Paul G. Published Printed in the United States of America. Library of Congress Cataloging-in-Publication Data. Mathews, Paul G. Includes bibliographical references and index. ISBN hardcover, case binding : alk.

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Published Printed in the United States of America 12 11 10 09 08 07 06 05 04 5 4 3 2 1. Mathews, Paul G. Includes bibliographical references and index. ISBN hardcover, case binding : alk.

Statistical hypothesis testing. Experimental design. ScienceStatistical methods. EngineeringStatistical methods. M Not for resale. No part of this publication may be reproduced in any form, including an electronic retrieval system, without the prior written permission of ASQ.

No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Publisher: William A. ASQ Mission: The American Society for Quality advances individual, organizational, and community excellence worldwide through learning, quality improvement, and knowledge exchange.

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Box , Milwaukee, WI Visit our Web site at www. Design of experiments DOE is a methodology for studying any response that varies as a function of one or more independent variables or knobs. By observing the response under a planned matrix of knob settings, a statistically valid mathematical model for the response can be determined. The resulting model can be used for a variety of purposes: to select optimum levels for the knobs; to focus attention on the crucial knobs and elim- inate the distractions caused by minor or insignificant knobs; to provide predictions for the response under a variety of knob settings; to identify and reduce the responses sen- sitivity to troublesome knobs and interactions between knobs; and so on.

Clearly, DOE is an essential tool for studying complex systems and it is the only rigorous replacement for the inferior but unfortunately still common practice of studying one variable at a time OVAT. Although Id had a mathematical statistics course as an undergraduate physics student, I found that my training in statistics was completely inadequate for survival in the GE organization. However, GE knew from experience that this was a major weakness of most if not all of the entry-level engineers coming from any science or engineering program and still is today , and dealt with the prob- lem by offering a wonderful series of internal statistics courses.

To tell the truth, we spent most of our time in that class solving DOE problems with pocket calculators because there was lit-. Although to some degree the calculations distracted me from the bigger DOE picture, that course made the power and efficiency offered by DOE methods very apparent.

During my twelve years at GE Lighting I was involved in about one experiment per week. Many of the systems that we studied were so complex that there was no other possible way of doing the work. While our experiments werent always successful, we did learn from our mistakes, and the designs and processes that we developed benefited greatly from our use of DOE methods. The proof of our success is shown by the longe- vity of our findingsmany of the designs and processes that we developed years ago are still in use today, even despite recent attempts to modify and improve them.

Although I learned the basic designs and methods of DOE at GE, I eventually real- ized that we had restricted ourselves to a relatively small subset of the available experi- ment designs. This only became apparent to me after I started teaching and consulting on DOE to students and corporate clients who had much more diverse requirements. I have to credit GE with giving me a strong foundation in DOE, but my students and clients get the credit for really opening my eyes to the true range of possibilities for designed experiments.

The textbooks that I chose for those classes were Montgomery, Design and Analysis of Experiments and Hicks, Fundamental Concepts in the Design of Experiments, however, I felt that both of those books spent too much time describing the calculations that the software took care of for us and not enough time presenting the full capabilities offered by the software. Since many students were still struggling to learn DOS while I was trying to teach them to use MINITAB, I supplemented their text- books with a series of documents that integrated material taken from the textbooks with instructions for using the software.

As those documents became more comprehensive they evolved into this textbook. I still have and occasionally use Montgomery; Box, Hunter, and Hunter, Statistics for Experimenters; Hicks; and other DOE books, but as my own book has become more complete I find that I am using those books less and less often and then only for reference.

I purposely limited the scope of this book to the basic DOE designs and methods that I think are essential for any engineer or scientist to understand. Ive left coverage of other experiment designs and analyses, including qualitative and binary responses, Taguchi methods, and mixture designs, to the other books. However, students who learn the material in this book and gain experience by running their own experiments will be well prepared to use those other books and address those other topics when it becomes necessary.

Obviously this is an important topic. Even if you choose the perfect experiment to study a particular problem, that experiment will waste time and resources if it uses too many runs and it will put you and your orga- nization at risk if it uses too few runs. Although the calculations are not difficult, the older textbooks present little or no instruction on how to estimate sample size.

To a large degree this is not their faultat the time those books were written the proba- bility functions and tables required to solve sample-size problems were not readily available. But now most good statistical and DOE software programs provide that information and at least a rudimentary interface for sample-size calculations. This book is unique in that it presents detailed instructions and examples of sample-size calculations for most common DOE problems.

