2
Getting Started
• To earn a Ph.D. in statistics, you must master a specific subject area in statistics.
• As you take courses and prepare for your Ph.D. comprehensive exams, the emphasis is
on (i) what you have learned and (ii) how you think. This is the first step to achieving
this mastery.
• After you pass your comprehensive exams, the focus will shift to what you will do to
address an original research question in statistics. This is the second step to achieving
this mastery.
• If successful, you will write and defend a dissertation that extends the existing knowledge base in statistics.
• The physical act of writing a dissertation is difficult for almost every Ph.D. student.
Here are several reasons that students find writing a dissertation very difficult.
– The dissertation is often the first major research experience a student is expected
to work independently.
– Because the goal of technical writing is focused on presentation rather than finding
new results, the process of writing is not very exciting.
– Technical writing requires a lot of time (writing paragraphs, correcting grammar,
forming arguments, using correct fonts, ...)
– Producing tables, figures, and reference lists can be tedious and boring.
– In your academic career, homework assignments and exams can be acceptable
even if they are incomplete or have minor errors. This is not the case with a
dissertation. You must address incomplete issues that you have been ignoring
and research results must be completely correct.
• You must realize that when writing a dissertation you will be required to think deeply
on an unsolved problem. This intellectual process will enable you to find a solution to
this problem. Then you have to prepare technical and structured presentations (the
written dissertation and the Ph.D. defense) to convince other statisticians that you
have truly solved this problem.
2.1
Selecting a research area in statistics and a topic for the dissertation.
• As a Ph.D. student in statistics, you are given the opportunity to intensively explore
a subject that interests you.
• Writing your dissertation will be a demanding experience. A dissertation represents a
personal academic achievement. Choosing a research topic is the first step in preparing
a dissertation. Hopefully, your dissertation topic in statistics is one that you are truly
interested in and that you will find personally rewarding.
• However, finding a topic for your dissertation research is not always easy. Possible
topics can originate from different sources such as
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– Suggestions from faculty members. I recommend that you talk to a faculty member
as early as possible about your interests in statistics. You should let them know
you just want to informally discuss your interests. Then, they may contact you at
a later date if they have a suitable research topic that agrees with your interests.
– Material you liked in previous courses. You may be able to find a topic based on
questions and problems that you would like to study further. You may be able to
take what you had read and learned from these courses to develop a dissertation
question.
– Something you read in a textbook, journal, or on the internet. If you become
familiar with searching available sources to find relevant scholarly research materials, you may get ideas for a research topic. I recommend that you first become
familiar with resources at your own university.
• The topic you eventually propose to your Ph.D. advisor must involve a research question that can be addressed within the time constraints of writing a dissertation. Often
this will be a requirement to complete the dissertation within two to three years after
passing your comprehensive exams.
• Here are several questions you need to consider when finding a dissertation topic:
– Will your topic sustain your interest for as long as it takes to complete a dissertation? (One year? Two years? Three years?)
– Will you have access to the existing literature (such as journals and texts) related
to your topic?
– Will you have sufficient time and resources (e.g., financial support)?
• Avoid a topic that is too broad or overly ambitious. In general, the recommendation is
to find a topic related to a question that can be thoroughly researched in a reasonable
time frame. If the topic is too large in scope, then it is likely that you would fail to
adequately address the topic.
• Consider the following questions for narrowing the scope and focus of a dissertation
topic:
– Are you able to formulate your research questions in written form?
– Are other statisticians currently work in this area?
– Can you identify publications (past and current) that are relevant to your topic?
– What is the extent of your knowledge regarding the research in the area of your
topic?
– How will acquire publications and texts to support your topic research?
My personal experience
• When selecting a research area and a topic for your dissertation, I recommend the
following:
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1. Make a list of area of statistics in the courses you have completed and rank them based
on your interest level. Also include whether or not a faculty member is interested in
research in that area. For example:
Area
Student Interest Level Faculty Interest
Response surface methodology
very high
yes
Experimental design
very high
yes
Sampling
very high
do not know
Statistical quality control
high
do not know
Linear models
medium
yes
Nonparametrics
medium
no
Categorical data analysis
medium
do not know
Generalized linear models
medium
yes
Bayesian statistics
low
do not know
Multivariate statistics
low
no
Time series
very low
no
Stochastic processes
very low
no
Probability theory
not at all
yes
When you assess your interest, be honest. If possible, you want to avoid working in a
research area for over a year that you have little interest. It is hard to stay motivated
when your interest level is low.
