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Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 1
Diploma in StatisticsDesign and Analysis of Experiments
Lecturer: Dr. Michael Stuart,
Department of Statistics
Office: LB 101
email: [email protected]
Too short; cover parts 1-3 in first half; need more for the 2nd half.
Try boys shoes, comparing two t-tests
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 2
Design and Analysis of ExperimentsCourse Outline
• Experimental and observational studies
• Basic design principles for experiments
– Randomisation
– Blocking (pairing)
– Factorial structure
• Standard designs, illustrated
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 3
Design and Analysis of ExperimentsCourse Outline
• Analysis of experimental data
– Exploratory data analysis
– Parameter estimation and significance testing
– Analysis of variance
– Model validation, diagnostics
• Computer laboratories
• Strategies for Experimenting
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 4
Design and Analysis of ExperimentsReferences
Mullins, E., Statistics for the Quality Control Chemistry Laboratory, Royal Society of Chemistry, 2003, particularly Chapters 4-5, 7-8.
Detailed coverage of much of the module, in a specific context.
Montgomery, D.C., Design and analysis of experiments, 6th ed., Wiley, 2005.
A comprehensive text, covers much more than this module, including statistical theory. Not always authoritative.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 5
Design and Analysis of ExperimentsFurther reading
Box, G.E.P, Hunter, J.S. and Hunter, W.G., Statistics for Experimenters, 2nd. ed., Wiley, 2005.Includes many gems of wisdom from these masters of the genre, though not a course text.
Daniel, C., Applications of Statistics to Industrial Experimentation, Wiley, 1976.Includes many gems of wisdom from this master of the genre, using methodology appropriate for an industrial setting.
Altman, D.G., Practical Statistics for Medical Research, Chapman & Hall / CRC, 1991.Does what it says on the tin!
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 6
Lecture 1.11. Introduction to Course
2. Case study on process improvement
statistical assessment of a process change
strategy for experimentation
3. Experimental vs observational study
another illustration
4. Multifactor designs
efficiency
interaction
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 7
2 Case study on process improvement
• Comparison of standard (old) and new processes for manufacture of electronic components
• Key issues
– homogeneity for valid comparison
– systematic allocation
– random allocation
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 8
Experimental design
• 50 components sampled per day,
• 6 days per week,
• 8 weeks,
• Systematic layout, as follows
Week Number
1 2 3 4 5 6 7 8
Monday Old New Old New Old New Old New
Tuesday New Old New Old New Old New Old
Wednesday Old New Old New Old New Old New
Thursday New Old New Old New Old New Old
Friday Old New Old New Old New Old New
Saturday New Old New Old New Old New Old
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 9
Results
Day Defectives Day Defectives Day Defectives Day Defectives 1 0 13 1 25 0 37 2 2 0 14 0 26 0 38 0 3 6 15 3 27 0 39 0 4 3 16 1 28 2 40 0 5 3 17 0 29 0 41 0 6 3 18 2 30 0 42 0 7 4 19 0 31 1 43 1 8 1 20 1 32 1 44 0 9 0 21 2 33 0 45 2 10 2 22 0 34 0 46 0 11 0 23 1 35 0 47 0 12 0 24 3 36 2 48 0
Numbers of defectives per daily sample of 50for 48 days (8 weeks)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 10
Comparison of two processes over eight weeks:data for first four weeks
Number of Defectives
in Samples of 50 Units
Week Day pair
Old Process
New Process
Difference (New – Old)
1 1 0 0 0 1 2 6 3 –3 1 3 3 3 0 2 4 1 4 +3 2 5 2 0 –2 2 6 0 0 0 3 7 1 0 –1 3 8 3 1 –2 3 9 0 2 +2 4 10 1 0 –1 4 11 0 2 +2 4 12 3 1 –2
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 11
Comparison of two processes over eight weeks:data for last four weeks, with eight week summary
Number of Defectives
in Samples of 50 Units
Week Day pair
Old Process
New Process
Difference (New – Old)
5 13 0 0 0 5 14 0 2 +2 5 15 0 0 0 6 16 1 1 0 6 17 0 0 0 6 18 2 0 –2 7 19 2 0 –2 7 20 0 0 0 7 21 0 0 0 8 22 0 1 +1 8 23 0 2 +2 8 24 0 0 0 Total 25 22 –3
8 week Average, per 50 units 1.04 0.92 –0.13 8 week Average, per cent 2.08 1.83 –0.25
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 12
Differences in numbers defective,with control limits
4 8 12 16 20 24
Day Pair
-8
-6
-4
-2
0
2
4
6
8
Difference
No statistical significance!
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 13
Alternative design(proposed by engineers)
Week Number
1 2 3 4 5 6 7 8
Monday Old Old Old Old New New New New
Tuesday Old Old Old Old New New New New
Wednesday Old Old Old Old New New New New
Thursday Old Old Old Old New New New New
Friday Old Old Old Old New New New New
Saturday Old Old Old Old New New New New
Assume this design was used;
check for no effect
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 14
Defect rates, per cent, with differences,for the first and second four week periods
First
Period Second Period
Difference
Both Processes 3.0 0.9 2.1
Old Process 3.3 0.8 2.5
New Process 2.7 1.0 1.7
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 15
75.3
56.0
1.2
1200
1.999.0
1200
973
9.00.3
n
)P̂100(P̂
n
)P̂100(P̂
P̂P̂Z
2
22
1
11
21
Defect rates, per cent, with differences,for the first and second four week periods
highly statistically significant!
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 16
Exercise
Assess the statistical significance of the difference in defect rates, %, between the first period and second period for the old process.
