Design and Analysis of Experiments

Preface......Page 6
Contents......Page 10
1.1.1. The Art of Experimentation......Page 22
1.1.2. Replication......Page 23
1.1.4. Randomization......Page 24
1.2. Analysis: Basic Principles and Techniques......Page 26
2.2. A Checklist for Planning Experiments......Page 28
2.3. A Real Experiment—Cotton-Spinning Experiment......Page 35
2.4. Some Standard Experimental Designs......Page 38
2.4.2. Block Designs......Page 39
2.4.3. Designs with Two or More Blocking Factors......Page 40
2.4.4. Split-Plot Designs......Page 42
2.5.1. Soap Experiment......Page 43
2.5.2. Battery Experiment......Page 47
2.5.3. Cake-Baking Experiment......Page 50
Exercises......Page 52
3.1. Introduction......Page 54
3.2. Randomization......Page 55
3.3. Model for a Completely Randomized Design......Page 56
3.4.2. Notation......Page 58
3.4.3. Obtaining Least Squares Estimates......Page 59
3.4.4. Properties of Least Squares Estimators......Page 61
3.4.5. Estimation of σ[sup(2)]......Page 63
3.4.6. Confidence Bound for σ[sup(2)]......Page 64
3.5.1. Testing Equality of Treatment Effects......Page 65
3.5.2. Use of p-Values......Page 69
3.6. Sample Sizes......Page 70
3.6.1. Expected Mean Squares for Treatments......Page 71
3.6.2. Sample Sizes Using Power of a Test......Page 72
3.7.1. Checklist, Continued......Page 74
3.7.2. Data Collection and Analysis......Page 75
3.7.4. Further Observations by the Experimenter......Page 77
3.8.1. Randomization......Page 78
3.8.2. Analysis of Variance......Page 79
Exercises......Page 82
4.1. Introduction......Page 88
4.2. Contrasts......Page 89
4.2.1. Pairwise Comparisons......Page 90
4.2.3. Difference of Averages......Page 91
4.2.4. Trends......Page 92
4.3.1. Confidence Interval for a Single Contrast......Page 94
4.3.3. Hypothesis Test for a Single Contrast or Treatment Mean......Page 96
4.4.1. Multiple Confidence Intervals......Page 99
4.4.2. Bonferroni Method for Preplanned Comparisons......Page 101
4.4.3. Scheffé Method of Multiple Comparisons......Page 104
4.4.4. Tukey Method for All Pairwise Comparisons......Page 106
4.4.5. Dunnett Method for Treatment-Versus-Control Comparisons......Page 108
4.4.6. Hsu Method for Multiple Comparisons with the Best Treatment......Page 110
4.4.7. Combination of Methods......Page 112
4.5. Sample Sizes......Page 113
4.6.1. Inferences on Individual Contrasts......Page 115
4.6.2. Multiple Comparisons......Page 117
Exercises......Page 118
5.1. Introduction......Page 124
5.2.1. Residuals......Page 125
5.2.2. Residual Plots......Page 126
5.4. Checking for Outliers......Page 128
5.5. Checking Independence of the Error Terms......Page 130
5.6. Checking the Equal Variance Assumption......Page 132
5.6.1. Detection of Unequal Variances......Page 133
5.6.2. Data Transformations to Equalize Variances......Page 134
5.6.3. Analysis with Unequal Error Variances......Page 137
5.7. Checking the Normality Assumption......Page 140
5.8.1. Using SAS to Generate Residual Plots......Page 143
5.8.2. Transforming the Data......Page 147
Exercises......Page 148
6.1. Introduction......Page 156
6.2.1. The Meaning of Interaction......Page 157
6.2.2. Models for Two Treatment Factors......Page 159
6.2.3. Checking the Assumptions on the Model......Page 161
6.3.1. Contrasts for Main Effects and Interactions......Page 162
6.3.2. Writing Contrasts as Coefficient Lists......Page 164
6.4. Analysis of the Two-Way Complete Model......Page 166
6.4.1. Least Squares Estimators for the Two-Way Complete Model......Page 167
6.4.2. Estimation of σ[sup(2)] for the Two-Way Complete Model......Page 168
6.4.3. Multiple Comparisons for the Complete Model......Page 170
