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 |  موضوع: كتاب Design and Analysis of Experiments الجمعة 01 أكتوبر 2021, 10:42 pm | |
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أحضرت لكم كتاب
Design and Analysis of Experiments
Ninth Edition
DOUGLAS C. MONTGOMERY
Arizona State University
و المحتوى كما يلي :
Contents
Preface iii
1
Introduction 1
1.1 Strategy of Experimentation 1
1.2 Some Typical Applications of Experimental Design 7
1.3 Basic Principles 11
1.4 Guidelines for Designing Experiments 13
1.5 A Brief History of Statistical Design 19
1.6 Summary: Using Statistical Techniques in Experimentation 20
1.7 Problems 21
2
Simple Comparative Experiments 23
2.1 Introduction 24
2.2 Basic Statistical Concepts 25
2.3 Sampling and Sampling Distributions 28
2.4 Inferences About the Differences in Means, Randomized Designs 33
2.4.1 Hypothesis Testing 33
2.4.2 Confidence Intervals 39
2.4.3 Choice of Sample Size 41
2.4.4 The Case Where ????2
1 ≠ ????22 44
2.4.5 The Case Where ????2
1 and ????22 Are Known 47
2.4.6 Comparing a Single Mean to a Specified Value 47
2.4.7 Summary 48
2.5 Inferences About the Differences in Means, Paired Comparison Designs 50
2.5.1 The Paired Comparison Problem 50
2.5.2 Advantages of the Paired Comparison Design 52
2.6 Inferences About the Variances of Normal Distributions 53
2.7 Problems 55
ixx Contents
3
Experiments with a Single Factor: The Analysis of Variance 64
3.1 An Example 65
3.2 The Analysis of Variance 67
3.3 Analysis of the Fixed Effects Model 69
3.3.1 Decomposition of the Total Sum of Squares 69
3.3.2 Statistical Analysis 72
3.3.3 Estimation of the Model Parameters 76
3.3.4 Unbalanced Data 78
3.4 Model Adequacy Checking 78
3.4.1 The Normality Assumption 79
3.4.2 Plot of Residuals in Time Sequence 81
3.4.3 Plot of Residuals Versus Fitted Values 81
3.4.4 Plots of Residuals Versus Other Variables 86
3.5 Practical Interpretation of Results 86
3.5.1 A Regression Model 87
3.5.2 Comparisons Among Treatment Means 88
3.5.3 Graphical Comparisons of Means 88
3.5.4 Contrasts 89
3.5.5 Orthogonal Contrasts 92
3.5.6 Scheffé’s Method for Comparing All Contrasts 93
3.5.7 Comparing Pairs of Treatment Means 95
3.5.8 Comparing Treatment Means with a Control 98
3.6 Sample Computer Output 99
3.7 Determining Sample Size 103
3.7.1 Operating Characteristic and Power Curves 103
3.7.2 Confidence Interval Estimation Method 104
3.8 Other Examples of Single-Factor Experiments 105
3.8.1 Chocolate and Cardiovascular Health 105
3.8.2 A Real Economy Application of a Designed Experiment 107
3.8.3 Discovering Dispersion Effects 109
3.9 The Random Effects Model 111
3.9.1 A Single Random Factor 111
3.9.2 Analysis of Variance for the Random Model 112
3.9.3 Estimating the Model Parameters 113
3.10 The Regression Approach to the Analysis of Variance 119
3.10.1 Least Squares Estimation of the Model Parameters 120
3.10.2 The General Regression Significance Test 121
3.11 Nonparametric Methods in the Analysis of Variance 123
3.11.1 The Kruskal–Wallis Test 123
3.11.2 General Comments on the Rank Transformation 124
3.12 Problems 125
4
Randomized Blocks, Latin Squares, and Related Designs 135
4.1 The Randomized Complete Block Design 135
4.1.1 Statistical Analysis of the RCBD 137
4.1.2 Model Adequacy Checking 145
4.1.3 Some Other Aspects of the Randomized Complete Block Design 145
4.1.4 Estimating Model Parameters and the General Regression Significance Test 150Contents xi
4.2 The Latin Square Design 153
4.3 The Graeco-Latin Square Design 160
4.4 Balanced Incomplete Block Designs 162