2. Within each of the areas that you are interested, make a list of topics you enjoyed most
in your coursework or read about. For example:
• Area: Response surface methodology.
designs, graphical methods.
Topics: Mixture experiments, optimal
• Area: Sampling.
Topics: Adaptive cluster sampling, bootstrapping methods
when sampling from finite populations.
• Very often a dissertation topic involves research in more than one area of statistics.
For example:
– Optimal designs for generalized linear models.
– Optimal bayesian designs.
Therefore, you should also think about combining areas of interest.
3. Next, talk to faculty members about the research areas and possible topics you are
interested in. The faculty member may be able to give you advice about who may be
available to work with you in your area of interest. If you are lucky, the faculty member
may have a potential research problem for you. As a faculty member, I appreciate when
a student has some ideas about potential research areas before he or she comes to see
me.
4. It will require some patience, but eventually, you will find a research area of interest
and an advisor. Once that happens, a specific topic in that research area will be
developed with the help of your advisor.
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• For my dissertation, I talked to several faculty and expressed interest in statistical
quality control, response surface methodology, and experimental design. During my
discussion with Dr. James Lucas, I was introduced to a problem in optimal response
surface design.
• After several more discussions with Dr. Lucas, he agreed to be my Ph.D. advisor and
that my dissertation area would be in optimal response surface design. The specific
topic would be the optimality properties of composite mixed resolution designs.
2.2
Preliminary reading of texts and professional papers.
• Once you have a potential topic for your dissertation research, you need to begin your
preliminary background research to become familiar with the prior research related
your topic and to increase your overall knowledge in that research area of statistics.
You need to determine how your dissertation research fits within the existing literature
in that area of statistics.
• To become a researcher in any subject, a Ph.D. student must search the published
literature with the goal of finding and then reading as much as you can about the
research area. In statistics research, a student begins by studying “general” references
and publications which include text textbooks and review articles.
• In general, textbooks provide an overall review of important results, give examples to
highlight these results, and may provide an extensive set of references that can help
you develop a reading list.
• Textbooks, however, often lack important details (such as having only outlines of statistical methodology, partial proofs of theorems, and motivation for results). You need
to go to the original sources (usually journal articles) to get the details.
• Then the student must expand his or her literature search to include more “specialized” publications. These will primarily be refereed statistical journals, conference
proceedings, technical reports, and monographs that are focused on your particular
research area in statistics. It is in these publications that statisticians contribute new
information on theory and methods in statistics.
• For your preliminary reading, you should ask your advisor first for a list of “general”
references and later for “specialized” references.
• You will need to read both older literature as well as current literature to determine
what has been published in your area of research. In English, there is an expression
“You do not want to reinvent the wheel.” which means you do not want to spend your
time on a project that already been done by someone else. If you limit yourself to
reading only the most current literature or only older literature, you may miss some
important results and then attempt to solve a problem that has already been solved.
• Therefore, you will need to read recent conference proceedings to see if anyone is
working in the same area and to be sure you are “not reinventing the wheel”. You do
not want to work on a dissertation for over a year and then find that the research has
already been done.
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• For organizational reasons, you should keep the full bibliographic citations of references.
For example, if you photocopy pages from a textbook or journal, you should record
author, title, date, pages, and other information that you will need in the Bibliography
/ References section of your dissertation.
• There is no specific length that a bibliography for a dissertation must be. It could
include as few as 25 references or as many as 150 references. The length depends on
how much prior research has been published in the area of your research.
My personal experience
• After I had a topic for my dissertation, I went to the library at my university and
found several books on experimental design. My advisor also gave me several articles
and his dissertation to read.
• In each of these sources, I looked at the set of references. From these references, I
prepared my own reference list of what I might need to read. I then asked my advisor
which of my references he would consider the most important for me to read first.
