael Stuart
"Randomised Blocks" analysis
of Fertilisers in Subplots
Minitab model: WholePlot Fertiliser
Source DF SS MS F P
WholePlot 35 61.08 1.75 11.89 0.000
Fertiliser 2 0.35 0.17 1.19 0.310
Error 70 10.27 0.15
Total 107 71.70
Split Plot Analysis
Source DF SS MS F P
Variety 11 43.64 3.97 5.46 0.000
WholePlot(Variety) 24 17.44 0.73 4.32 0.000
Fertiliser 2 0.35 0.17 1.04 0.362
Variety*Fertiliser 22 2.19 0.10 0.59 0.909
Error 48 8.10 0.17
Total 107 71.70
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Lecture 5.1 38
© 2016 Michael Stuart
"Randomised Blocks" analysis
of Fertilisers in Subplots
Minitab model: WholePlot Fertiliser
Source DF SS MS F P
WholePlot 35 61.08 1.75 11.89 0.000
Fertiliser 2 0.35 0.17 1.19 0.310
Error 70 10.27 0.15
Total 107 71.70
Split Plot Analysis
Source DF SS MS F P
Variety 11 43.64 3.97 5.46 0.000
WholePlot(Variety) 24 17.44 0.73 4.32 0.000
Fertiliser 2 0.35 0.17 1.04 0.362
Variety*Fertiliser 22 2.19 0.10 0.59 0.909
Error 48 8.10 0.17
Total 107 71.70
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Lecture 5.1 39
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
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Design and Analysis of Experiments
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Lecture 5.1 40
© 2016 Michael Stuart
Diagnostic Plots
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Lecture 5.1 41
© 2016 Michael Stuart
Diagnostics for Whole Plots Analysis
Unremarkable
Postgraduate Certificate in Statistics
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3
2
1
0
-1
-2
-3210-1-2
Dele
ted
Resi
du
al
Normal Score
Normal Probability Plot
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Lecture 5.1 42
© 2016 Michael Stuart
Refit with Ajax deleted
Source DF SS MS F P
Variety 10 42.66 4.27 6.78 0.000
WholePlot(Variety) 22 13.84 0.63 5.02 0.000
Fertiliser 2 0.17 0.09 0.69 0.508
Variety*Fertiliser 20 1.80 0.09 0.72 0.788
Error 44 5.51 0.13
Total 98 63.99
No substantial change
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Lecture 5.1 43
© 2016 Michael Stuart
Refit Diagnostics
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
4
3
2
1
0
-1
-2
-3
-43210-1-2-3
Dele
ted
Resi
du
al
Score
Normal Probability Plot
Unremarkable
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Lecture 5.1 44
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
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Design and Analysis of Experiments
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Lecture 5.1 45
© 2016 Michael Stuart
Split Plots in Blocks 1
Cambridge Grassland Experiment
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.
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Lecture 5.1 46
© 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
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Lecture 5.1 47
© 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.
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Lecture 5.1 48
© 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
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Lecture 5.1 49
© 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
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Lecture 5.1 50
© 2016 Michael Stuart
Plot structure
72 subplots
grouped in 18 whole plots
grouped in 6 blocks
3 grassland treatments randomly assigned
to whole plots within blocks
4 fertilisers randomly assigned
to subplots within whole plots
least variation
in between variation
most variation
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Lecture 5.1 51
© 2016 Michael Stuart
Unit / Treatment Structure Diagram
Factors
Grassland
Treatment
Fertiliser
Units
Blocks
↓
Whole Plots
↓
Subplots
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Lecture 5.1 52
© 2016 Michael Stuart
Split Plots Analysis
• Treatments vary between whole plots,
– Treatment effects evaluated with reference to
chance variation between whole plots
• Fertilisers vary between subplots
– fertiliser effects evaluated with reference to
chance variation between subplots
• Whole plots nested in Blocks
• Three sources of chance variation
• Implications for Analysis of Variance
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Lecture 5.1 53
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
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Lecture 5.1 54
© 2016 Michael Stuart
Split Plots in Blocks 2
Electronic components baked in an oven.
Two factors thought to influence life times:
oven temperature
bake time.
Trial settings:
Oven Temperature (T), °F, 580, 600, 620, 640,
Baking time (B), min, 5, 10, 15.
To save on costly runs,
three components baked together at each temperature,
one withdrawn at each of the set times.
This plan was replicated 3 times (in 3 blocks).
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Lecture 5.1 55
© 2016 Michael Stuart
Results of accelerated life time tests
for electronic components Baking Time (min.)
