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 |  موضوع: كتاب Design and Analysis of Experiments الجمعة 08 أكتوبر 2021, 10:12 pm | |
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أخواني في الله
أحضرت لكم كتاب
Design and Analysis of Experiments
Eighth Edition
Douglas C. Montgomery
Arizona State University
و المحتوى كما يلي :
Contents
Preface v
1
Introduction 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
Simple Comparative Experiments 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
2 1 2 2
2 1 Z 2 2
xi3
Experiments with a Single Factor:
The 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 Scheffé’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
Randomized Blocks, Latin Squares,
and Related Designs 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
xii Contents4.1.3 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
Introduction to Factorial Designs 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
The 2k Factorial Design 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 in the 2k
Factorial Design 304
7.1 Introduction 304
7.2 Blocking a Replicated 2k Factorial Design 305
7.3 Confounding in the 2k Factorial Design 306
Contents xiii7.4 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
Additional Design and Analysis Topics for 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
xiv Contents9.4 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
Fitting Regression Models 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 Methods and Designs 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
Contents xv12
Robust Parameter Design and Process
Robustness Studies 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
Experiments with Random Factors 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
Nested and Split-Plot Designs 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 Design and Analysis 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
xvi Contents15.2 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
Contents xvii1
INDEX
Index Terms Links
22 factorial design 5 234
23 factorial design 7 241
24 factorial design 7
2k factorial design 233 253
2k fractional factorial design 320
2k-1 fractional factorial design 321
2k-2 fractional factorial design 333
2k-p fractional factorial design 340
32 factorial design 188 396
33 factorial design 397
3k factorial designs 395 402
3k-1 fractional factorial design 408
3k-p fractional factorial design 410
A
Additivity of the RCBD model 150
Adjusted R2 464
Advantages of factorials 186
Alias matrix 249 417
Aliases 322 349 358
Alternate fraction 323
Alternate mixed models 583
Alternative hypothesis 37 38
Analysis of covariance versus blocking 664
Analysis of covariance 16 139 655
665 667
Analysis of crossed array designs 558
Analysis of variance (ANOVA) 69 71 73
74 117 142
160 169 170
189 236Index Terms Links
Analysis of variance method for estimating variance
components 118 575
ANOVA for a BIBD 169 170
ANOVA for a Latin square 160
ANOVA for a single-factor random model 117
ANOVA for a two-factor factorial design 189
ANOVA for the RCBD 142
A-optimality 513
Approximate confidence intervals on variance
components 598
Approximate F-tests in ANOVA 592
Assumptions 41 69 73
Average prediction variance 283
Axial points in mixture designs 534
Axial runs in a central composite design 288
B
Balanced incomplete block designs (BIBD) 168
Balanced nested design 605
Bartlett’s test for equal variances 84
Basic design 324 333 409
Best-guess approach to experimentation 4
Binomial distribution 646
Blocking 12 13 56
139 219 304
313 331 344
356 507 655
Blocking a replicated 2k factorial design 305
Blocking and noise reduction 313
Blocking and repeated measures 679
Blocking in a factorial design 219 304
Blocking in a fold-over design 356
Blocking in a fractional factorial design 331 344
Blocking in response surface designs 507
Boundary point mixture designs 534
Box plot (box and whisker plot) 27
Box-Behnken designs 503
Box-Cox method 643Index Terms Links
C
Canonical analysis of a response surface 489
Canonical variables 494
Cause-and-effect diagram 17
Cell plot 416
Center points in a 2k design 285
Center points in the central composite design 503
Central composite design 288 489 501
503 504 65
568
Central limit theorem 33
Characterization experiments 9
see also screening experiments
Characterization of a response surface 488
Chi-square distribution 33 57
Cochran’s theorem 74
Coded design factors 290
Coding data in ANOVA 76
Combined array design 558 561 567
Complete randomization 12
Completely randomized design (CRD) 66 69 188
233 234
Component axis 534
Component of interaction 397 398 408
Components of variance model 70 574
Computer models 523