This book is appropriate for a one-quarter or one-semester course in DOE. Although the book contains a few references to calculus methods, in most cases alternative methods based on simple algebra are also presented. Students are expected to have good algebra skillsno calculus is required. As prerequisites, students should have completed either: 1 a one-quarter or semes- ter course in statistical methods for quality engineering such as with Ostle, Turner, Hicks, and McElrath, Engineering Statistics: The Industrial Experience or 2 a one- quarter or semester course in basic statistics such as with one of Freunds books and a one-quarter or semester course in statistical quality control covering SPC and accep- tance sampling such as with Montgomerys Statistical Quality Control.

Although most DOE textbooks now present and describe the solutions to DOE prob- lems using one or more software packages, I find that they still tend to be superficial and of little real use to readers and students.

There are many other powerful programs available that dont get used much because they are so difficult to run. Why buy, learn, and maintain multiple software packages when one will suffice? Most graph attributes are easy to configure and can be edited after a graph is created.

MINITAB has a simple but powerful integrated sample-size calculation inter- face that can solve the most common sample-size problems. This eliminates the need to buy and learn another program that is dedicated to sample-size calculations. MINITAB can also be used to solve many more complex sample- size problems that are not included in the standard interface. All of the custom analysis macros that are described in this book are provided on the CD-ROM included with the book.

Preface xvii. Variable names are capitalized and displayed in the standard font. Since many readers and students who would consider this book have rusty statistical skills, a rather detailed review of graphical data presentation methods, descriptive sta- tistics, and inferential statistics is presented in the first three chapters.

Sample-size calculations for basic confidence intervals and hypothesis tests are also presented in Chapter 3. This is a new topic for many people and this chapter sets the stage for the sample-size calculations that are presented in later chapters.

Chapter 4 provides a qualitative introduction to the language and concepts of DOE. This chapter can be read superficially the first time, but be prepared to return to it fre- quently as the topics introduced here are addressed in more detail in later chapters. Chapters 5 through 7 present experiment designs and analyses for one-way and multi-way classifications. Chapter 7 includes superficial treatment of incomplete designs, nested designs, and fixed, random, and mixed models. Chapter 8 provides detailed coverage of linear regression and the use of variable transformations.

Polynomial and multivariable regression and general linear models are introduced in preparation for the analysis of multivariable designed experiments. Chapters 9, 10, and 11 present two-level full factorial, fractional factorial, and response-surface experiment designs, respectively.

Although the two-level plus centers designs are not really response- surface designs, they are included in the beginning of Chapter 11 because of the new concepts and issues that they introduce.

Descriptions of simple experiments with toys that could be performed at home or in a DOE class. Paper helicopter templates are provided on graph paper to simplify the construction of helicopters to various specifications. Microsoft Excel experiment design files with integrated simulations. In many ways, the material in this book is easy and the hard thingsthe ones no book can captureare only learned through experience.

But dont rush into performing experiments at work where the results could be embarrassing or worse. Rather, take the time to perform the simple experiments with toys that are described in the documents on the supplementary CD-ROM. If you can, recruit a DOE novice or child to help you perform these experiments.

Observe your assistant carefully and honestly note the mistakes that you both make because then youll be less likely to commit those mistakes again under more important circumstances. And always remem- ber that you usually learn more from a failed experiment than one that goes perfectly.

Table of Contents. Chapter 10 Fractional Factorial Experiments. Chapter 11 Response-Surface Experiments. Appendix A Statistical Tables.

A plot permits you to explore a data set visually, and you will often see things in a plot that you would have missed otherwise. For example, a simple histogram of measurement data can show you how the data are centered, how much they vary, if they fall in any special pattern, and if there are any outliers present.

These char- acteristics are not obvious when data are presented in tabular form. Usually we plot data with a specific question in mind about the distribution loca- tion, variation, or shape. But plotting data also lets us test assumptions about the data that weve knowingly or unknowingly made. Only after these assumptions are validated can we safely proceed with our intended analysis. When theyre not valid, alternative methods may be necessary.

Qualitative data characterize things that are sorted by type, such as fruit apples, oranges, pears,. Qualitative data are usually summa- rized by counting the number of occurrences of each type of event. Quantitative data characterize things by size, which requires a system of measurement. Examples of quantitative data are length, time, and weight. Design of experiments DOE problems involve both types of data, and the distinction between them is important.

A bar chart is constructed by first deter- mining the different ways the subject can be categorized and then determining the num- ber of occurrences in each category. The number of occurrences in a category is called the frequency and the category or type is called the class.

A bar chart is a plot of frequency versus class. Bar lengths correspond to frequencies, that is, longer bars correspond to higher frequencies.

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## Design of Experiments with Minitab

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## Design of Experiments with MINITAB. Paul Mathews

Includes bibliographical references and index. ISBN hardcover, case binding : alk. Statistical hypothesis testing. Experimental design. M Not for resale. No part of this publication may be reproduced in any form, including an electronic retrieval system, without the prior written permission of ASQ.