• Do not expect everything you read will be important to your research. I read most
articles quickly. That is, I only wanted to determine what were the most important
results in each article. I was not concerned about understanding all of the details
(such as following every step in a proof).
– If the most important results were directly related to my dissertation topic, I
would spend more time later understanding all of the details.
– If the most important results were not directly related to my dissertation topic, I
would spend no more time on that article.
• As I continued reading the statistical literature recommended by my advisor, I began
to make my own decisions about what to read.
• If I believed the reference would be included in my dissertation, I kept notes about
that reference. I will discuss organization and taking notes in the next section.
• I recommend beginning with a search for review articles or edited books on specialized
topics. This will give you a good starting point for creating a reading list.
• For example, suppose you want to read more about “variance dispersion graphs”. The
following pages contain the references from a review I wrote on graphical methods
used to display prediction variance properties of response surface design. This review
appeared in
Borkowski, J.J. (2006) Book Chapter: “Graphical Methods for Assessing the Prediction
Capability of Response Surface Designs”, Response Surface Methodology and Related
Topics, Editor: Andre Khuri. Publisher: World Scientific, Chapter 14: pages 349-378.
• The references containing the keywords variance dispersion graphs or dispersion graphs
have those keywords highlighted in boxes. The next step would be to find and quickly
review these articles.
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2.3
Keeping notes and being organized.
• Many students are very excited when they begin their dissertation research. This
excitement must be matched with good organizational skills to have the greatest benefit
to completing your dissertation more quickly.
• It is not possible to remember all the information you have read, and as time passes
you tend to forget things. Therefore, start writing notes from the very beginning.
• Record information as soon as possible. Many people find it useful to keep a research
notebook that will contain
– Research information that you have found useful such as notes on publications
you have read.
– A thorough record of textbooks you will include as references in case you have to
read them again.
– Your own ideas and observations including possible ways to address the research
problem.
– Problems that you encounter.
– Write your questions before you meet with your advisor.
– New questions or ideas that interest you. This could be useful for future research
after the dissertation.
– References to read at a later date.
• To be successful writing your dissertation requires that you develop good organization
skills. The goal is to develop these skills as early as possible in the process.
• You need to develop note-taking skills that are efficient with respect to time. That
is, you need to learn to focus on recording only what is directly relevant to your
dissertation and ignore what is not relevant.
• As stated in Section 2.2, you need to keep bibliographical records of what you have
read. Keeping accurate records from the beginning will save a lot of time during the
dissertation writing process.
• You also want to keep two copies of the records with at least one of those copies in
electronic format, and keep copies in different locations. For example, what good is
keeping both copies on your computer and the your computer breaks down? None.
• Therefore, even though most of your research material will be saved in computer files,
I strongly recommend that you keep a paper copy. Also, make regular back-ups of your
work as you work on the computer. You do not want to waste time having to retype
your notes.
• Warning: When you refer directly to the work of another author in your dissertation,
it is essential to keep very accurate records. You want to be sure that you avoid
plagiarism which is the improper use of another researchers’ work without giving them
credit. That is, you are taking credit for someone else’s hard work.
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• It does not matter whether this is done intentionally or unintentionally. It is considered
unprofessional and unethical. We will discuss plagiarism in Section 7.5 of the course
notes.
My personal experience
• As I mentioned in Section 2.1, I kept notes of important results from published research
and textbooks. I also created a list of keywords corresponding to major topics discussed
in the publication.
• For example, the following are a partial list of keywords I created for my dissertation:
A-efficiency, D-efficiency, G-efficiency, Taguchi methods, central composite designs,
fractional-factorial designs, restrictions on randomization, ...
• I would then assign one or more of these keywords throughout my notes if the content
was related to that keyword. This helped me later when I began to write the chapters
of my dissertation (and the literature review, in particular). By organizing my notes
with keywords, I could easily find and group subsets of my notes related to a common
topic.
• I now scan my notes and save them as pdf files. This is a quick way to have an
electronic copy of my written notes without having to type them.
• When something is unclear while I am reading a research paper or textbook, I like
to attach simple notes to remind me of what the problems are. I like to write simple
questions such as
– Why is this true?
– Are there any practical applications of this result?
– Is there a simpler way to approach this problem?