Conditional inference chart 264
Conference matrices 521
Confidence coefficient 43
Confidence intervals 36 43 57
59 78 109
251 251 467
468 597 600
Confidence interval on contrasts 93
Confidence interval on the mean response 468
Confidence intervals on effects 251 252
Confidence intervals on regression model coefficients 467
Confirmation experiments 15 20 333Index Terms Links
Confounding 306 402
Confounding in the 2k factorial design 306 313 315
Confounding in the 3k factorial design 402
Construction of optimal designs 514
Continuous probability distribution 28
Contour plots 185 496
Contrasts 92
Contrasts and preplanned comparisons 95
Contrasts in a two-level design 236 242
Control-by-noise interaction in robust design 557
Controllable factors 3 16
Coordinate exchange algorithm for design construction 514
Correlation between residuals 82
Correlation matrix 416
Covariance 30
Covariance matrix 116 454
Covariate 655
Critical region for a statistical test 37
Crossed array designs 56
Crossed factors, see factorial design
Crossover designs 164
Cubodial versus spherical region of interest 504
D
Data snooping 95
Defining contrast for a blocked design 308 314 403
Defining relation for a fold-over design 356
Defining relation for a fractional factorial design 322 334 408
Definitive screening designs 520
Degrees of freedom 32
Design generator 321 334
Design resolution 323 340
Designs balanced for residual effects 164
Designs for robust design 567
Desirability function optimization in RSM 498
Deterministic versus stochastic computer (simulation)
models 523
Different error structures in the split-plot design 623Index Terms Links
Discovery experiments 15
Discrete probability distribution 28
Dispersion effects 114 253 271
338
D-optimal designs 283 513
Dot diagram 26
Dunnett’s test for comparing means with a control 101
Duplicate measurements on the response 274
E
Effect heredity 326
Effect magnitude and direction 236
Effect of a factor 183
Effect of outliers in unreplicated designs 267
Effects coding 238 242
Effects model 69 141 166
188
Effects model for a Graeco-Latin square design 166
Effects model for a two-factor factorial design 188
Effects model for the Latin square design 160
Effects model for the RCBD 141
Empirical model 2 19 20
89
Engineering method 2
Equiradial designs 505
Estimate 31
Estimating missing values in the RCBD 154
Estimating model parameters in a two-factor factorial 198
Estimating model parameters in the BIBD 172
Estimating model parameters in the RCBD 155
Estimating the overall mean in a random model 122
Estimating variance components 118 152
see also residual
maximum likelihood method (REML)
Estimation of parameters in ANOVA models 78
Estimator 31
Evolutionary operation (EVOP) 540
Expected mean squares 73Index Terms Links
Expected value 29
Expected value operator 29
Experiment 1
Experimental error 27
Experimental units 69 140
Experiments with computer models 10 523
Experimentwise error rates 98
Exponential distribution 646
Exponential family of distributions 646
Extra sum of squares method 465
F
Face-centered cube design 504
Factor effect 5 6 234
Factorial design 5 7 183
187 206 233
Factorial experiment in a Latin square 223
Factorial experiment in a randomized complete block
(RCBD) 219 304
Factorial experiments with covariates 667
Family of fractional factorial designs 323
F-distribution 35
First-order model 19
see also models for data from experiments
First-order response surface designs 501
Fisher LSD procedure for comparing all pairs of means 99
Fixed factor effect 69 189 573
Fold over of a design 353 354 356
366
Fold over of a resolution IV design 368
Fold over of resolution III designs 353
Follow-up runs 20
see also confirmation experiments
Formulation experiments 11
Fraction of design space plot 285 506
Fractional factorial design 7 320
Full cubic mixture model 533
Full fold over 354Index Terms Links
G
Gamma distribution 646 651
Gaussian process model 525 527
General factorial designs 206
Generalized interaction 314 334 406
Generalized linear models 645
G-optimal designs 283 433 513
Graeco-Latin square designs 165 411
Graphical comparison of means 91
Graphical evaluation of designs 506
Guidelines for designing experiments 14
H
Hadamard matrix designs 375
Half-normal plot of effects 262
Hall designs 418
Hat matrix in regression 470
Hidden replication 260
Hierarchical designs, see nested designs
Histogram 27
Hybrid designs 506
Hypothesis testing 26 36
Hypothesis tests on variances 57 58 84
85
I
I and J components of interaction 397
Identity element 245
Identity link 646
Immediacy 21
Incomplete block design 168 306
Independence assumption in ANOVA 82
Independent random variables 30
Influence on regression coefficients 473