4.4.1 Statistical Analysis of the BIBD 163
4.4.2 Least Squares Estimation of the Parameters 167
4.4.3 Recovery of Interblock Information in the BIBD 169
4.5 Problems 171
5
Introduction to Factorial Designs 179
5.1 Basic Definitions and Principles 179
5.2 The Advantage of Factorials 182
5.3 The Two-Factor Factorial Design 183
5.3.1 An Example 183
5.3.2 Statistical Analysis of the Fixed Effects Model 186
5.3.3 Model Adequacy Checking 191
5.3.4 Estimating the Model Parameters 194
5.3.5 Choice of Sample Size 196
5.3.6 The Assumption of No Interaction in a Two-Factor Model 197
5.3.7 One Observation per Cell 198
5.4 The General Factorial Design 201
5.5 Fitting Response Curves and Surfaces 206
5.6 Blocking in a Factorial Design 215
5.7 Problems 220
6
The 2k Factorial Design 230
6.1 Introduction 230
6.2 The 22 Design 231
6.3 The 23 Design 240
6.4 The General 2k Design 252
6.5 A Single Replicate of the 2k Design 254
6.6 Additional Examples of Unreplicated 2k Designs 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 308
7.1 Introduction 308
7.2 Blocking a Replicated 2k Factorial Design 309
7.3 Confounding in the 2k Factorial Design 311
7.4 Confounding the 2k Factorial Design in Two Blocks 311
7.5 Another Illustration of Why Blocking Is Important 319
7.6 Confounding the 2k Factorial Design in Four Blocks 320xii Contents
7.7 Confounding the 2k Factorial Design in 2p Blocks 322
7.8 Partial Confounding 323
7.9 Problems 325
8
Two-Level Fractional Factorial Designs 328
8.1 Introduction 329
8.2 The One-Half Fraction of the 2k Design 329
8.2.1 Definitions and Basic Principles 329
8.2.2 Design Resolution 332
8.2.3 Construction and Analysis of the One-Half Fraction 332
8.3 The One-Quarter Fraction of the 2k Design 344
8.4 The General 2k−p Fractional Factorial Design 351
8.4.1 Choosing a Design 351
8.4.2 Analysis of 2k−p Fractional Factorials 354
8.4.3 Blocking Fractional Factorials 355
8.5 Alias Structures in Fractional Factorials and Other Designs 360
8.6 Resolution III Designs 362
8.6.1 Constructing Resolution III Designs 362
8.6.2 Fold Over of Resolution III Fractions to Separate Aliased Effects 364
8.6.3 Plackett–Burman Designs 367
8.7 Resolution IV and V Designs 376
8.7.1 Resolution IV Designs 376
8.7.2 Sequential Experimentation with Resolution IV Designs 377
8.7.3 Resolution V Designs 383
8.8 Supersaturated Designs 384
8.9 Summary 385
8.10 Problems 386
9
Additional Design and Analysis Topics for Factorial
and Fractional Factorial Designs 405
9.1 The 3k Factorial Design 406
9.1.1 Notation and Motivation for the 3k Design 406
9.1.2 The 32 Design 407
9.1.3 The 33 Design 408
9.1.4 The General 3k Design 413
9.2 Confounding in the 3k Factorial Design 413
9.2.1 The 3k Factorial Design in Three Blocks 413
9.2.2 The 3k Factorial Design in Nine Blocks 416
9.2.3 The 3k Factorial Design in 3p Blocks 417
9.3 Fractional Replication of the 3k Factorial Design 418
9.3.1 The One-Third Fraction of the 3k Factorial Design 418
9.3.2 Other 3k−p Fractional Factorial Designs 421
9.4 Factorials with Mixed Levels 422
9.4.1 Factors at Two and Three Levels 422
9.4.2 Factors at Two and Four Levels 424
9.5 Nonregular Fractional Factorial Designs 425Contents xiii
9.5.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs 427
9.5.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs 436
9.5.3 Analysis of Nonregular Fractional Factorial Designs 441
9.6 Constructing Factorial and Fractional Factorial Designs Using
an Optimal Design Tool 442
9.6.1 Design Optimality Criterion 443
9.6.2 Examples of Optimal Designs 443