6.4.4. Analysis of Variance for the Complete Model......Page 173
6.5.1. Least Squares Estimators for the Main-Effects Model......Page 179
6.5.2. Estimation of σ[sup(2)] in the Main-Effects Model......Page 183
6.5.3. Multiple Comparisons for the Main-Effects Model......Page 184
6.5.5. Analysis of Variance for Equal Sample Sizes......Page 186
6.6. Calculating Sample Sizes......Page 189
6.7.2. Analysis Based on Orthogonal Contrasts......Page 190
6.7.3. Tukey’s Test for Additivity......Page 193
6.7.4. A Real Experiment—Air Velocity Experiment......Page 194
6.8. Using SAS Software......Page 196
6.8.1. Contrasts and Multiple Comparisons......Page 198
6.8.2. Plots......Page 202
6.8.3. One Observation per Cell......Page 203
Exercises......Page 204
7.1. Introduction......Page 214
7.2.1. Models......Page 215
7.2.2. The Meaning of Interaction......Page 216
7.2.3. Separability of Factorial Effects......Page 218
7.2.4. Estimation of Factorial Contrasts......Page 220
7.3. Analysis—Equal Sample Sizes......Page 222
7.4. A Real Experiment—Popcorn–Microwave Experiment......Page 226
7.5.1. Analysis Assuming That Certain Interaction Effects Are Negligible......Page 232
7.5.2. Analysis Using Normal Probability Plot of Effect Estimates......Page 234
7.5.3. Analysis Using Confidence Intervals......Page 236
7.6. Design for the Control of Noise Variability......Page 238
7.6.1. Analysis of Design-by-Noise Interactions......Page 239
7.6.2. Analyzing the Effects of Design Factors on Variability......Page 242
7.7. Using SAS Software......Page 244
7.7.2. Voss–Wang Confidence Interval Method......Page 245
7.7.3. Identification of Robust Factor Settings......Page 247
7.7.4. Experiments with Empty Cells......Page 248
Exercises......Page 252
8.1. Introduction......Page 264
8.2. Models......Page 265
8.3.2. Least Squares Estimates for Simple Linear Regression......Page 269
8.4. Test for Lack of Fit......Page 270
8.5. Analysis of the Simple Linear Regression Model......Page 272
8.6.1. Analysis of Variance......Page 276
8.6.2. Confidence Intervals......Page 278
8.7.1. Simple Linear Regression......Page 279
8.7.2. Quadratic Regression......Page 281
8.7.3. Comments......Page 282
8.8.1. Checklist......Page 283
8.8.2. One-Way Analysis of Variance and Multiple Comparisons......Page 285
8.8.3. Regression Analysis......Page 288
8.9. Using SAS Software......Page 289
Exercises......Page 294
9.1. Introduction......Page 298
9.2. Models......Page 299
9.2.2. Model Extensions......Page 300
9.3.1. Normal Equations (Optional)......Page 301
9.3.2. Least Squares Estimates and Adjusted Treatment Means......Page 302
9.4. Analysis of Covariance......Page 303
9.5.1. Individual Confidence Intervals......Page 307
9.5.2. Multiple Comparisons......Page 308
9.6. Using SAS Software......Page 309
Exercises......Page 313
10.1. Introduction......Page 316
10.2. Blocks, Noise Factors or Covariates?......Page 317
10.3.1. Block Sizes......Page 318
10.3.2. Complete Block Design Definitions......Page 319
10.3.3. The Randomized Complete Block Design......Page 320
10.3.4. The General Complete Block Design......Page 321
10.4.1. Model and Analysis of Variance......Page 322
10.4.2. Multiple Comparisons......Page 326
10.5.1. Design Details......Page 327
10.5.3. Analysis of the Cotton-Spinning Experiment......Page 328
10.6.1. Model and Analysis of Variance......Page 330
10.6.2. Multiple Comparisons for the General Complete Block Design......Page 333
10.6.3. Sample-Size Calculations......Page 336
10.7. Checking Model Assumptions......Page 337
10.8. Factorial Experiments......Page 338
10.9. Using SAS Software......Page 341