Inner array in a crossed array 556
Integrated variance 28Index Terms Links
Interaction 4 6 184
234 244
Interaction and curvature 186
Interaction between treatments and blocks 150
Interblock analysis of the BIBD 174
Interclass correlation coefficient 121
Intrablock analysis of the BIBD 174
I-optimal design 283 433
Irregular design regions 511
Iterative experimentation 20
see also sequential experimentation
J
J component of interaction 397
K
Kruskal-Wallis test 128
L
Lack of fit 251 473
Latin hypercube designs 524
Latin square designs 158 223 409
Latin square designs and Sudoku puzzles 159
Least squares normal equations 125 126 452
Lenth’s method for analyzing unreplicated 2k designs 262
Levels of a factor 25 36 66
see also treatments
Levene’s test for equal variances 85
Leverage points 473
Linear mixture model 532
Linear predictor 646
Linear statistical model 69
see also models for data from experiments
Link function 646
Log link 646 651
Logistic regression model 647
Logit link 647Index Terms Links
M
Main effect of a factor 183 234
Maximum entropy designs 525
Maximum likelihood estimation of variance components 123
see also residual maximum likelihood method (REML)
Mean of a distribution 29
Mean squares 72
Means model for a two-factor factorial design 189
Means model for the RCBD 141
Means model 69 141 189
Measurement systems capability study 575 582
Mechanistic model 2
Method of least squares 89 125 451
Method of unweighted means 56
Minimum aberration design 341 415
Minimum run resolution IV designs 435
Minimum run resolution V designs 366 438
Minimum variance estimator 31
Missing value problems in the RCBD 154 158
Mixed level fractional factorials 412 414
Mixed model 581
Mixture designs for constrained regions 535
Mixture experiments 530
Model adequacy checking, see residual plots
Model independent estimate of error 474
Models for data from experiments 36 53 69
89 141 160
166 188 189
238 247 285
479 533 534
574 581 583
646 657 667
Modified large-sample method for finding confidence
intervals on variance components 600
Moment estimators of variance components 119 575
m-stage nested designs 614
Multiple comparisons 90 98
Multiple comparisons in a factorial experiment 194Index Terms Links
Multiple comparisons in the RCBD 146
Multiple linear regression model 450
see also regression models
Multiple responses in RSM 496 498
N
Nested and factorial factors 616
Nested designs 574 604 605
612 614 616
No interaction in a factorial model 202
No-confounding designs 420 425
Noise factors 16 556
Noise reduction from blocking 56 146
Nongeometric designs 357
Nonisomorphic designs 418
Nonlinear programming 498
Nonnormal response distributions 84 87 269
643
Nonparametric ANOVA 128
Nonregular fractional factorial designs 359 374 415
425
Nonstandard models 512
Normal distribution 32 646
Normal probability plot 41
Normal probability plot of effects 257
Normal probability plot of residuals 81
Normality assumption in ANOVA 80
Nuisance factors 13 139
Null hypothesis 37
O
Ockham’s razor 326
Odds ratio 647
One replicate of a factorial experiment 203 255
One-factor-at-a-time (OFAT) experiments 4
One-half fraction 7 321
One-sided alternative hypothesis 38
One-step RSM designs 520Index Terms Links
Operating characteristic curve 45 105 153
201 587
Optimal designs with covariates 672
Optimal designs 21 280 374
431 511 535
672
Optimal response surface designs 511
Optimization experiment 9 14 17
Optimization with contour plots 496
Orthogonal blocking 507
Orthogonal coding 238 242
Orthogonal contrasts 94
Orthogonal design 238 242 459
Orthogonal Latin squares 165 396
Outer array in a crossed array 556
Outliers 82 267
P
Paired comparison tests 53
Partial aliasing 358 411
Partial confounding 309 316
Partial F test 466
Partial fold over 371
Path of steepest ascent 481 485 486
Placket-Burman designs 357
Point exchange algorithms for design construction 513
Poisson distribution 646
Pooled estimate of variance 72
Power curve 45
Power family transformations 643
Power of a statistical test 37 107
Prediction interval on a future observation 468
Prediction profile plot 264
Prediction variance profiler 283
Pre-experimental planning 18
PRESS statistic 251 470 471
Principal block 308 403
Principal fraction 323Index Terms Links
Probability distributions 28
Process robustness study 544
Product design 10
Projection of a 2k 260
Projection of fractional factorial designs 325 343 359
Projection of Plackett-Burman designs 359
Propagation of error 563
Proportional data in ANOVA 652
Pseudocomponents 536
Pure quadratic curvature 286
P-values 40
Q
Quadratic mixture model 532
Quadratic model 90
see also second-order model
Qualitative factors 233 289 399
Quantitative factors 185 233 285
395 399
R
R2 251
