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 |  موضوع: كتاب A First Course in Design and Analysis of Experiments السبت 08 يناير 2022, 1:16 am | |
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أخواني في الله
أحضرت لكم كتاب
A First Course in Design and Analysis of Experiments
Gary W. Oehlert
University of Minnesota
و المحتوى كما يلي :
Contents
Preface xvii
1 Introduction 1
1.1 Why Experiment? 1
1.2 Components of an Experiment . 4
1.3 Terms and Concepts . 5
1.4 Outline . 7
1.5 More About Experimental Units 8
1.6 More About Responses . 10
2 Randomization and Design 13
2.1 Randomization Against Confounding . 14
2.2 Randomizing Other Things . 16
2.3 Performing a Randomization 17
2.4 Randomization for Inference 19
2.4.1 The paired t-test 20
2.4.2 Two-sample t-test . 25
2.4.3 Randomization inference and standard inference . 26
2.5 Further Reading and Extensions 27
2.6 Problems 28
3 Completely Randomized Designs 31
3.1 Structure of a CRD . 31
3.2 Preliminary Exploratory Analysis . 33
3.3 Models and Parameters . 34viii CONTENTS
3.4 Estimating Parameters . 39
3.5 Comparing Models: The Analysis of Variance . 44
3.6 Mechanics of ANOVA . 45
3.7 Why ANOVA Works 52
3.8 Back to Model Comparison . 52
3.9 Side-by-Side Plots . 54
3.10 Dose-Response Modeling 55
3.11 Further Reading and Extensions 58
3.12 Problems 60
4 Looking for Specific Differences—Contrasts 65
4.1 Contrast Basics . 65
4.2 Inference for Contrasts . 68
4.3 Orthogonal Contrasts 71
4.4 Polynomial Contrasts 73
4.5 Further Reading and Extensions 75
4.6 Problems 75
5 Multiple Comparisons 77
5.1 Error Rates . 78
5.2 Bonferroni-Based Methods . 81
5.3 The Scheff´e Method for All Contrasts . 85
5.4 Pairwise Comparisons 87
5.4.1 Displaying the results . 88
5.4.2 The Studentized range . 89
5.4.3 Simultaneous confidence intervals . 90
5.4.4 Strong familywise error rate 92
5.4.5 False discovery rate 96
5.4.6 Experimentwise error rate . 97
5.4.7 Comparisonwise error rate . 98
5.4.8 Pairwise testing reprise 98
5.4.9 Pairwise comparisons methods that do not control
combined Type I error rates 98
5.4.10 Confident directions 100CONTENTS ix
5.5 Comparison with Control or the Best 101
5.5.1 Comparison with a control . 101
5.5.2 Comparison with the best . 104
5.6 Reality Check on Coverage Rates . 105
5.7 A Warning About Conditioning . 106
5.8 Some Controversy 106
5.9 Further Reading and Extensions 107
5.10 Problems 108
6 Checking Assumptions 111
6.1 Assumptions 111
6.2 Transformations . 113
6.3 Assessing Violations of Assumptions 114
6.3.1 Assessing nonnormality 115
6.3.2 Assessing nonconstant variance 118
6.3.3 Assessing dependence . 120
6.4 Fixing Problems . 124
6.4.1 Accommodating nonnormality 124
6.4.2 Accommodating nonconstant variance 126
6.4.3 Accommodating dependence . 133
6.5 Effects of Incorrect Assumptions 134
6.5.1 Effects of nonnormality 134
6.5.2 Effects of nonconstant variance 136
6.5.3 Effects of dependence . 138
6.6 Implications for Design . 140
6.7 Further Reading and Extensions 141
6.8 Problems 143
7 Power and Sample Size 149
7.1 Approaches to Sample Size Selection . 149
7.2 Sample Size for Confidence Intervals 151
7.3 Power and Sample Size for ANOVA 153
7.4 Power and Sample Size for a Contrast . 158
7.5 More about Units and Measurement Units . 158x CONTENTS
7.6 Allocation of Units for Two Special Cases . 160
7.7 Further Reading and Extensions 161
7.8 Problems 162
8 Factorial Treatment Structure 165
8.1 Factorial Structure 165
8.2 Factorial Analysis: Main Effect and Interaction 167
8.3 Advantages of Factorials 170
8.4 Visualizing Interaction . 171
8.5 Models with Parameters . 175
8.6 The Analysis of Variance for Balanced Factorials . 179
8.7 General Factorial Models 182
8.8 Assumptions and Transformations . 185
8.9 Single Replicates 186
8.10 Pooling Terms into Error 191
8.11 Hierarchy 192
8.12 Problems 197
9 A Closer Look at Factorial Data 203
9.1 Contrasts for Factorial Data . 203
9.2 Modeling Interaction 209
9.2.1 Interaction plots 209
9.2.2 One-cell interaction 210
9.2.3 Quantitative factors 212
9.2.4 Tukey one-degree-of-freedom for nonadditivity 217
9.3 Further Reading and Extensions 220
9.4 Problems 222
10 Further Topics in Factorials 225
10.1 Unbalanced Data 225
10.1.1 Sums of squares in unbalanced data 226
10.1.2 Building models 227
10.1.3 Testing hypotheses . 230
10.1.4 Empty cells . 233
10.2 Multiple Comparisons . 234CONTENTS xi
10.3 Power and Sample Size . 235
10.4 Two-Series Factorials 236
10.4.1 Contrasts 237
10.4.2 Single replicates 240
10.5 Further Reading and Extensions 244
10.6 Problems 245
11 Random Effects 253
11.1 Models for Random Effects . 253