Replicate Temperature of Oven (°F)
5 10 15
1 580 217 233 175 600 158 138 152 620 229 186 155 640 223 227 156 2 580 188 201 195 600 126 130 147 620 160 170 161 640 201 181 172 3 580 162 170 213 600 122 185 180 620 167 181 182 640 182 201 199
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Lecture 5.1 56
© 2016 Michael Stuart
Classwork
What are the whole units?
What are the whole unit treatment factors?
What are the whole unit treatments?
What are the sub units?
What are the sub unit treatment factors?
What are the sub unit treatments?
What is the unit structure?
What is the treatment structure?
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Lecture 5.1 57
© 2016 Michael Stuart
Classwork
What are the whole units?
What are the whole unit treatment
factors?
What are the whole unit treatments
What are the sub units?
What are the sub unit treatment
factors?
What are the sub unit treatments?
Bakes
Oven temperature
580, 600, 620, 640
Single components
Baking time
5, 10, 15
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Lecture 5.1 58
© 2016 Michael Stuart
Unit / Treatment Structure
Baking Time (min.)
Replicate Temperature of Oven (°F)
5 10 15
1 580 217 233 175 600 158 138 152 620 229 186 155 640 223 227 156
2 580 188 201 195 600 126 130 147 620 160 170 161 640 201 181 172
3 580 162 170 213 600 122 185 180 620 167 181 182 640 182 201 199
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Lecture 5.1 59
© 2016 Michael Stuart
Unit structure
36 components
nested in 12 bakes
nested in 3 replicates
Treatment structure
4 temperatures randomly allocated to bakes within
replicates
3 baking times randomly allocated to components within
bakes
Classwork: Produce a Unit/Treatment Structure Diagram
least variation
in between variation
most variation
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Lecture 5.1 60
© 2016 Michael Stuart
Experimental
Units
Replicates
Bakes
Components
Unit / Treatment Structure Diagram
Treatment
Factors
Temperature
Baking time
ANOVA
MS(Replicates)
MS(Temperature)
MS(Bakes Error)
MS(Baking time)
MS(TxB)
MS(Components Error) Postgraduate Certificate in Statistics
Design and Analysis of Experiments
↓
↓
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Lecture 5.1 61
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
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Lecture 5.1 62
© 2016 Michael Stuart
Replication (Blocking) as a Random Effects
Factor
Day to day changes in set-up conditions depend on
• environmental factors,
• staff factors,
• timing of critical tasks,
– and more,
varying more or less unpredictably
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Lecture 5.1 63
© 2016 Michael Stuart
Recall: Block by Treatment Interaction
as Error term
Ref: Laboratory 1, Lecture 4.1
Postgraduate Certificate in Statistics
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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
.
Soybean seed germination rates
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Lecture 5.1 64
© 2016 Michael Stuart
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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Lecture 5.1 65
© 2016 Michael Stuart
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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Lecture 5.1 66
© 2016 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,
UNLESS no interaction effect.
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Lecture 5.1 67
© 2016 Michael Stuart
A Randomized Blocks Analysis for
Temperature effects
Results of accelerated life time tests
for electronic components, with oven run means Baking Time (min.)
Replicate Temperature of Oven (°F)
5 10 15 Mean
1 580 217 233 175 208.3 600 158 138 152 149.3 620 229 186 155 190.0 640 223 227 156 202.0 2 580 188 201 195 194.7 600 126 130 147 134.3 620 160 170 161 163.7 640 201 181 172 184.7 3 580 162 170 213 181.7 600 122 185 180 162.3 620 167 181 182 176.7 640 182 201 199 194.0
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Lecture 5.1 68
© 2016 Michael Stuart
A Randomized Blocks Analysis for
Temperature effects
Results of accelerated life time tests
for electronic components, with oven run means Baking Time (min.)
Replicate Temperature of Oven (°F)
5 10 15 Mean
1 580 217 233 175 208.3 600 158 138 152 149.3 620 229 186 155 190.0 640 223 227 156 202.0 2 580 188 201 195 194.7 600 126 130 147 134.3 620 160 170 161 163.7 640 201 181 172 184.7 3 580 162 170 213 181.7 600 122 185 180 162.3 620 167 181 182 176.7 640 182 201 199 194.0
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Lecture 5.1 69
© 2016 Michael Stuart
A Randomized Blocks Analysis for
Temperature effects
Results of accelerated life time tests
for electronic components, with oven run means
Replicate Temperature of Oven (°F)
Mean Lifetime
1 580 208.3 600 149.3 620 190.0 640 202.0 2 580 194.7 600 134.3 620 163.7 640 184.7 3 580 181.7 600 162.3 620 176.7 640 194.0
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Lecture 5.1 70
© 2016 Michael Stuart
A Randomized Blocks Analysis for
Temperature effects
Mean results of accelerated life time tests
for electronic components for each oven run
Minitab model: R T
Temperature Replicate of Oven (°F) 1 2 3
580 208.3 194.7 181.7 600 149.3 134.3 162.3 620 190.0 163.7 176.7 640 202.0 184.7 194.0
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Lecture 5.1 71
© 2016 Michael Stuart
Minitab results
Analysis of Variance for Lifetime
Source DF SS MS F P
R 2 654.24 327.12 3.32 0.107
T 3 4164.77 1388.26 14.09 0.004
Error 6 591.31 98.55
Total 11 5410.32
Conclusions:
the effect of changing Temperature is highly statistically
significant,
blocking appears to have been effective.
