The History of a Research Project

Report
Preliminaries
• Thank you John McCool
– Good excuse to review, think about and give an
overview of a 25 year research project
• Stat Day: A worthwhile conference
– I have attended it frequently
• What can/should we do to advance statistics
Last years Stat Day: Follow-up
• Drove up with Ron Snee and Steve Bailey
• Good Discussion when driving home with two
major topics:
– 1) Should we use and teach Definitive Screening
Designs? Do they add much to Plackett-Burman
Designs for Screening?
– 2) Status of “Living A Great Life: Mathematical
Models for Living” –a book that I may write
Definitive Screening Designs
• Jones, B and Nachtsheim, C. J. (2011) “A Class of
Three-Level Designs for Definitive Screening in
the Presence of Second-Order Effects” JQT, 43,1
pp 1- 15
• Jones, B and Nachtsheim, C. J. (2013) “Definitive
Screening Designs with Added Two-Level
Categorical Factors” JQT, 45, 2, pp 121-129
• Xiao, L., Lin, D. K. J. and Bai, F. (2012)
“Constructing Definitive Screening Designs Using
Conference Matrices” JQT 44, 1, pp 2-8
Properties:
Estimates linear and quadratic terms
For each factor
Design can be saturated:
e.g. 6 factors in 13 runs
Can be blocked
Add center points for each block
Can study 2-level categorical factors
Replace 0s with ±1
Near orthogonal
Should use Conference Matrix
Computer search with 10,000 starts
Will not find best design
Even Number of Factors
Use Dummy columns
Recommendation:
Use instead or reflected
Plackett-Burman Design
e.g. 24(vs 25) points to study 11(12)factors
Analysis Table for The 12-Run Plackett Burman Design (L12)
Trial
Avg
X1
X2
X3
X4
X5
X6
X7
X8
X9
X10
X11
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12
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Sum +
Sum Check
Differ.
Effect
Y
Living A Great Life:
Mathematical Models for Living
• How to live the best life given your starting status
– Life is a process not a destination
– Always have a Plan B
• Time is the universal limiting resource
• Living well is the best revenge
• There are good mathematical models for many
aspects of life:
– Buying a house
– Finding a spouse/life partner
RSM & Optimum Design Theory
• Kiefer, J. C. (1959) “Optimum Experimental Designs” JRSS B
21, 272-319
• Lucas, J.M. (1976) “Which Response Surface Design is Best”
Technometrics 16, 411-417
• Original Paper was rejected by Technometrics – just a
calculation exercise
• Closed RSM to Optimum Design Theory:
– Composite and Box-Behnken Designs are so efficient that there
is little possibility of better designs.
– Bill Notz of OSU, a Kiefer student, confirmed this in his talk at
the 2013 FTC
• Split-Plot Designs Re-Opened the door
• Possible Ph.D. Topics here
Large Response Surface Designs
– Crosier, R.B (1991) “Some New Three-Level Response
Surface Designs” Technical Report CRDEC-TR-308, US
Army Chemical Research, Development and
Engineering Center (Available at Stat. Lab Server,
Carnegie Mellon University) Up to 15 factors
• I recommended this paper; however, it was
rejected by Technometrics
– Mee, R. W. (2007) Optimal Three-Level Designs for
Response Surfaces in Spherical Experimental Regions.
(Up to 16 factors and references a 14 factor
application)
– This paper includes Crosier’s designs
Letter Results
• Crosier paper was rejected, decision was
unchanged
– No Longer an Associate Editor of Technometrics
– Would have become longest term AE
The History of a Research Project
By
James M. Lucas
2014 National Quality Month
Statistical Symposium
Penn State Great Valley
October 17, 2014
Abstract
•
•
•
I describe how a simple observation lead to a long term research project studying
experiments with Hard-to-Change and Easy-to-Change Factors.
The observation was that most experiments I had been involved in were not
randomized even though the experimental runs were conducted using a random
run order. Randomization requires that each experimental unit be treated the
same way. So for a randomized experiment a random run order is necessary but it
is not sufficient to achieve randomization. Resetting of all experimental factors is
also required before randomization is achieved.
