Design and Analysis of Experiments

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Preface v 1 Introduction to Designed Experiments 1 1.1 Strategy of Experimentation 1 1.2 Some Typical Applications of Experimental Design 8 1.3 Basic Principles 11 1.4 Guidelines for Designing Experiments 14 1.5 A Brief History of Statistical Design 21 1.6 Summary: Using Statistical Techniques in Experimentation 22 1.7 Problems 23 2 Basic Statistical Methods 25 2.1 Introduction 25 2.2 Basic Statistical Concepts 27 2.3 Sampling and Sampling Distributions 30 2.4 Inferences About the Differences in Means, Randomized Designs 36 2.4.1 Hypothesis Testing 36 2.4.2 Confidence Intervals 43 2.4.3 Choice of Sample Size 44 2.4.4 The Case Where 48 2.4.5 The Case Where and Are Known 50 2.4.6 Comparing a Single Mean to a Specified Value 50 2.4.7 Summary 51 2.5 Inferences About the Differences in Means, Paired Comparison Designs 53 2.5.1 The Paired Comparison Problem 53 2.5.2 Advantages of the Paired Comparison Design 56 2.6 Inferences About the Variances of Normal Distributions 57 2.7 Problems 59 3 Analysis of Variance 65 3.1 An Example 66 3.2 The Analysis of Variance 68 3.3 Analysis of the Fixed Effects Model 70 3.3.1 Decomposition of the Total Sum of Squares 71 3.3.2 Statistical Analysis 73 3.3.3 Estimation of the Model Parameters 78 3.3.4 Unbalanced Data 79 3.4 Model Adequacy Checking 80 3.4.1 The Normality Assumption 80 3.4.2 Plot of Residuals in Time Sequence 82 3.4.3 Plot of Residuals Versus Fitted Values 83 3.4.4 Plots of Residuals Versus Other Variables 88 3.5 Practical Interpretation of Results 89 3.5.1 A Regression Model 89 3.5.2 Comparisons Among Treatment Means 90 3.5.3 Graphical Comparisons of Means 91 3.5.4 Contrasts 92 3.5.5 Orthogonal Contrasts 94 3.5.6 Scheffe s Method for Comparing All Contrasts 96 3.5.7 Comparing Pairs of Treatment Means 97 3.5.8 Comparing Treatment Means with a Control 101 3.6 Sample Computer Output 102 3.7 Determining Sample Size 105 3.7.1 Operating Characteristic Curves 105 3.7.2 Specifying a Standard Deviation Increase 108 3.7.3 Confidence Interval Estimation Method 109 3.8 Other Examples of Single-Factor Experiments 110 3.8.1 Chocolate and Cardiovascular Health 110 3.8.2 A Real Economy Application of a Designed Experiment 110 3.8.3 Discovering Dispersion Effects 114 3.9 The Random Effects Model 116 3.9.1 A Single Random Factor 116 3.9.2 Analysis of Variance for the Random Model 117 3.9.3 Estimating the Model Parameters 118 3.10 The Regression Approach to the Analysis of Variance 125 3.10.1 Least Squares Estimation of the Model Parameters 125 3.10.2 The General Regression Significance Test 126 3.11 Nonparametric Methods in the Analysis of Variance 128 3.11.1 The Kruskal Wallis Test 128 3.11.2 General Comments on the Rank Transformation 130 3.12 Problems 130 4 Experiments with Blocking Factors 139 4.1 The Randomized Complete Block Design 139 4.1.1 Statistical Analysis of the RCBD 141 4.1.2 Model Adequacy Checking 149 4.1.3 Some Other Aspects of the Randomized Complete Block Design 150 4.1.4 Estimating Model Parameters and the General Regression Significance Test 155 4.2 The Latin Square Design 158 4.3 The Graeco-Latin Square Design 165 4.4 Balanced Incomplete Block Designs 168 4.4.1 Statistical Analysis of the BIBD 168 4.4.2 Least Squares Estimation of the Parameters 172 4.4.3 Recovery of Interblock Information in the BIBD 174 4.5 Problems 177 5 Factorial Experiments 183 5.1 Basic Definitions and Principles 183 5.2 The Advantage of Factorials 186 5.3 The Two-Factor Factorial Design 187 5.3.1 An Example 187 5.3.2 Statistical Analysis of the Fixed Effects Model 189 5.3.3 Model Adequacy Checking 198 5.3.4 Estimating the Model Parameters 198 5.3.5 Choice of Sample Size 201 5.3.6 The Assumption of No Interaction in a Two-Factor Model 202 5.3.7 One Observation per Cell 203 5.4 The General Factorial Design 206 5.5 Fitting Response Curves and Surfaces 211 5.6 Blocking in a Factorial Design 