– Is there other references that addresses this problem?
– Is this relevant to my research?
Later, when I return to the reference, I am reminded quickly of what the issues are.
• You must also be prepared that some attempts to solve a problem (such as a proof)
will be unsuccessful. It is easy to become frustrated when you cannot find a solution
to a problem after several attempts.
• Despite the initial lack of success, do not throw away the notes containing the unsuccessful attempts. If you give yourself time (i.e, “step away from the problem”) and
then look at what you had written at a later date, you may gain new insight on how
to approach the problem.
• I found this approach to work when trying to write computer algorithms. It is frustrating to know what you should do, but cannot get the computer to do it. I just put
the code aside for awhile, and then reviewed it one or two days later after reducing
the stress. It was easy to discover what the mistake was now that I was no longer
frustrated.
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• In summary, you will find that you will have many questions as you read and write.
Keep detailed notes.
• I still keep notes every time I get an idea for a new research topic. If I did not keep
notes, I would probably forget the ideas I had.
2.4
Efficient management of your work time.
• Throughout the research and dissertation process, you will have numerous tasks. Some
tasks will be simple, others will be complex. Some tasks can be performed at any time
(such as typing) while other tasks have a specific order (such as the PhD proposal after
preparing a literature review).
• Some tasks are complex because a dissertation is a lengthy piece of professional work.
Lengthy not only in terms of how much you write, but lengthy in terms of the hours of
work required. Thus, strong organizational and time-management skills are required.
• The dissertations process is long. Therefore, it is essential for a PhD student to continually think about the amount of work that is required to complete the components
of the dissertation, and to develop good work habits as soon as possible.
• As you work on your dissertation, the weeks seem to pass by more quickly. Once again,
this means that you will need to be well-organized from the start so that your progress
toward a completion date is planned.
• Unfortunately, many Ph.D. students do not follow a well-organized work schedule.
They have the “I will do it tomorrow” attitude. In English, we call this “procrastination”. The lack of structure will seriously affect the quality of the dissertation. It will
almost certainly lead to increasing stress levels as time passes.
• Many Ph.D. students prepare a work schedule. Some Ph.D. advisors will require a
work schedule as a way to assess student progress.
• Once you have a schedule, it is much easier to set targets for the completion of the
separate parts of the dissertation.
• One challenge is making efficient your use of your time in gathering research information. The availability of information from internet sources can create problems with
respect to location (Can I find it?) and quality (Is is trustworthy?). Not everything
you find on the internet will be acceptable for use in a dissertation. Therefore, having
good internet search skills may be essential.
• Some publications (such as technical reports) can not be accessed quickly. You may
need to (i) request copies using inter-library loan services, (ii) contact the author
directly, (iii) need someone to translate the paper. In any of these cases, you will need
to allow extra time.
• It is unlikely that you will understand everything you read the first time you read
it. You should expect to have to read some references multiple times. This is another
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time demand. I remember having to spend several days reading and re-reading a paper
on optimal designs before I finally understood the argument and all of the technical
details.
• Finally, you must allow enough time for several revisions of your writing. Do not
expect your advisor and committee members to be happy after a single revision of
your writing. Expect for them to make suggestions and require changes several times.
• Remember that your dissertation needs to be completed along with other commitments
such as a job, time with family, observing holidays, ... To develop a structured work
schedule, you will need to estimate the time it will take to complete each research task.
Be realistic, not optimistic, when estimating time.
My personal experience
• When you begin working on a dissertation, the question is no longer ‘Are smart enough?’.
The question is now ‘Can you do independent research?’. Being smart is a necessary
but not sufficient condition for earning a Ph.D.
• You will no longer be taking courses and working and studying with your friends. You
will be be working alone. Therefore, I recommend having a structured work schedule.
• When working on my dissertation, I had a structured work schedule. To support
myself, I worked 20 to 30 hours a week as a statistician for the chemical company
DuPont. I also worked around 20-30 hours a week on my dissertation in the evenings
and on weekends. The point is ‘Do not expect to continue to have the same life style
once you begin working on the dissertation’.
• Because English is a second language for PhD students at Thammasat University, you
will have to deal with this issue and the management of your time.