I will give an overview of experiments with Hard-to-Change and Easy-to-Change
Factors, and describe recent research and practical applications. I will discuss the
contributions of my students and me, describe personal interactions that occurred
with other researchers and journal editors during this project, and show how they
advanced or slowed down scientific progress. I will discuss the problems and
advantages of early disclosure of research results including the disadvantage of
preemption by others and the advantage of faster dissemination of scientific
results. I will describe current research efforts and current applications in this very
active area and tell what problems need to be solved to best help experimenters.
Audience Questions
• Why?
• To be sure I am telling you something useful
• To be on the same page that you are on
DOE Questions
• Are you involved with running experiments?
• How often do your experiments involve hardto-change factors? (First asked at ‘92 JSM)
– Seldom (<10% of the time)
– Sometimes (<50% of the time)
– Often (>50% of the time)
• Software developer surprised at large fraction
of “Often” answers
DOE Questions
• How many of you are involved with running
experiments?
• How many of you “randomize” to guard
against trends or other unexpected events?
• If the same level of a factor such as
temperature is required on successive runs,
how many of you set that factor to a neutral
level and reset it?
15
Additional Questions
• How many of you have conducted
experiments on the same process on which
you have implemented a Quality Control
Procedure?
– What did you find?
16
MY OBSERVATIONS
• Comparing residual standard deviation from
an experiment with residual standard
deviation from an in-control process.
– Experimental standard deviation is larger. 1.5X to
3X is common.
• Why? You are inducing variation by making
changes in the process
17
Background
• Why I made the observation that most
experiments I had been involved in were not
randomized even though the experimental runs
were conducted in a random run order
• I had the mixed model tools from the 1980s
– DuPont ASG’s Mixed Model Program based on Fellner,
William H. (1986) “Robust Estimates of Variance
Components” Technometrics 58, pp 51-60
• Commercially Available much later
– SAS PROC MIXED in 1996
– JMP6 in 2005
Introduction
• Experiments with H-T-C and/or E-T-C factors
occur frequently
• Industrial experiments are seldom really
randomized
– They often shouldn’t be randomized
• Split-Plot experiments (intentionally or
inadvertently) are often conducted
• Proper Split-Plot Blocking is often the answer
• Good Reference:
– Jones, B. and Nachtsheim, C. J. (2009) “Split-Plot
Designs: What, Why and How” JQT 41, 4 pp 340-361
Good Quote
• Cuthbert Daniel (1976) comes to split-plot experiments
from an affordability perspective when he says that “In
nearly all experimental situations some factors are
hard to vary, whereas others, if not easy, are at least
amenable to deliberate variation. Most industrial
experiments are, then, split-plot in their design. The
total number of runs is largely determined by the
number of combinations of the hard-to-vary factors
that can be afforded.”
• Compare this quote to J&N (2009):
• All industrial experiments are split-plot experiments
Framework
(For much of my research in Design of Experiments)
• DuPont Applied Statistics Group’s Strategy of
Experimentation (SOE) Course
– 2½ day course
– Taken by a large fraction of DuPont’s
professionals
• Managerial familiarity with DOE
– Sells statistics
• Designs Used:
– Plackett-Burman Screening Designs
– Two level Factorials for process
improvement
– Box-Bhnken and Composite Response Surface
Designs for Process Optimization
My DOE Course
• Based on the DuPont Course
• Includes recommendations for Hard-to-Cange
and Easy-to-Change factors
• Screening will also include definitive Screening
Designs
• Also Emphasize “Bold Experimentation”
Statistical Systems: Introduction
• Statistical Systems
– Take a Systems Approach to solve problems that
have statistical aspects
– Tie together statistical techniques and
engineering/scientific knowledge
– Many were developed following WW II
• SOE is a useful statistical system
• Good Example of Statistical Engineering
23
Experiments with H-T-C Factors:
Approach
• Get Help: Four Ph. D Dissertations:
• Huey Ju (U of D)
– Expected Variance over all Randomizations
• Jeetu Ganju (U of D)
– Bias of RNR Experiments
• Frank Anbari (Drexel)
– Optimum blocking with One H-T-C factor
• Derek Webb (Montana State)
– More than one H-T-C factor
Aside: My Decision for LAGL
• U of D Adjunct from 1970s
– Organized team taught QC Course
– Previously directed 3 Ph.D. Dissertations
• Why I am not an academic
– Vince Laricca told me: “I fought very hard to keep you from
joining the department because you would have Made the
department too applied.”