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Lecture 5.1 72
© 2016 Michael Stuart
Minitab results
Minitab model: R T
Source DF SS MS F P
R 2 654.24 327.12 3.32 0.107
T 3 4164.77 1388.26 14.09 0.004
Error 6 591.31 98.55
Total 11 5410.32
Minitab model: R T R*T
Source DF SS MS F P
R 2 654.24 327.12 * *
T 3 4164.77 1388.26 * *
R*T 6 591.31 98.55 * *
Error * ** **
Total 11 5410.32
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Lecture 5.1 73
© 2016 Michael Stuart
Block by Treatment Interaction
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Lecture 5.1 74
© 2016 Michael Stuart
Diagnostic analysis
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Lecture 5.1 75
© 2016 Michael Stuart
Diagnostic analysis
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
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Lecture 5.1 76
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
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Design and Analysis of Experiments
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Lecture 5.1 77
© 2016 Michael Stuart
Split units model in Minitab
R
T R*T
B T*B R*B
• R is designated as a Random effect factor
• R*T serves as error term at Oven/Bake level
• (R*T*B could serve as error term at Component
level)
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
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Lecture 5.1 78
© 2016 Michael Stuart
Analysis of Variance for Lifetime
Source DF SS MS F P
R 2 1962.7 981.4 0.54 0.618 x
T 3 12494.3 4164.8 14.09 0.004
R*T 6 1773.9 295.7 1.22 0.362
B 2 566.2 283.1 0.16 0.856
T*B 6 2600.4 433.4 1.79 0.185
R*B 4 7021.3 1755.3 7.23 0.003
Error 12 2912.1 242.7
Total 35 29331.0
x Not an exact F-test.
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Design and Analysis of Experiments
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Lecture 5.1 79
© 2016 Michael Stuart
Analysis of Variance for Lifetime
Source DF SS MS F P
R 2 1962.7 981.4 0.54 0.618 x
T 3 12494.3 4164.8 14.09 0.004
R*T 6 1773.9 295.7 1.22 0.362
B 2 566.2 283.1 0.16 0.856
T*B 6 2600.4 433.4 1.79 0.185
R*B 4 7021.3 1755.3 7.23 0.003
Error 12 2912.1 242.7
Total 35 29331.0
x Not an exact F-test.
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
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Lecture 5.1 80
© 2016 Michael Stuart
Diagnostics
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Design and Analysis of Experiments
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Lecture 5.1 81
© 2016 Michael Stuart
Diagnostics
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Design and Analysis of Experiments
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Lecture 5.1 82
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
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Lecture 5.1 83
© 2016 Michael Stuart
Graphical and numerical summaries
Temperature 580 600 620 640
Mean Lifetime 195 149 177 194
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Lecture 5.1 84
© 2016 Michael Stuart
Graphical and numerical summaries
The experimenter "may want to look into his
technique and records quite carefully".
C. Daniel (1976), p. 270
Postgraduate Certificate in Statistics
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Baking Time
Replicate 1 2 3
5 207 169 158 10 196 171 184 15 160 169 194
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Lecture 5.1 85
© 2016 Michael Stuart
Design and Analysis of Experiments
Lecture 5.1
1. Review of Lecture 4.2
2. Fisher's Potato Study
− Decomposing the split plot analysis
− Multi level diagnostics
3. Split Plots in Blocks
− Cambridge Grassland Study; Review
− Baking Electronic Components; Analysis
− Block*Treatment Interaction as Error
− Split plot analysis
− An unexpected interaction
Postgraduate Certificate in Statistics
Design and Analysis of Experiments
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Lecture 5.1 86
© 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
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Lecture 5.1 87
© 2016 Michael Stuart
Reading
Lecture Notes:
Split Units Design and Analysis
Split Plots Analysis Case Study
Postgraduate Certificate in Statistics
Design and Analysis of Experiments