Homework
Assess the statistical significance of the difference in defect rates, %, between the first period and second period for the new process.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 17
Numbers defective in time order
Long term downward trend,
systematic bias
How can this be?
6 12 18 24 30 36 42 48
Day
0
1
2
3
4
5
6
Defectives
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 18
How to avoid systematic bias
• Make comparisons under
homogeneous experimental conditions
• 1 Systematic arrangement, as implemented:
avoids known biases
• 2 Random allocation:
within each day pair, allocate old and new processes at random
avoids known and unknown biases
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 19
Two design principles
• Blocking
– identify homogeneous blocks of experimental units
– assess effects of experimental change within homogeneous blocks
– average effects across blocks
• Randomisation
– allocate experimental conditions to units at random
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 20
Strategy for ExperimentationThe SIPOC Process Model
Cu
stom
ers
Process
Su
pplie
rs Inputs Outputs
S I P O C
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 21
Strategy for ExperimentationStatistical Thinking
Cu
stom
er
Process
Su
pplie
r Inputs Outputs
Process management and
improvement
Input measures
Process measures
Process changes Output
measures
Supplier performance
Customer Feedback
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 22
Strategy for ExperimentationShewhart's PDCA Cycle
Check
Act
Plan
Do
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 23
Strategy for ExperimentationShewhart's PDCA Cycle
• Plan: Plan a change to the process, predict its effect, plan to measure the effect
• Do: Implement the change as an experiment and measure the effect
• Check: Analyse the results to learn what effect the change had, if any
• Act: If successful, make the change permanent, proceed to plan the next improvement
or
if not, proceed to plan an alternative change
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 24
Strategy for Experimentation:new vs old manufacturing process
Plan:
• Compare defect rates for old process and new (cheaper) process
– predict reduction, or no increase, in number of defectives using new process
• Sample output over an eight week period, six days per week
– select 50 components at random per day
• Count number of defectives per sample
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 25
Do:
• Implement plan
• Record daily numbers of defectives
Assessing experimental process for manufacturing electronic components
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 26
Check:
• Analyse data
• test statistical significance of the change
Assessing experimental process for manufacturing electronic components
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 27
Act:
• If no worse, make the change permanent,
– proceed to plan the next improvement
or
• if not, proceed to plan an alternative change
Assessing experimental process for manufacturing electronic components
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 28
3 Observational vs Experimental study
Alternative design:
• sample 1200 components from old process inventory,
• sample 1200 components from new process inventory,
• compare
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 29
Example 2: walking babies
•How long does it take a baby to walk?
•Can this be affected by special training programs?
4 "training" programs:
1. special exercises
2. normal daily exercise
3. weekly check
4. end of study check
each of 24 babies allocated at random to groups of 6 in each program.
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 30
Example 2: walking babies
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 31
Example 2: walking babies
Alternative design:
each of 4 different consultants prescribes one of the four training programs,
select a sample randomly from babies assigned to each program.
Problems:
assignment of babies to programsequivelent to
assignment of mothers to consultants
lurking variables!
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 32
Walking babiesvs
Defective components
Level of control:
less control
means
more variation
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 33
4 Multi-factor experiments
• Traditional versus statistical design
– efficiency
– interaction
• Several levels
• Several factors
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 34
Illustration of a traditional design,with 12 experimental runs
Pressure
Temperature
High
High
Low
Low4321 YYYY
8765 YYYY
1211109 YYYY
(best)
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 35
Illustration of a full factorial design,with 12 experimental runs
Pressure
Temperature
High
High
Low
Low321 YYY
121110 YYY987 YYY
654 YYY
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 36
Interaction between the factors
Pressure
Temperature
High
High
Low
Low65
75
70
60
5
15
5 5
best
best
best
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 37
Multilevel Interaction:Emotional Arousal
160 subjects,
– 80 male (M),– 80 female (F)
shown one of 4 pictures:
– nude female,– nude male,– infant,– landscape.
Response variable:
– level of emotional arousal
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 38
Infa
nt
Land
sdca
pe
Nud
e F
emal
e
Nud
e M
ale
10
15
20
25
Male
PicturesIn
fant
Land
sdca
pe
Nud
e F
emal
e
Nud
e M
ale
10
15
20
25
Female
Pictures
Levels of Arousal of Males and Females to Different Visual Stimuli
Interaction between FactorsCase study: Emotional Arousal
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 39
Non-linear response:Optimisation vs Improvement
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 40
Optimising performance; hill climbing
50 52 54 56 58 60 62 64
Temperature
65
66
67
68
69
70
Yield
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 41
50 52 54 56 58 60 62 64
Temperature
65
66
67
68
69
70
Yield
Optimising performance; hill climbing
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 42
50 52 54 56 58 60 62 64
Temperature
65
66
67
68
69
70
Yield
Optimising performance; hill climbing
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 43
50 52 54 56 58 60 62 64
Temperature
65
66
67
68
69
70
Yield
Optimising performance; hill climbing
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 44
50 52 54 56 58 60 62 64
Temperature
65
66
67
68
69
70
Yield
Optimising performance; hill climbing
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 45
Several factors
2 - level factors: 2 factors: 22 = 4 runs
3 factors: 23 = 8 runs
4 factors: 24 = 16 runs
5 factors: 25 = 32 runs
6 factors: 26 = 64 runs
7 factors: 27 = 128 runs
Multi-level: 2 × 3 × 4 × 5 = 120 runs
Diploma in StatisticsDesign and Analysis of Experiments
Lecture 1.1 46
Reading
SA Sections 1.9, 11.4 - 11.6
EM Sections 4.3, 4.5.1, 5.2
DCM Section 2.5, 3.1 - 3.3