• How did that work out for him?
• Other Academic Opportunities:
– Drexel-did not like the commute
– Penn State-move was too disruptive
• I love to Consult
Once Bitten: Conducting the wrong
experiment
• 3-Factor Box-Behnken design for Mylar
– One factor easy-to-change
– Factor had “wrong” sign
– Factor was not significant
• Learning: Better Consulting
• Better Experiment
–
–
–
–
3 x 3 Factorial
3 settings of the easy-to-change factor
27 versus 15 runs
Larger experiment can be cheaper
26
RSM: One Easy-to-Change Factor
• Design good experiment in H-T-C factors
– Examine all levels of E-T-C Factor at each setting of
H-T-C Factors (and often use more levels)
– Gives a Classic Split-Plot Experiment
• Examples:
– Pharmaceutical Pill experiments: compression
– Plating Process: Current ratio
• E-T-C factor often has the largest effect
Example:
Experiments
Used to
Develop
The Ni/Pd
Plating
Process
One Hard-to-Change Factor
• Plan good Experiment ignoring H-T-C aspect
• Very Expensive to change factor
– Completely restrict running H-T-C factor
– Sometimes the only feasible experiment
• Visit each level of the H-T-C factor twice using
Standard Blocking:
– 4 Factor Composite (FCC) Show.
– Box-Behnken often has 2 (or 3) Orthogonal Blocks
– Completely restrict each block
–Good Ph.D. Project
Good Standard
Blocking
Are more Blocks
needed?
Potential Ph. D.
Topic
Fewer center points
Ju and Lucas (2002) History
• Originally submitted to Technometrics in 1994
– Associate Editor: Dick DeVeaux
– Reviewers: Ray Myers, Tom Lorenzen
• Revisions got no closer to publication
– AE personal letter
• Advanced Science by motivating papers published before it
was:
– Letsinger, J. D., Myers, R. H. and Lentner, M. (1996) Response
Surface Methods for Bi-Randomization Structures” JQT 28, pp
381-397
– Ray said that he expected our paper to be published before his
was
• Ju and Lucas Response Surface Paper was not published
Comments on Letsinger et. al. (1996)
• High Impact paper
– Recommended REML for analysis of unbalanced
Split-Plot experiments
• Visited each level of the H-T-C factor only once
– A definite weakness in the designs considered
– Minimum-Aberation Designs
– Split-Plots vs Generalized S-Ps
Randomized Not Reset (RNR) Experiments;
Also Called Random Run Order (RRO)
• A large fraction (perhaps a large majority) of
industrial experiments are Randomized not Reset
(RNR) experiments
• Properties of RNR experiments and a discussion
of how experiments should be conducted:
– “Lk Factorial Experiments with Hard-to-Change and
Easy-to-Change Factors” Ju and Lucas, 2002, JQT 34,
411-421 [studies one H-T-C factor and uses Random
Run Order (RRO) rather than RNR]
– “Factorial Experiments when Factor Levels Are Not
Necessarily Reset” Webb, Lucas and Borkowski, 2004,
JQT 36, 1, pp 1-11
33
Not Resetting Factors
• Common practice
• Has had many successes!
– Complete randomization may be impractical
• Not addressed by the classical definition
– Gives a split-plot blocking structure with the blocks
determined at random
• May be cost effective
• Causes biased hypothesis tests (Ganju and Lucas
1997, 1999, 2005)
34
Ganju and Lucas References
• Ganju, J., and Lucas, J. M. (1997). “Bias in Test
Statistics when Restrictions on Randomization are
Caused by Factors”. Communications in Statistics –
Theory and Methods, 26, pp. 47-63.