Exercises......Page 345
11.1. Introduction......Page 360
11.2.2. Design Plans and Randomization......Page 361
11.2.3. Estimation of Contrasts (Optional)......Page 363
11.2.4. Balanced Incomplete Block Designs......Page 364
11.2.5. Group Divisible Designs......Page 366
11.2.6. Cyclic Designs......Page 367
11.3.1. Contrast Estimators and Multiple Comparisons......Page 369
11.3.2. Least Squares Estimation (Optional)......Page 372
11.4.1. Multiple Comparisons and Analysis of Variance......Page 375
11.4.2. A Real Experiment—Detergent Experiment......Page 376
11.5.1. Multiple Comparisons and Analysis of Variance......Page 381
11.7. A Real Experiment—Plasma Experiment......Page 383
11.8. Sample Sizes......Page 389
11.9.1. Factorial Structure......Page 390
11.10.1. Analysis of Variance and Estimation of Contrasts......Page 393
11.10.2. Plots......Page 398
Exercises......Page 399
12.1. Introduction......Page 408
12.2.1. Selection and Randomization of Row–Column Designs......Page 409
12.2.2. Latin Square Designs......Page 410
12.2.3. Youden Designs......Page 412
12.2.4. Cyclic and Other Row–Column Designs......Page 413
12.3. Model for a Row–Column Design......Page 415
12.4.1. Least Squares Estimation (Optional)......Page 416
12.4.2. Solution for Complete Column Blocks (Optional)......Page 418
12.4.3. Formula for ssE (Optional)......Page 419
12.4.4. Analysis of Variance for a Row–Column Design (Optional)......Page 420
12.5.1. Analysis of Variance for Latin Square Designs......Page 422
12.5.2. Confidence Intervals for Latin Square Designs......Page 424
12.5.3. How Many Observations?......Page 426
12.6.1. Analysis of Variance for Youden Designs......Page 427
12.6.3. How Many Observations?......Page 428
12.7. Analysis of Cyclic and Other Row–Column Designs......Page 429
12.8. Checking the Assumptions on the Model......Page 430
12.10. Using SAS Software......Page 431
12.10.1. Factorial Model......Page 434
Exercises......Page 436
13.1. Introduction......Page 442
13.2.2. Confounding......Page 443
13.2.3. Analysis......Page 444
13.3.1. Contrasts......Page 445
13.3.2. Experiments in Two Blocks......Page 446
13.3.3. Experiments in Four Blocks......Page 451
13.3.4. Experiments in Eight Blocks......Page 453
13.4.1. Experiments in Two Blocks......Page 454
13.4.2. Experiments in More Than Two Blocks......Page 456
13.5. A Real Experiment—Mangold Experiment......Page 458
13.7. Multireplicate Designs......Page 462
13.8.1. A Real Experiment—Decontamination Experiment......Page 463
13.9. Partial Confounding......Page 467
13.10. Comparing the Multireplicate Designs......Page 470
13.11. Using SAS Software......Page 473
Exercises......Page 475
14.1. Introduction......Page 482
14.2.1. Contrasts......Page 483
14.2.2. Confounding Using Contrasts......Page 484
14.2.3. Confounding Using Equations......Page 485
14.2.4. A Real Experiment—Dye Experiment......Page 488
14.2.5. Plans for Confounded 3[sup(p)] Experiments......Page 491
14.3.1. Confounding in 4[sup(p)] Experiments......Page 492
14.4. Designing Confounded Asymmetrical Experiments......Page 493
14.5. Using SAS Software......Page 496
Exercises......Page 498
15.1. Introduction......Page 504
15.2.1. Half-Fractions of 2[sup(p)] Experiments; 2[sup(p–1)] Experiments......Page 505
15.2.3. A Real Experiment—Soup Experiment......Page 508
15.2.4. Quarter-Fractions of 2[sup(p)] Experiments; 2[sup(p–2)] Experiments......Page 511
15.2.5. Smaller Fractions of 2[sup(p)] Experiments......Page 515
15.3.1. One-Third Fractions of 3[sup(p)] Experiments; 3[sup(p–1)] Experiments......Page 517
15.4.1. 2[sup(p)] X 4[sup(q)] Experiments......Page 522
