DESIGN AND ANALYSIS OF EXPERIMENTS Third Edition Douglas C. Montgomery ARIZONA STATE UNIVERSITY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore
Contents Chapter 1. Introduction 1-1 What Is Experimental Design? 1-2 Applications of Experimental Design 1-3 Basic Principles 1-4 Guidelines for Designing Experiments 1-5 Historical Perspective 1-6 Using Statistical Techniques in Experimentation 1 3 8 9 12 12 Chapter 2. Simple Comparative Experiments 2-1 Introduction 2-2 Basic Statistical Concepts 2-3 Sampling and Sampling Distributions Inferences About the Differences in Means, Randomized 2-5 I 2-6 2-7 Designs Hypothesis Testing Choice of Sample Size Confidence Intervals 1 r The Case Where a\ and a\ Are Known Comparing a Single Mean to a Specified Value Summary nferences About the Difference in Means, Paired Comparison Designs 2-5.1 The Paired Comparison Problem 2-5.2 Advantages of the Paired Comparison Design Inferences About the Variances of Normal Distributions u 2 14 14 15 20 27 28 31 33 34 35 36 37 38 38 41 42 45
XÜ Contents Chapter 3. Experiments with a Single Factor: The Analysis of Variance 3-1 An Example 3-2 The Analysis of Variance 3-3 Analysis of the Fixed Effects Model 3-3.1 3-3.2 3-3.3 3-3.4 3-3.5 Decomposition of the Total Sum of Squares Statistical Analysis Estimation of the Model Parameters Model Adequacy Checking: Preview The Unbalanced Case 3-4 Comparison of Individual Treatment Means 3-4.1 Graphical Comparison of Means 3-4.2 Contrasts 3-4.3 Orthogonal Contrasts 3-4.4 Scheffe's Method for Comparing All Contrasts 3-4.5 Comparing Pairs of Treatment Means 3-4.6 Comparing Treatments with a Control 3-5 The Random Effects Model 3-6 Sample Computer Output 3-7 50 51 53 55 55 59 63 66 67 67 67 69 70 72 73 79 81 87 87 Chapter 4. More About Single-Factor Experiments 95 4-1 Model Adequacy Checking 95 4-1.1 The Normality Assumption 96 4-1.2 Plot of Residuais in Time Sequence 99 4-1.3 Plot of Residuais Versus Fitted Values y 100 4-1.4 Selecting a Variance-Stabilizing Transformation 103 4-1.5 Plots of Residuais Versus Other Variables 108 4-1.6 Discovering Dispersion Effects 109 4-2 Choice of Sample Size 110 4-2.1 Operating Characteristic Curves 110 4-2.2 Specifying a Standard Deviation Increase 114 4-2.3 Confidence Interval Estimation Method 115 4-3 Fitting Response Curves in the Single-Factor One-Way Model 116 4-3.1 General Regression Approach 116 4-3.2 Orthogonal Polynomials 118 4-4 The Regression Approach to the Analysis of Variance 121 4-5 Nonparametric Methods in the Analysis of Variance 126 4-5.1 The Kruskal-Wallis Test 126 4-5.2 General Comments on the Rank Transformation 127 4-6 Repeated Measures 128 4-7 130
Contents XÜi Chapter 5. Randomized Blocks, Latin Squares, and Related Designs 134 5-1 The Randomized Complete Block Design 134 5-1.1 Statistical Analysis 135 5-1.2 Model Adequacy Checking 146 5-1.3 Estimating Missing Values 148 5-1.4 Estimating Model Parameters and the General Regression Significance Test 151 5-1.5 Sample Computer Output 154 5-2 The Latin Square Design 156 5-3 The Graeco-Latin Square Design 166 5-4 169 Chapter 6. 6-1 6-2 6-3 6-4 6-5 6-6 6-7 Incomplete Block Designs 176 Introduction 176 Balanced Incomplete Block Designs 176 6-2.1 Statistical Analysis 177 6-2.2 Least Squares Estimation of the Parameters 183 Recovery of Interblock Information in the Balanced Incomplete Block Design 184 Partially Balanced Incomplete Block Designs 187 Youden Squares 190 Lattice Designs 193 194 Chapter 7. Introduction to Factorial Designs 197 7-1 Basic Definitions and Principles 197 7-2 The Advantage of Factorials 199 7-3 The Two-Factor Factorial Design 201 7-3.1 An Example 201 7-3.2 Statistical Analysis of the Fixed Effects Model 203 7-3.3 Model Adequacy Checking 210 7-3.4 Estimating the Model Parameters 213 7-3.5 Choice of Sample Size 216 7-3.6 The Assumption of No Interaction in a Two-Factor Model 217 7-4 7-5 7-6 7-7 7-3.7 One Observation per Cell 218 Random and Mixed Models 222 7-4.1 The Random Effects Model 222 7-4.2 Mixed Models 224 7-4.3 Choice of Sample Size 228 The General Factorial Design 228 Fitting Response Curves and Surfaces 237 Dealing with Unbalanced Data 244