• Ganju, J., and Lucas, J. M. (1999). “Detecting
Randomization Restrictions Caused by Factors”.
Journal of Statistical Planning and Inference, 81, pp.
129-140.
• Ganju, J., and Lucas, J. M. (2000). “Analysis of
Unbalanced Data from an Experiment with Random
Block Effects and Unequally Spaced Factor Levels”.
The American Statistician, 54, 1, pp. 5-11.
• Ganju, J., and Lucas, J. M. (2005). “Randomized and
Random Run Order Experiments” Journal of
Statistical Planning and Inference 133, pp.199-210.
35
An Essential Element of Randomized
Not Reset (RNR) Experiments
(DuPont Applied Statistics Group)
37
Randomization Questions?
• Should Industrial Experiments be Randomized?
– Historically most have not been randomized even if a
random run order was used
• How large is the experimental Error?
– RSM tools assume that the error is small
• Can confirmatory runs be made?
• Experimental error is increased by setting factor
levels
– Cost benefit analysis is needed
• Discussion paper – not a Ph.D topic
Analysis Comments
• Consultants Rule: 90% of the information
gained from Appropriate Plot.
• Academics are heavily into the formal analysis
– Even though is often adds little
• Wayne Nelson (2007) said “I have often
designed and conducted experiments as splitplots but I have seldom bothered to analyze
them as split-plots.”
Experiment With Random Block Effects
• 3 Temperatures x 3 Times x 12 Days
– Factorial with 10 extra center points – Almost balanced
•
At ’92 JSM in Toronto told Andre he had missed significant
DayxTime Interaction in:
– Kuhri, A. I. (1992) “RS Models with Random Block Effects”
Technometrics 34, 26-37
• He Wrote:
– Khuri, A. I. (1994) “RS Models with Mixed Effects” JQT 28, 177-186
• This used RSM approach rather than model structure so we played
off of his analysis and wrote:
Ganju, J. and Lucas, J. M. “Analysis of Unbalanced Data From an
Experiment with Random Block Effects and Unequally Spaced Factor
Levels” (2000) The American Statistician 54, 1, 5-11
The
Essential
Plot
Main Effects
>90% of SS
Shows significant
high order
interactions
The Formal Analysis: Adds Little
Comments:
Coding Used
_1, 0, 1
Not equally
spaced levels
Other analyses
conducted
Current Research Project-Partial Confounding
Purpose: To describe partial confounding and to evaluate the advantages it provides when a
blocked 2-level factorial is used to estimate a main effects plus two-factor interaction model.
Abstract: Traditionally blocked 2-level factorial experiments may confound some 2- factor
interactions with blocks. Increased precision can be obtained by the use of partial
confounding. Partial confounding uses fractional factorials and a different confounding
relationship for each fraction to increase the precision of estimates. We describe partial
confounding for 2-level factorials for traditional and split-plot blocking. We show the
increases in precision obtained by partial confounding and show when it is useful. We give
many examples of blocked and split-plot experiments where partial confounding provides
increased precision. The precision of partially confounded designs is compared with the
precision obtained using computer generated designs.
A simple example illustrates partial confounding. Consider a 24 experiment that is run in 4blocks. The traditional blocking procedure is to use blocking generators ABC and ABD so that
the 2-factor interaction CD is also confounded with blocks. Partial confounding uses the two
half fractions with defining contrast I = ABC and uses a different blocking procedure in each
half fraction. In one half- fraction ABD and CD are confounded with blocks and in the other
half fraction ACD and BD are confounded with blocks. Increased overall precision is obtained
because the confounded interaction terms are partially estimated within blocks.
43
New Observation: Improved Blocking
for Factorial Designs
• Why Block?
– Increase precision of the experiment
– Reduce bias
– Better answers to the questions of interest
• Consider an experiment when 4 runs can
conveniently be done in a shift. With Block size 4,
the shift-to-shift variation is placed in Blocks.
• How should we do the blocking?
44
24 in Four Blocks - Notes
• Confounded Blocking Relationship:
• I = ABC = ABD = CD
– Two-Factor Interaction is also confounded
• Model of Interest: 4 main Effects plus 6 Two-factor
interactions
• Conducting the experiment:
– Randomize Block Order
– Randomize Run Order in Each Block
• Can we do better?