• At first, it will probably take longer for you to read journal articles and textbooks
because of translation. This will also be true when writing in English. You may have
to allow for extra time for your advisor to make corrections to English grammar as
well as the statistical content.
• You need to set up a schedule that has you working for at least 5 days per week for a
total of at least 20 hours per week. If you have a scholarship and do not have to work,
you should be working at least 30-40 hours a week on your dissertation.
• Expect to spend at least one year researching and writing your dissertation.
• Learn to work without interruptions (phone calls, visits from friends, text messaging,
watching television...).
• Everything I discussed in Section 2.3.1 regarding keeping notes and being organized
also takes time. However, it is an efficient use of time because investing time in being
organized will save time later in the process.
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• You do not have to approach the dissertation linearly. For example, you do not have to
complete Chapter 1 (Topic 1) before starting Chapter 2 (Topic 2), or complete Chapter
2 (Topic 2) before starting Chapter 3 (Topic 3), and so on.
– Most often Ph.D. research will involve computation and theory. You do not have
to work out all of the theoretical results before proceeding to the computational
results. You can alternate working between both.
– Generating numerical or computational results will often give insight into theoretical results.
– Generating theoretical results will often give insight into numerical or computational results.
– That is why it is not always an efficient use of time to focus only on the theory
or only on the computation at one time.
• If you are frustrated or tired of staring at the computer, you should take a break for
awhile and work on something else (such as typing parts of the dissertation). You will
be more productive if you can lower your stress level.
• There will always be stress when working on Ph.D. research, and the stress levels tend
to increase over time. You just need to control the stress and not let it slow down your
progress. Good management of time helps to control stress.
• Everything I have said is advice is focused on avoiding being unproductive. However,
if you are being productive, do not change what you are doing.
– Most of my Ph.D. research was done in the evening after I finished working during
the day at DuPont. If I was being productive, I would not let myself be distracted
(no phone calls, no television, no meetings with friends, ...).
– Do not let distractions prevent you from working on your dissertation.
There are times you have to say ‘No’.
∗ “No, I cannot meet you for lunch today.”
∗ “No, I cannot go to Future Park.”
∗ “No, I cannot watch this movie.”
• Yes, there will be time to relax with friends and family. This is about time management.
Just expect to spend less time relaxing while working on the dissertation. I still enjoyed
time with friends and family because I did not waste time on other less important
activities. In general, staying focused and avoiding distractions help to guarantee
success.
2.5
Overview of the contents of a dissertation.
• All PhD dissertations in statistics have the following elements:
– A clear research topic and specific problems that your dissertation will solve.
– A comprehensive review of the statistical literature related to your topic.
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– Performing independent research related to your topic.
– Justification of the methodology you will use in your research.
– A discussion of your research results, and how the results relate to your original
question.
– A professional presentation of the dissertation in the form of a Ph.D. defense.
• Dissertations vary in format and structure across universities. It is important that
you become familiar with the specific requirements of Thammasat University and the
requirements of the Ph.D. degree program in statistics.
• A common format for a dissertation is
– Title Page.
– Abstract. The abstract is a brief statement about the dissertation research.
– Table of Contents.
– List of Tables.
– List of Figures.
– List of Abbreviations/Notation. (optional) which are alphabetically ordered.
– Introduction. The Introduction to your dissertation should describe the topic
and scope of the dissertation. It should explicitly state what new statistical
knowledge you will be presenting and why it is important. You want to clearly
define the objectives of your research.
– Literature Review. I will be discussing this later in the course.
– Methodology. You must describe the methodology you will be using to generate
your research results. You need to demonstrate to your audience that you have a
good understanding of the methods to be used. It should include descriptions of
any computational methods you used.
– Results. Your results can be presented in either one long chapter or several
shorter chapters. Computational results are often presented in the form of figures,
tables, and data analysis. Theoretical results are usually in the form of proofs of
theorems and corollaries.
– Discussion: A discussion is a summary of the Results, how the Results are
related to the content in the literature review, and should address the research
questions stated in the Introduction. It also contains recommendations and ideas
for future research.
– Bibliography. This is a list of all references you have cited in your dissertation.