XIV Contents 7-7.1 Proportional Data: An Easy Case 7-7.2 Approximate Methods 7-7.3 The Exact Method 245 247 249 249 Chapter 8. Ruies for Sums of Squares and Expected Mean Squares 257 8-1 8-2 8-3 8-4 Chapter 9. 9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 Chapter 10. 10-1 10-2 10-3 10-4 10-5 10-6 Chapter 11. 11-1 11-2 11-3 11-4 11-5 11-6 11-7 11-8 Chapter 12. 12-1 Ruies for Sums of Squares Ruies for Expected Mean Squares Approximate F Tests The 2 k Factorial Design Introduction The 2 2 Design The 2 3 Design The General 2 k Design A Single Replicate of the 2 k Design The Addition of Center Points to the 2 k Design Yates' Algorithm for the 2 k Design Confounding in the 2 k Factorial Introduction The 2 k Factorial Design in Two Blocks The 2 k Factorial Design in Four Blocks The 2 k Factorial Design in 2 P Blocks Partial Confounding Two-Level Fractional Factorial Designs Introduction The One-Half Fraction of the 2 k Design The One-Quarter Fraction of the 2 k Design The General 2 k ~ p Fractional Factorial Design Resolution III Designs Resolution IV and V Designs Summary Some Other Topics Regarding Factorial and Fractional Factorial Designs The 3^ Factorial Design 12-1.1 Notation and Motivation for the 3 k Design 257 259 262 268 270 270 270 278 288 289 304 309 310 319 319 319 326 329 329 333 335 335 336 349 358 367 375 378 378 387 387 387
12-2 12-3 1 12-5 12-6 Chapter 13. 13-1 13-2 13-3 13-4 13-5 12-1.2 The 3 2 Design 12-1.3 The 3 3 Design 12-1.4 The General 3 k Design 12-1.5 Yates' Algorithm for the 3 k Design Confounding in the 3 k Factorial Design 12-2.1 The 3 k Factorial Design in Three Blocks 12-2.2 The 3 k Factorial Design in Nine Blocks 12-2.3 The 3 k Factorial Design in 3 P Blocks Fractional Replication of the 3 k Factorial Design 12-3.1 The One-Third Fraction of the 3 k Design 12-3.2 Other 3 k ~ p Fractional Factorial Designs Factorials with Mixed Levels 1.1 Factors at Two and Three Levels 1.2 Factors at Two and Four Levels Taguchi's Contributions to Experimental Design and Quality Engineering 12-5.1 The Taguchi Philosophy 12-5.2 The Taguchi Approach to Parameter Design Nested or Hierarchial Designs Introduction The Two-Stage Nested Design 13-2.1 Statistical Analysis 13-2.2 Diagnostic Checking 13-2.3 Estimation of the Model Parameters The General m-stage Nested Design Designs with Nested and Crossed Factors Contents XV 388 391 395 397 399 399 402 404 405 405 408 410 410 412 414 415 417 433 439 439 440 440 445 446 450 452 456 Chapter 14. Multifactor Experiments with Randomization Restrictions 461 14-1 Randomized Blocks and latin Squares as Multifactor Designs 461 14-2 The Split-Plot Design 468 14-3 The Split Split-Plot Design 472 14-4 475 Chapter 15. Regression Analysis 15-1 Introduction 15-2 Simple Linear Regression 15-3 Hypothesis Testing in Simple Linear Regression 15-4 Interval Estimation in Simple Linear Regression 15-5 Model Adequacy Checking 479 479 479 486 489 493
XVi Contents 15-5.1 Residual Analysis 15-5.2 The Lack-of-Fit Test 15-5.3 The Coefficient of Determination 15-6 Multiple Linear Regression 15-7 Hypothesis Testing in Multiple Linear Regression 15-8 Other Linear Regression Models 15-9 Sample Computer Printout 15-10 493 493 497 498 507 512 515 515 Chapter 16. 16-1 16-2 16-3 16-4 16-5 16-6 16-7 Chapter 17. 17-1 17-2 17-3 17-4 17-5 Response Surface Methods and Designs 521 Introduction to Response Surface Methodology 521 The Method of Steepest Ascent 523 Analysis of a Second-Order Model 531 16-3.1 Location of the Stationary Point 531 16-3.2 Characterizing the Response Surface 532 16-3.3 Ridge Systems 538 Experimental Designs for Fitting Response Surfaces 541 16-4.1 Designs for Fitting the First-Order Model 541 16-4.2 Designs for Fitting the Second-Order Model 542 16-4.3 Blocking in Response Surface Designs 548 Mixture Experiments 551 Evolutionary Operation 558 563 Analysis of Covariance 569 Introduction 569 A Single-Factor Design with One Covariate 569 Development by the General Regression Significance Test 581 Other Covariance Models 584 587 Bibliography 590 Appendix Table I. Table II. Table III. Table IV. Table V. Table VI. Cumulative Standard Normal Distribution Percentage Points of the t Distribution Percentage Points of the x 2 Distribution Percentage Points of the F Distribution Operating Characteristic Curves for the Fixed Effects Model Analysis of Variance Operating Characteristic Curves for the Random Effects Model Analysis of Variance 597 598 600 601 602 607 611
Contents XVÜ Table VII. Significant Ranges for Duncan's Multiple Range Test 615 Table VIII. Percentage Points of the Studentized Range Statistic 617 Table IX. Critical Values for Dunnett's Test for Comparing Treatments with a Control 619 Table X. Coefficients of Orthogonal Polynomials 623 Table XI. Random Numbers 624 Table XII. Alias Relationships for 2 k ~ p Fractional Factorial Designs with k < 11 and n < 64. 626 Index 645