15.4.2. 2[sup(p)] X 3[sup(q)] Experiments......Page 523
15.5. Blocked Fractional Factorial Experiments......Page 524
15.6.1. 2[sup(p)] Orthogonal Arrays......Page 527
15.6.2. Saturated Designs......Page 533
15.6.3. 2[sup(p)] X 4[sup(q)] Orthogonal Arrays......Page 534
15.6.4. 3[sup(p)] Orthogonal Arrays......Page 535
15.7. Design for the Control of Noise Variability......Page 536
15.7.1. A Real Experiment—Inclinometer Experiment......Page 537
15.8.1. Fractional Factorials......Page 542
15.8.2. Design for the Control of Noise Variability......Page 545
Exercises......Page 550
16.1. Introduction......Page 568
16.2.1. Models......Page 570
16.2.2. Standard First-Order Designs......Page 572
16.2.3. Least Squares Estimation......Page 573
16.2.5. Analysis of Variance......Page 574
16.2.6. Tests for Lack of Fit......Page 575
16.2.7. Path of Steepest Ascent......Page 580
16.3.1. Models and Designs......Page 582
16.3.2. Central Composite Designs......Page 583
16.3.4. Analysis of Variance for a Second-Order Model......Page 585
16.3.5. Canonical Analysis of a Second-Order Model......Page 587
16.4.1. Rotatability......Page 590
16.4.2. Orthogonality......Page 591
16.4.3. Orthogonal Blocking......Page 592
16.5. A Real Experiment: Flour Production Experiment, Continued......Page 594
16.6. Box–Behnken Designs......Page 597
16.7.1. Analysis of a Standard First-Order Design......Page 600
16.7.2. Analysis of a Second-Order Design......Page 603
Exercises......Page 607
17.1. Introduction......Page 614
17.2. Some Examples......Page 615
17.3.1. The Random-Effects One-Way Model......Page 617
17.3.2. Estimation of σ[sup(2)]......Page 618
17.3.3. Estimation of σ[sup(2)][sub(T)]......Page 619
17.3.4. Testing Equality of Treatment Effects......Page 622
17.3.5. Confidence Intervals for Variance Components......Page 624
17.4. Sample Sizes for an Experiment with One Random Effect......Page 628
17.6.1. Models and Examples......Page 631
17.6.3. Estimation of σ[sup(2)]......Page 634
17.6.4. Estimation of Variance Components......Page 635
17.6.5. Confidence Intervals for Variance Components......Page 637
17.6.6. Hypothesis Tests for Variance Components......Page 641
17.7.1. Expected Mean Squares and Hypothesis Tests......Page 643
17.7.2. Confidence Intervals in Mixed Models......Page 646
17.8.1. Rules—Equal Sample Sizes......Page 648
17.8.2. Controversy (Optional)......Page 649
17.9. Block Designs and Random Blocking Factors......Page 651
17.10.1. Checking Assumptions on the Model......Page 653
17.10.2. Estimation and Hypothesis Testing......Page 656
Exercises......Page 660
18.1. Introduction......Page 666
18.2. Examples and Models......Page 667
18.3.1. Least Squares Estimates......Page 669
18.3.2. Estimation of σ[sup(2)]......Page 670
18.3.4. Hypothesis Testing......Page 671
18.4.1. Expected Mean Squares......Page 675
18.4.2. Estimation of Variance Components......Page 677
18.4.3. Hypothesis Testing......Page 678
18.4.4. Some Examples......Page 679
18.5.1. Voltage Experiment......Page 683
Exercises......Page 688
19.1. Introduction......Page 696
19.2. Designs and Models......Page 697
19.3.1. Split-Plot Analysis......Page 699
19.3.2. Whole-Plot Analysis......Page 701
19.3.4. A Real Experiment—Oats Experiment......Page 702
19.4. Split-Split-Plot Designs......Page 705
19.5. Split-Plot Confounding......Page 707
19.6. Using SAS Software......Page 708
Exercises......Page 712
A. Tables......Page 716
Bibliography......Page 746
H......Page 752
Y......Page 753
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Z......Page 755
C......Page 756
D......Page 757
L......Page 758
P......Page 759
S......Page 760
Y......Page 761