– Appendices. Appendices contain additional material not required in the chapters but was necessary in the research. Appendices would include computer code,
long tables of results, data sets, ...
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My personal experience
• The following pages were taken from the beginning of my dissertation.
– There is the cover page.
– The Table of Contents: Note that my chapters are not Literature Review, Methods, Results, Discussion. The format will not be the same at every university.
– The Abstract
– Lists of Tables and Figures
• Be sure to know the rules for Thammasat University or what your advisor expects. For
example, as a advisor, I require my Ph.D. students to type the dissertation in LaTeX
(which I am using for these course notes). I will not accept a dissertation typed in
Microsoft Word.
• At my university, Microsoft Word is not allowed for a dissertation in statistics. Several
reasons for this requirement:
– It is the standard for publishing in most statistical journals. Many journals will
no longer accept Word documents and require articles LaTex files be submitted.
– The numbering of chapters, theorems, equations, tables, and figures is automated.
For example, if you add or remove an equation, LaTeX will automatically renumber all the equations in your document.
– LaTeX is more flexible than Word for mathematics and statistics. Equations,
matrices, systems of equations,... are all easy to make. Also, LaTeX contains sets
of mathematical fonts that are not available in Word.
– LaTeX documents look more professional than Word documents.
• I now do all of my professional word processing using LaTeX. You will be given an
introduction to LaTeX during this short course.
• Check with your advisor what he or she accepts. You do not want to spend a lot of
time entering notes or typing the dissertation in Word and then find out months later
that you have to use LaTeX.
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TABLE OF CONTENTS
THE EVALUATION OF
MIXED RESOLUTION DESIGNS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
viii
x
Chapter
1 RESPONSE SURFACE METHODOLOGY . . . . . . . . . . . . .
1.1
1.2
by
John J. Borkowski Jr.
1.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quadratic Response Surface Designs . . . . . . . . . . . . . . . . .
1
2
1.2.1
1.2.2
Two-Level Fractional Factorial Designs . . . . . . . . . . . .
Composite Designs . . . . . . . . . . . . . . . . . . . . . . .
4
6
Taguchi System of Experimental Design . . . . . . . . . . . . . . .
9
1.3.1
1.3.2
1.3.3
A dissertation submitted to the Faculty of the University of Delaware in
partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Statistics
c 1992 John J. Borkowski Jr.
All Rights Reserved
3
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3.1
3.2
3.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Variance . . . . . . . . . . . . . . . . . . . . . . . . . .
Prediction Variance . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
39
47
4 OPTIMALITY PROPERTIES OF MIXED RESOLUTION
DESIGNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Introduction . . . . . . . . . . . . . . . . . . . . . . .
The Design Evaluation Criteria . . . . . . . . . . . .
The Support of Optimum Designs on the Hypercube
The Support of Mixed Resolution Designs . . . . . .
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57
62
64
Invariant Transformations . . . . . . . . . . . . . . . . . . .
65
Support of Optimum Designs on the Complete Barycentric Set J .
Support of Optimum Designs on Barycentric Subsets of J . . . . . .
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4.6.1
4.6.2
The Prediction Variance d((x, z), ξ) . . . . . . . . . . . . . .
The Generalized Variance |M−1 (ξ)| . . . . . . . . . . . . . .
79
84
Efficiencies of Mixed Resolution Designs . . . . . . . . . . . . . . .
90
4.4.1
4.5
4.6
4.7
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5 COMMENTS AND RELATED RESEARCH PROBLEMS . . . 103
5.1
5.2
5.3
5.4
5.5
5.6
9
11
15
2 MIXED RESOLUTION DESIGNS . . . . . . . . . . . . . . . . . .
18
Introduction . . . . . . . . . . . . . . . . . . . . .
Robustness vs Product Differentiation . . . . . .
Single Factor Arrays . . . . . . . . . . . . . . . .
Designs of Mixed Resolution . . . . . . . . . . . .
Smallest Composite Designs of Mixed Resolution
2.5.1
2.5.2
THE GENERALIZED VARIANCE AND PREDICTION
VARIANCE OF MIXED RESOLUTION DESIGNS . . . . . . .