9.6.3 Extensions of the Optimal Design Approach 453
9.7 Problems 454
10
Fitting Regression Models
(online at www.wiley.com/college/montgomery) 460
10.1 Introduction 461
10.2 Linear Regression Models 461
10.3 Estimation of the Parameters in Linear Regression Models 462
10.4 Hypothesis Testing in Multiple Regression 473
10.4.1 Test for Significance of Regression 473
10.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients 475
10.5 Confidence Intervals in Multiple Regression 478
10.5.1 Confidence Intervals on the Individual Regression Coefficients 478
10.5.2 Confidence Interval on the Mean Response 478
10.6 Prediction of New Response Observations 479
10.7 Regression Model Diagnostics 480
10.7.1 Scaled Residuals and PRESS 480
10.7.2 Influence Diagnostics 483
10.8 Testing for Lack of Fit 483
10.9 Problems 485
11
Response Surface Methods and Designs 489
11.1 Introduction to Response Surface Methodology 490
11.2 The Method of Steepest Ascent 492
11.3 Analysis of a Second-Order Response Surface 497
11.3.1 Location of the Stationary Point 497
11.3.2 Characterizing the Response Surface 499
11.3.3 Ridge Systems 505
11.3.4 Multiple Responses 506
11.4 Experimental Designs for Fitting Response Surfaces 511
11.4.1 Designs for Fitting the First-Order Model 511
11.4.2 Designs for Fitting the Second-Order Model 511
11.4.3 Blocking in Response Surface Designs 518
11.4.4 Optimal Designs for Response Surfaces 521
11.5 Experiments with Computer Models 535
11.6 Mixture Experiments 542
11.7 Evolutionary Operation 553
11.8 Problems 558xiv Contents
12
Robust Parameter Design and Process Robustness
Studies (online at www.wiley.com/college/montgomery) 569
12.1 Introduction 569
12.2 Crossed Array Designs 571
12.3 Analysis of the Crossed Array Design 573
12.4 Combined Array Designs and the Response Model Approach 576
12.5 Choice of Designs 582
12.6 Problems 585
13
Experiments with Random Factors 589
13.1 Random Effects Models 589
13.2 The Two-Factor Factorial with Random Factors 590
13.3 The Two-Factor Mixed Model 597
13.4 Rules for Expected Mean Squares 602
13.5 Approximate F-Tests 605
13.6 Some Additional Topics on Estimation of Variance Components 609
13.6.1 Approximate Confidence Intervals on Variance Components 609
13.6.2 The Modified Large-Sample Method 613
13.7 Problems 615
14
Nested and Split-Plot Designs 618
14.1 The Two-Stage Nested Design 619
14.1.1 Statistical Analysis 619
14.1.2 Diagnostic Checking 624
14.1.3 Variance Components 626
14.1.4 Staggered Nested Designs 626
14.2 The General m-Stage Nested Design 628
14.3 Designs with Both Nested and Factorial Factors 630
14.4 The Split-Plot Design 634
14.5 Other Variations of the Split-Plot Design 640
14.5.1 Split-Plot Designs with More Than Two Factors 640
14.5.2 The Split-Split-Plot Design 645
14.5.3 The Strip-Split-Plot Design 649
14.6 Problems 650
15
Other Design and Analysis Topics
(online at www.wiley.com/college/montgomery) 656
15.1 Nonnormal Responses and Transformations 657
15.1.1 Selecting a Transformation: The Box–Cox Method 657
15.1.2 The Generalized Linear Model 659Contents xv
15.2 Unbalanced Data in a Factorial Design 666
15.2.1 Proportional Data: An Easy Case 667
15.2.2 Approximate Methods 668
15.2.3 The Exact Method 670
15.3 The Analysis of Covariance 670
15.3.1 Description of the Procedure 671
15.3.2 Computer Solution 679
15.3.3 Development by the General Regression Significance Test 680
15.3.4 Factorial Experiments with Covariates 682
15.4 Repeated Measures 692
15.5 Problems 694
Appendix (online at www.wiley.com/college/montgomery) 697