219 5.7 Problems 225 6 Two-Level Factorial Designs 233 6.1 Introduction 233 6.2 The 22 Design 234 6.3 The 23 Design 241 6.4 The General 2k Design 253 6.5 A Single Replicate of the 2k Design 255 6.6 Additional Examples of Unreplicated 2k Design 268 6.7 2k Designs are Optimal Designs 280 6.8 The Addition of Center Points to the 2k Design 285 6.9 Why We Work with Coded Design Variables 290 6.10 Problems 292 7 Blocking and Confounding Systems for Two-Level Factorials 304 7.1 Introduction 304 7.2 Blocking a Replicated 2k Factorial Design 305 7.3 Confounding in the 2k Factorial Design 306 7.4 Confounding the 2k Factorial Design in Two Blocks 306 7.5 Another Illustration of Why Blocking Is Important 312 7.6 Confounding the 2k Factorial Design in Four Blocks 313 7.7 Confounding the 2k Factorial Design in 2p Blocks 315 7.8 Partial Confounding 316 7.9 Problems 319 8 Two-Level Fractional Factorial Designs 320 8.1 Introduction 320 8.2 The One-Half Fraction of the 2k Design 321 8.2.1 Definitions and Basic Principles 321 8.2.2 Design Resolution 323 8.2.3 Construction and Analysis of the One-Half Fraction 324 8.3 The One-Quarter Fraction of the 2k Design 333 8.4 The General 2kp Fractional Factorial Design 340 8.4.1 Choosing a Design 340 8.4.2 Analysis of 2kp Fractional Factorials 343 8.4.3 Blocking Fractional Factorials 344 8.5 Alias Structures in Fractional Factorials and other Designs 349 8.6 Resolution III Designs 351 8.6.1 Constructing Resolution III Designs 351 8.6.2 Fold Over of Resolution III Fractions to Separate Aliased Effects 353 8.6.3 Plackett-Burman Designs 357 8.7 Resolution IV and V Designs 366 8.7.1 Resolution IV Designs 366 8.7.2 Sequential Experimentation with Resolution IV Designs 367 8.7.3 Resolution V Designs 373 8.8 Supersaturated Designs 374 8.9 Summary 375 8.10 Problems 376 9 Other Topics on Factorial and Fractional Factorial Designs 394 9.1 The 3k Factorial Design 395 9.1.1 Notation and Motivation for the 3k Design 395 9.1.2 The 32 Design 396 9.1.3 The 33 Design 397 9.1.4 The General 3k Design 402 9.2 Confounding in the 3k Factorial Design 402 9.2.1 The 3k Factorial Design in Three Blocks 403 9.2.2 The 3k Factorial Design in Nine Blocks 406 9.2.3 The 3k Factorial Design in 3p Blocks 407 9.3 Fractional Replication of the 3k Factorial Design 408 9.3.1 The One-Third Fraction of the 3k Factorial Design 408 9.3.2 Other 3kp Fractional Factorial Designs 410 9.4 Factorials with Mixed Levels 412 9.4.1 Factors at Two and Three Levels 412 9.4.2 Factors at Two and Four Levels 414 9.5 Nonregular Fractional Factorial Designs 415 9.5.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs 418 9.5.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs 425 9.5.3 Analysis of Nonregular Fractional Factorial Designs 427 9.6 Constructing Factorial and Fractional Factorial Designs Using an Optimal Design Tool 431 9.6.1 Design Optimality Criteria 433 9.6.2 Examples of Optimal Designs 433 9.6.3 Extensions of the Optimal Design Approach 443 9.7 Problems 444 10 Regression Modeling 449 10.1 Introduction 449 10.2 Linear Regression Models 450 10.3 Estimation of the Parameters in Linear Regression Models 451 10.4 Hypothesis Testing in Multiple Regression 462 10.4.1 Test for Significance of Regression 462 10.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients 464 10.5 Confidence Intervals in Multiple Regression 467 10.5.1 Confidence Intervals on the Individual Regression Coefficients 467 10.5.2 Confidence Interval on the Mean Response 468 10.6 Prediction of New Response Observations 468 10.7 Regression Model Diagnostics 470 10.7.1 Scaled Residuals and PRESS 470 10.7.2 Influence Diagnostics 472 10.8 Testing for Lack of Fit 473 10.9 Problems 475 11 Response Surface Methodology 478 11.1 Introduction to Response Surface Methodology 478 11.2 The Method of Steepest Ascent 480 11.3 Analysis of a Second-Order Response Surface 486 11.3.1 Location of the Stationary Point 486 11.3.2 Characterizing the Response