– Computer Generated Design
– Partially Confounded design
45
Confounded Blocking:
24 Factorial in Four Blocks
Obs.
A
B
C
D
Blk
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Parameter Variances
• V(β) = σb2 /4 + σe2/16
– Constant and CD terms
• V(β) = σe2/16
– All other model terms
• Design is Orthogonal
• Maximum Variance of Prediction is the sum of
the parameter variances
47
Properties of Confounded 24 Design
Prediction Variance Profile
Average Variance = 0.465278
Maximum Variance = (2/4)σb2 + (11/16)σe2
= 0.5 + 0.6875 = 1.1875 when σ = 1
48
Globally Optimum Design
(A design that may unachievable)
• Power-Of-Orthogonality (POO) Theorem
– Orthogonal with maximum diagonals of
– V-, G-, A- & D Optimal
X’V-1X
Optimal Max. Var. = (1/4)σb2 + (11/16)σe2
= 0.25 + 0.6875 = 0.9375 when σ = 1
Corollary: 2k-p is globally optimum when no
model terms are confounded with blocks
49
Computer Generated Design:
JMP Custom Design with 10,000 Starts
Run Random Block
1
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1
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Design Evaluation
Prediction Variance Profile
X1
-1
1
-1
1
-1
-1
-1
1
1
1
1
-1
1
-1
-1
1
X2
-1
-1
1
1
1
-1
-1
-1
1
1
-1
1
-1
-1
1
1
X3
1
-1
-1
-1
1
-1
1
1
-1
1
1
1
-1
-1
-1
1
X4
1
-1
-1
1
1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
Y
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Average Variance = 0.454653
50
Partially Confounded Blocking:
24 Factorial in Four Blocks
Obs.
A
B
C
D
Blk
1
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1
2
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3
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4
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32
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32
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51
Partial Confounding Relationship
I=-ABC Half Fraction block on ABD
So Blocks 1 and 4 are exactly the same
as for confounded blocking
I=ABC Half Fraction block on ACD
So half the items in Blocks 2 and 3
will change signs
Average Variance = 0.446759
Prediction Variance Profile
52
Illigitemi non carborundum
Questions?
Comments?
53
Partial Confounding Discussion
• Should a computer program beat a Grand
Master Chess player?
– Yes – Chess is a deterministic game
• Should a computer program be able to beat an
experienced experimental designer?
– No – Science is open-ended
• Opportunity for research where a computer
generated design is current best practice
– Extend to more factors
– Workable Ph.D. Dissertation
54
Classical Definition: A
Completely Randomized Design
• “Completely randomized designs are
designs in which the assignment of factorlevel combinations to a test run sequence or
to experimental units (physical entities on
which measurements are taken) is made by
a random process where all assignments are
equally likely” Gunst(QP, Feb. 2000)
55
The Classical Definition is
Inadequate
• It does not address resetting so it does not
address how industrial and scientific
experiments are conducted
• It does not address the inherent split-plot
aspects of experiments using equipment
– This effects the desired inferences
– New edition of MGH changes definition
Letter to the editor on “Randomization is the Key to
Experimental Design Structure” by R. F. Gunst,
Quality Progress (2000), May, 14.
56
Operational Definition: A
Completely Randomized Design
• Observation = Model + Error
• A completely randomized design is
achieved by using a process that makes the
errors independent
– A random order is necessary but not sufficient
to achieve a CRD
– Do what is needed so that each experimental
unit is treated I the same way
– Consistent with Fisher
57
Split-Plot Experiments
• Main Effects Are (Partially or Fully)
confounded with Blocks (Cochran and Cox
1957)
• Others use a less general definition
Purpose of a Ph. D. Advisor
• Provide a workable topic that can be
completed within a year
Types of Factors
• Require Resetting for Randomization
– Hard-to-Change (HTC)
• Temperature
– Easy-to-Change (ETC)
• Current density
• Not requiring resetting
• Surfactant type
• Determined by experimental situation
60

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