Table I. Cumulative Standard Normal Distribution 698
Table II. Percentage Points of the t Distribution 700
Table III. Percentage Points of the ????2 Distribution 701
Table IV. Percentage Points of the F Distribution 702
Table V. Percentage Points of the Studentized Range Statistic 707
Table VI. Critical Values for Dunnett’s Test for Comparing Treatments
with a Control 709
Table VII. Coefficients of Orthogonal Polynomials 711
Table VIII. Alias Relationships for 2k−p Fractional Factorial Designs
with k ≤ 15 and n ≤ 64 712
Bibliography (online at www.wiley.com/college/montgomery) 724
Index 73
I n d e x
22 factorial design, 231
23 factorial design, 240
2k factorial design, 230, 252
2k−1fractional factorial design, 329
2k−2 design, 344
2k−p fractional factorial design, 351
32 factorial design, 406
33 factorial design, 407
3k factorial design, 405, 413
3k−1 fractional factorial design, 418
3k−p fractional factorial designs, 431
A
Additivity of the Latin square, 154
Additivity of the RCBD, 145
Adjusted R2, 265, 475
Agricultural era of experimentation, 19
Agricultural versus industrial experiments, 19
Alias matrix, 360, 427
Aliases, 330
Allowed-to-vary factors, 15
Alternate fraction, 331
Alternative hypothesis, 34
Analysis of combined array designs, 576
(online, Chapter 12)
Analysis of covariance as an alternative to blocking, 670,
679 (online, Chapter 15)
Analysis of covariance, 136, 670 (online, Chapter 15)
Analysis of variance (ANOVA), 67, 69, 73, 112
Analysis of variance identity for the RCBD, 138
Analysis of variance partition of the total sum
of squares, 70
ANOVA F-test, 73
ANOVA method for estimating variance components,
113, 591
Approximate F-tests, 605
Assumptions in the t-test, 37
B
Balanced incomplete block designs (BIBD), 162
Bartlett’s test for equality of variances, 82
Bayesian D-optimal designs, 453
Best-guess approach to experimentation, 4
Blocking, 11, 12, 135, 153, 215, 308, 518
Blocking fractional factorials, 355, 367
Blocking in a 2k design, 215, 308, 311
Blocking in response surface designs, 518
Box plot, 25, 66
Box-Behnken designs, 513
Box-Cox method for choosing a transformation, 657
(online, Chapter 15)
Break through innovation, 2
C
Canonical form of the second-order model, 499
Cause-and-effect diagram, 16
Center points in the 2k design, 285, 513
Central composite design (CCD), 288, 512, 513
Central limit theorem, 31
Chi-square distribution, 31
Chi-square test on the variance of a normal
distribution, 53
Coded and natural variables, 236
Coded variables, 236, 290
Coding the data in ANOVA, 75
Column generators, 329, 344
Combined array designs, 572, 576
(online, Chapter 12)
Comparison of means, 88, 95
Complete randomization, 11
Completely randomized design, 68
Components of interaction, 408
Components of variance model, 68, 111
Confidence interval for a contrast, 91
Confidence interval on the mean response in regression, 478
(online, Chapter 10)
Confidence interval, 40, 41
Confidence intervals on means in ANOVA, 77
Confirmation runs, 18, 343
Confounding, 311, 320, 413
Constant variance assumption in ANOVA, 81
Constrained optimization, 508
Construction of optimal designs, 524
Contour plot, 499, 506
Contrasts, 89, 93
Controllable variables, 3
Cook’s distance, 483 (online, Chapter 10)
731732 Index
Corrected sum of squares, 29
Critical region, 34
Crossed array designs, 571 (online Chapter 12)
Crossover designs, 159
D
Defining relation for a fractional factorial, 329
Definitive screening designs, 530
Degrees of freedom, 31
Design factors, 15
Design generators. 344
Design projection, 258