R2 for prediction 251
Random effects model 116 573 574
Random factor effect 69 116
Random sample 30
Random treatments and blocks 151
Random variable 27
Randomization 12 139 141
143 159
Randomization tests 43 77
Randomized block design 56
Randomized complete block design (RCBD) 140
Rank transformation in ANOVA 130
Ranks 128
Recovery of interblock information in the BIBD 174
Reference distribution 38
Regression approach to ANOVA 125Index Terms Links
Regression model for a factorial 238
Regression models 89 185 238
449 450 451
645
Regular fractional factorial designs 359
Relationship between coded and natural variables 128
REML 123 152 222
579
Repeated measures designs 677
Replicated design 5
Replication 5 12 13
66 106
Replication of Latin squares 163
Replication versus repeated measurements 13
Residual plots 80 81 82
83 88 146
149 198 239
261 609 662
Residuals 80 146 198
239 260 261
453 609 662
Resolution III designs 323 351 353
408 415
Resolution IV designs 324 366 415
435
Resolution V designs 324 373 415
438
Response curves 211
Response model approach to robust design 562
Response surface designs 395 479 500
501 520
Response surface methodology (RSM) 478 481 486
488 496
Response surface plots 185 211 214
240 261
Response variable 3 15
Restricted form of the mixed model 581
Ridge systems in response surfaces 495Index Terms Links
Rising ridge 495
Robust parameter design 554 557 567
Robustness 15
Rotatability 502
Rotatable central composite design 503
R-student 472
Rules for determining expected mean squares 588
Run 1
S
Sample mean 30
Sample size determination 4 106 108
109 153 201
587
Sample standard deviation 31
Sample variance 30
Sampling distribution 30 32
Saturated fractional factorial design 351
Scaled prediction variance (SPV) 506
Scatter diagram 67
Scheffe’s method for comparing all contrasts 96
Scientific method 2
Screening experiments 14 17 233
Second-order model 19 90
see also models for data from experiments
Second-order response surface model 285 479
Sequences of fractional factorials 331 332
Sequential experimentation 15 20 21
23 288 331
367 480 501
524
Signal-to-noise ratios 558
Significance level of a statistical test 37 38
Simplex centroid design 532
Simplex design in RSM 501
Simplex lattice design 531
Simplex mixture designs 531
Simultaneous confidence intervals 79 96Index Terms Links
Single factor experiment 68
Single replicate of a 2k 255
Single-factor fold over 354
Small composite designs 505
Space-filling designs 524
Sparsity of effects principle 255
Special cubic mixture model 533
Sphere packing designs 525
Spherical central composite design 503
Split-plot designs 574 621 625
627 632
Split-split-plot designs 632
Staggered nested designs 612
Standard error 38 96
Standard error of a regression coefficient 454
Standard Latin square 162
Standard normal distribution 33
Standard order in a 2k design 237 253
Standardized contrasts 94
Standardized residual 470
Stationary point on a response surface 486
Stationary ridge 495
Statistic 30
Statistical approach to designing experiments 11
Steepest ascent 480
Strategy of experimentation 3
Strip-split-plot designs 636
Strong heredity 326
Studentized range statistic 98
Studentized residual 470 471
Subplot error 622
Subplot treatments 621
Subplots 621
Subsampling 626
Supersaturated designs 374
Symmetric BIBD 169Index Terms Links
T
t-distribution 34 35
Test for significance of regression 462
Test statistic 37
Tests of hypotheses on regression model coefficients 46
Total effect of a factor 236
Transformations to correct
violations of assumptions 84 87 269
643
Transmission of error 561
Treatments 25 68
Trilinear coordinates 531
Tukey’s additivity test 204
Tukey’s test for comparing all pairs of means 98
Tukey-Kramer test 98
Two-factor factorial design 187
Two-sample t-test 38 41
Two-sample t-test with unequal variances 48
Two-sided alternative hypothesis 37
Two-stage nested designs 604
Types of factors in experiments 16
U
Unbalanced data in ANOVA 79 652
Unbiased estimator 31
Uncontrollable factors 3 16 556
Uniform designs 525
Unreplicated 2k designs 255
Unrestricted form of the mixed model 583
Unscaled prediction variance 283
Unusual sample size requirements 513
V
Variability 27
Variance components 116 574 611
Variance dispersion graph 506Index Terms Links
Variance modeling 559
Variance of a distribution 29 57
Variance operator 29
V-optimality 513
W
W component of interaction 398
Weak heredity 327
Weighted squares of means method 655
Whole plot error 622
Whole plot treatments 621
Whole plots 621
X
X component of interaction 398
Y
Y component of interaction 398
Yates’s order 237
Z
Z component of interaction 398
Z-tests on means 50
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