4.1
4.2
4.3
4.4
Screening Designs . . . . . . . . . . . . . . . . . . . . . . . .
Taguchi Designs . . . . . . . . . . . . . . . . . . . . . . . . .
Taguchi Designs as Response Surface Designs . . . . . . . .
2.1
2.2
2.3
2.4
2.5
May 1992
Signal-to-Noise Ratios and Mixed Resolution Designs . .
The Development of Computer Aided Designs . . . . . .
Computer Aided Designs for the Mixed Resolution Model
Posterior Efficiencies . . . . . . . . . . . . . . . . . . . .
Irregular Fractions . . . . . . . . . . . . . . . . . . . . .
D−Optimality vs. Ds −Optimality . . . . . . . . . . . .
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107
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116
Appendix
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
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Minimum Aberration Designs . . . . . . . . . . . . . . . . .
Finding Minimum Aberration Mixed Resolution Designs . .
31
33
LIST OF FIGURES
1.1
A 15-Point Composite Design for Three Factors . . . . . . . . . .
8
2.1
Robustness vs Product Differentiation . . . . . . . . . . . . . . . .
22
LIST OF TABLES
1.1
The 9-Point 3-Level 4-Factor Screening Design (or Taguchi’s L9
Design) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2
The 36-Point Taguchi Design in 7 Factors . . . . . . . . . . . . . .
14
2.1
Mean Response by Level: The Byrne-Taguchi Example . . . . . .
19
2.2
Robustness vs Product Differentiation . . . . . . . . . . . . . . . .
20
2.3
Taguchi vs Composite Design Size Comparison . . . . . . . . . . .
25
2.4
Minimum Aberration: An Example . . . . . . . . . . . . . . . . .
32
2.5
Taguchi vs (Composite) Mixed Resolution Design Size Comparison
36
3.1
The Design Matrix X . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.2
The Modified Design Matrix XΔ
42
XΔ XΔ
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3.3
The
Matrix . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.4
The X X Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.5
The (X X)−1 Matrix . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.1
The Moment Matrix M(ξ) . . . . . . . . . . . . . . . . . . . . . .
81
−1
4.2
The Inverse Moment Matrix M
(ξ) . . . . . . . . . . . . . . . .
81
4.3
Summary Table of D-Optimum Weights . . . . . . . . . . . . . . .
87
4.4
Summary of D-efficiencies and G-efficiencies . . . . . . . . . . . .
97
4.5
G-Efficiencies for Replicated Star-Point Mixed Resolution Designs
100
5.1
D-efficiency and G-efficiency Comparison . . . . . . . . . . . . . .
114
A.1
Reference Table for Mixed Resolution Designs . . . . . . . . . . .
119
A.2
Table of Defining Relations for Mixed Resolution Designs . . . . .
122
The final chapter will include comments on related research problems with
emphasis on computer generated designs and their application to the mixed resolution model and the use of signal-to-noise ratios as a response surface analysis tool
for achieving robustness using mixed resolution designs.
ABSTRACT
Many industries have adopted the use of statistically designed experiments
for achieving a robust process, i.e., a process that is insensitive to changes and
perturbations in the uncontrollable process variables. In Japan, and then within
the United States and other countries, Dr. Genichi Taguchi popularized the use of
experimental design for achieving robustness. The Taguchi designs are generated
by crossing two orthogonal designs called the inner and outer array designs. As an
improvement to the Taguchi system of designs, a number of authors have suggested
alternative designs which are based on a single factor array. This dissertation will
describe and evaluate a new class of response surface designs, designs of “mixed
resolution,” which are based on a single factor array and can be adopted for achieving
a robust process.
In the evaluation of mixed resolution designs, the problem of finding globally
optimum designs for achieving a robustness process is addressed and solved. D− and
G−optimal designs for the mixed resolution model are defined and their optimality
properties are studied. The D− and G−efficiencies of the mixed resolution designs
are then calculated.
A discussion of model building considerations is presented as the different
experimental design approaches allow for the estimation of different sets of model
terms. The Taguchi system of experimental design in contrast with mixed resolution
designs will be the primary focus in this comparative discussion. This will include
an experimental run size comparison between single array and crossed orthogonal
arrays.
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