Design resolution, 332
Designs with nested and factorial factors, 630
Desirability function optimization, 508
Deterministic computer models, 535
Dispersion effects, 109
Distance based designs for mixture experiments, 551
D−optimal designs, 281
Dot diagram, 24,
Dunnett’s test to compare means with a control, 98
E
Eigenvalues, 499
Empirical models, 2
Equiradial design, 515
Estimating variance components, 113
Estimator, 28
Evolutionary operation (EVOP), 553
Expected mean squares, 71, 112
Expected value of a random variable, 27
Experiments with computer models, 535
Extra sum of squares method, 476
Extreme vertices designs for mixture experiments, 551
F
Face-centered CCD, 514
Factor screening experiment, 13, 15
Factorial experiment, 4, 179
Factorial experiments with covariates, 682
(online, Chapter 15)
F-distribution, 33
First-order model, 17
Fisher’s least significant difference (LSD) method
to compare pairs of means, 97
Fixed effects model, 68
Fold over of fractional factorial designs, 364, 366, 377
Fold over of resolution III designs, 364, 366
Fraction of design space plot, 284, 517
Fractional factorial experiment, 7
Fractional factorial split-plot designs, 644
F-test on two variances of independent normal
distributions, 54
Full model, 121, 476
G
Gaussian process model, 537
Generalized interaction, 320, 344
Generalized linear model (GLM), 659
G-optimal designs, 282
Graeco-Latin square design, 160
Guidelines for designing experiments, 13
H
Half-normal probability plot, 261
Hall designs, 427
Held-constant factors, 15
Histogram, 25
Hybrid designs, 516
Hypothesis testing, 24, 33
I
Incremental innovation, 2
Influential observations in regression, 482
(online, Chapter 10)
Innovation and designed experiments, 2
Interaction, 17, 180
Interblock information in the BIBD, 169
Interpretation of ANOVA results, 87
I-optimal designs, 282
K
Kruskal-Wallis test, 123
L
Lack of fit testing in regression, 483 (online, Chapter 10)
Latin square designs, 153, 218
Lenth’s method, 262
Levene’s test for equality of variances, 83
Leverage points, 483
Linear predictor in the GLM, 660 (online, Chapter 15)
Linear statistical model, 68
M
Main effect of a factor, 17, 180
Mean of a distribution, 27
Means model, 67
Mechanistic models, 2
Method of least squares, 463
Method of steepest ascent, 239, 492
Minimal resolution IV designs, 376
Minimum variance estimator, 29
Missing values in the Latin square design, 157
Missing values in the RCBD, 149
Mixed level factorial designs, 422
Mixed models, 597
Mixture experiment, 10, 542
Multifactor split plot designs, 640
Multiple comparisons following ANOVA, 87, 93, 95, 142
Multiple response optimization, 506Index 733
N
Nested designs, 619
Nested designs with m stages, 628
No-confounding designs, 428
Nonregular fractional factorial design, 369, 425, 427,
436, 447
Normal distribution, 30
Normal probability plot, 37
Normal probability plot of residuals, 79
Normal probability plotting of effects, 254
Normality assumption in ANOVA, 79
Nuisance factors, 135
Null hypothesis, 34
O
One-factor-at-a-time (OFAT) approach to experimentation, 4
Operating characteristic curve, 42, 103
Optimal designs, 19, 280, 281, 282, 442, 522, 524
Optimal designs for mixture experiments, 547
Optimal designs for robustness studies, 582
Optimization experiment, 14
Orthogonal contrasts, 92
Orthogonal design, 233, 469
Outliers, 266, 480
P
Paired comparison design, 47, 52
Paired t-test, 47
Partial aliasing, 36
Partial confounding, 314, 323
Partial fold over, 381
Partial F-test, 477
Path of steepest ascent, 492