Surface 488 11.3.3 Ridge Systems 495 11.3.4 Multiple Responses 496 11.4 Experimental Designs for Fitting Response Surfaces 500 11.4.1 Designs for Fitting the First-Order Model 501 11.4.2 Designs for Fitting the Second-Order Model 501 11.4.3 Blocking in Response Surface Designs 507 11.4.4 Optimal Designs for Response Surfaces 511 11.5 Experiments with Computer Models 523 11.6 Mixture Experiments 530 11.7 Evolutionary Operation 540 11.8 Problems 544 12 Robust Design 554 12.1 Introduction 554 12.2 Crossed Array Designs 556 12.3 Analysis of the Crossed Array Design 558 12.4 Combined Array Designs and the Response Model Approach 561 12.5 Choice of Designs 567 12.6 Problems 570 13 Random Effects Models 573 13.1 Random Effects Models 573 13.2 The Two-Factor Factorial with Random Factors 574 13.3 The Two-Factor Mixed Model 581 13.4 Sample Size Determination with Random Effects 587 13.5 Rules for Expected Mean Squares 588 13.6 Approximate F Tests 592 13.7 Some Additional Topics on Estimation of Variance Components 596 13.7.1 Approximate Confidence Intervals on Variance Components 597 13.7.2 The Modified Large-Sample Method 600 13.8 Problems 601 14 Experiments with Nested Factors and Hard-to-Change Factors 604 14.1 The Two-Stage Nested Design 604 14.1.1 Statistical Analysis 605 14.1.2 Diagnostic Checking 609 14.1.3 Variance Components 611 14.1.4 Staggered Nested Designs 612 14.2 The General m-Stage Nested Design 614 14.3 Designs with Both Nested and Factorial Factors 616 14.4 The Split-Plot Design 621 14.5 Other Variations of the Split-Plot Design 627 14.5.1 Split-Plot Designs with More Than Two Factors 627 14.5.2 The Split-Split-Plot Design 632 14.5.3 The Strip-Split-Plot Design 636 14.6 Problems 637 15 Other Topics 642 15.1 Nonnormal Responses and Transformations 643 15.1.1 Selecting a Transformation: The Box Cox Method 643 15.1.2 The Generalized Linear Model 645 15.2 Unbalanced Data in a Factorial Design 652 15.2.1 Proportional Data: An Easy Case 652 15.2.2 Approximate Methods 654 15.2.3 The Exact Method 655 15.3 The Analysis of Covariance 655 15.3.1 Description of the Procedure 656 15.3.2 Computer Solution 664 15.3.3 Development by the General Regression Significance Test 665 15.3.4 Factorial Experiments with Covariates 667 15.4 Repeated Measures 677 15.5 Problems 679 Appendix 683 Table I. Cumulative Standard Normal Distribution 684 Table II. Percentage Points of the t Distribution 686 Table III. Percentage Points of the 2 Distribution 687 Table IV. Percentage Points of the F Distribution 688 Table V. Operating Characteristic Curves for the Fixed Effects Model Analysis of Variance 693 Table VI. Operating Characteristic Curves for the Random Effects Model Analysis of Variance 697 Table VII. Percentage Points of the Studentized Range Statistic 701 Table VIII. Critical Values for Dunnett s Test for Comparing Treatments with a Control 703 Table IX. Coefficients of Orthogonal Polynomials 705 Table X. Alias Relationships for 2kp Fractional Factorial Designs with k 15 and n 64 706 Bibliography 719 Index 725

**Douglas C. Montgomery**, Regents' Professor of Industrial
Engineering and Statistics at Arizona State University, received
his B.S., M.S., and Ph.D. degrees from Virginia Polytechnic
Institute, all in engineering. From 1969 to 1984, he was a faculty
member of the School of Industrial & Systems Engineering at the
Georgia Institute of Technology; from 1984 to 1988, he was at the
University of Washington, where he held the John M. Fluke
Distinguished chair of Manufacturing Engineering, was Professor of
Mechanical Engineering, and Director of the Program in industrial
Engineering. He has authored and coauthored many technical papers
as well as twelve other books. Dr. Montgomery is a Stewart Medalist
of the American Society for Quality, and has also received the
Brumbaugh Award, the William G. Hunter Award, and the Shewell
Award(twice) from the ASQ.

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