Plackett-Burman designs, 367
Power curve, 42, 103
Power family of transformations, 657 (online, Chapter 15)
Power of a test, 34
Prediction interval, 479
Prediction of new observations in regression, 479
(online, Chapter 10)
Pre-experimental planning, 17
PRESS statistic, 99, 482 (online, Chapter 10)
Principal block, 314, 321
Principal fraction, 330
Probability distributions, 26
Projection of fractional factorials, 354
Projection property, 329
P-value, 36
Q
Quantitative versus qualitative factors in ANOVA, 87
R2R
, 474 (online, Chapter 10)
Random effects model, 68, 111, 589
Random error term, 67
Randomization, 11, 135
Randomization test, 39
Randomization tests and ANOVA, 76
Randomized complete block design (RCBD), 135, 136
Rank transformation in ANOVA, 124
RCBD with random treatments and blocks, 147
Reduced model, 122, 476
Reference distribution, 35
Regression approach to ANOVA, 119
Regression models, 87
REML method for estimating variance components, 118,
147, 595
Repeated measures designs, 692
Replicated design, 5
Replication versus repeat run, 12
Replication, 11
Residual plotting, 79, 81, 86, 145
Residuals, 79
Resolution III designs, 362
Resolution IV designs, 376
Resolution V designs, 383
Response surface, 206, 490
Response surface methodology, 490
Response surface models, 490
Response variable, 3
Restricted form of the mixed model, 597
Rising ridge systems, 506
Robust design, 20, 569
Rotatability, 512
Rotatable CCD, 513
R-student, 482 (online, Chapter 10)
Rules for expected mean squares, 602
S
Sample mean, 28
Sample size in ANOVA, 103
Sample standard deviation, 28
Sample variance, 28
Sampling distributions, 30
Scatter diagram, 66
Scheffé’s method for comparing all contrasts, 93
Scientific or engineering method, 2
Second-order model, 17
Sequences of fractional factorials, 341
Sequential experimentation, 14, 21, 341, 491
Signal-to-noise ratios, 573
Simplex centroid design, 542
Simplex design for fitting a first-order model, 511
Simplex designs for mixtures, 542
Simplex lattice design, 542
Single replicate of the 2k design, 254
Single-factor fold over, 364
Small composite designs, 515734 Index
Space filling designs, 536
Space filling designs for mixture experiments, 551
Sparsity of effects principle, 329, 331
Spherical CCD, 513
Split-plot designs, 634
Split-split-plot designs, 645
Standard error, 35
Standard Latin square, 157
Standard normal distribution, 30
Standardized contrast, 91
Standardized residuals, 80, 480
Stationary point, 497
Stationary ridge systems, 505
Stochastic computer models, 535
Strategy of experimentation, 1, 3
Strip-split plot designs, 649
Studentized residuals, 481 (online, Chapter 10)
Subplot error, 635
Subplots or split-plots, 635
Supersaturated designs, 384
Tt−
distribution, 32
Test statistic, 34
Testing significance of regression, 473
Tests on individual terms in regression, 475
Transformations, 657 (online, Chapter 15)
Tukey’s test to compare pairs of means, 95
Two-sample t-test, 35, 50
Two-sample t-test with unequal variances, 44
Two-stage nested designs, 619
Type I error, 34
Type II error, 34
Types of experiments, 13
U
Unbalanced data in a factorial, 666 (online, Chapter 15)
Unbalanced data in ANOVA, 78
Unbiased estimator, 29
Uncontrollable variables, 3
Unrestricted form of the mixed model, 599
V
Variance components, 111
Variance dispersion graph, 517
Variance of a distribution, 27
Variance stabilizing transformations, 82, 85, 657
(online, Chapter 15)
W
Whole plot error, 635
Whole plots, 635
ZZ
-test, 47
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