11.2 Why Use Random Effects? . 256
11.3 ANOVA for Random Effects 257
11.4 Approximate Tests . 260
11.5 Point Estimates of Variance Components 264
11.6 Confidence Intervals for Variance Components 267
11.7 Assumptions 271
11.8 Power 272
11.9 Further Reading and Extensions 274
11.10 Problems 275
12 Nesting, Mixed Effects, and Expected Mean Squares 279
12.1 Nesting Versus Crossing 279
12.2 Why Nesting? 283
12.3 Crossed and Nested Factors . 283
12.4 Mixed Effects 285
12.5 Choosing a Model 288
12.6 Hasse Diagrams and Expected Mean Squares . 289
12.6.1 Test denominators . 290
12.6.2 Expected mean squares 293
12.6.3 Constructing a Hasse diagram . 296
12.7 Variances of Means and Contrasts . 298
12.8 Unbalanced Data and Random Effects . 304
12.9 Staggered Nested Designs . 306
12.10 Problems 307xii CONTENTS
13 Complete Block Designs 315
13.1 Blocking . 315
13.2 The Randomized Complete Block Design . 316
13.2.1 Why and when to use the RCB 318
13.2.2 Analysis for the RCB . 319
13.2.3 How well did the blocking work? . 322
13.2.4 Balance and missing data . 324
13.3 Latin Squares and Related Row/Column Designs . 324
13.3.1 The crossover design 326
13.3.2 Randomizing the LS design 327
13.3.3 Analysis for the LS design . 327
13.3.4 Replicating Latin Squares . 330
13.3.5 Efficiency of Latin Squares 335
13.3.6 Designs balanced for residual effects . 338
13.4 Graeco-Latin Squares 343
13.5 Further Reading and Extensions 344
13.6 Problems 345
14 Incomplete Block Designs 357
14.1 Balanced Incomplete Block Designs 358
14.1.1 Intrablock analysis of the BIBD 360
14.1.2 Interblock information . 364
14.2 Row and Column Incomplete Blocks 368
14.3 Partially Balanced Incomplete Blocks . 370
14.4 Cyclic Designs . 372
14.5 Square, Cubic, and Rectangular Lattices 374
14.6 Alpha Designs 376
14.7 Further Reading and Extensions 378
14.8 Problems 379CONTENTS xiii
15 Factorials in Incomplete Blocks—Confounding 387
15.1 Confounding the Two-Series Factorial . 388
15.1.1 Two blocks . 389
15.1.2 Four or more blocks 392
15.1.3 Analysis of an unreplicated confounded two-series 397
15.1.4 Replicating a confounded two-series . 399
15.1.5 Double confounding 402
15.2 Confounding the Three-Series Factorial 403
15.2.1 Building the design 404
15.2.2 Confounded effects 407
15.2.3 Analysis of confounded three-series 408
15.3 Further Reading and Extensions 409
15.4 Problems 410
16 Split-Plot Designs 417
16.1 What Is a Split Plot? 417
16.2 Fancier Split Plots 419
16.3 Analysis of a Split Plot . 420
16.4 Split-Split Plots . 428
16.5 Other Generalizations of Split Plots 434
16.6 Repeated Measures . 438
16.7 Crossover Designs . 441
16.8 Further Reading and Extensions 441
16.9 Problems 442
17 Designs with Covariates 453
17.1 The Basic Covariate Model . 454
17.2 When Treatments Change Covariates 460
17.3 Other Covariate Models . 462
17.4 Further Reading and Extensions 466
17.5 Problems 466xiv CONTENTS
18 Fractional Factorials 471
18.1 Why Fraction? 471
18.2 Fractioning the Two-Series . 472
18.3 Analyzing a 2k−q 479
18.4 Resolution and Projection 482
18.5 Confounding a Fractional Factorial . 485
18.6 De-aliasing . 485
18.7 Fold-Over 487
18.8 Sequences of Fractions . 489
18.9 Fractioning the Three-Series 489
18.10 Problems with Fractional Factorials 492
18.11 Using Fractional Factorials in Off-Line Quality Control 493
18.11.1 Designing an off-line quality experiment . 494
18.11.2 Analysis of off-line quality experiments 495
18.12 Further Reading and Extensions 498
18.13 Problems 499
19 Response Surface Designs 509
19.1 Visualizing the Response 509
19.2 First-Order Models . 511
19.3 First-Order Designs . 512
19.4 Analyzing First-Order Data . 514
19.5 Second-Order Models 517
19.6 Second-Order Designs . 522
19.7 Second-Order Analysis . 526
19.8 Mixture Experiments 529
19.8.1 Designs for mixtures 530
19.8.2 Models for mixture designs 533
19.9 Further Reading and Extensions 535
19.10 Problems 536CONTENTS xv
20 On Your Own 543
20.1 Experimental Context 543
20.2 Experiments by the Numbers 544
20.3 Final Project . 548
Bibliography 549
A Linear Models for Fixed Effects 563
A.1 Models . 563
A.2 Least Squares 566
A.3 Comparison of Models . 568
A.4 Projections . 570
A.5 Random Variation 572
A.6 Estimable Functions . 576
A.7 Contrasts 578
A.8 The Scheff´e Method . 579
A.9 Problems 580
B Notation 583
C Experimental Design Plans 607
C.1 Latin Squares 607
C.1.1 Standard Latin Squares 607
C.1.2 Orthogonal Latin Squares . 608
C.2 Balanced Incomplete Block Designs 609
C.3 Efficient Cyclic Designs 615
C.4 Alpha Designs 616
C.5 Two-Series Confounding and Fractioning Plans 617
D Tables 621
Index 64
Index
0/1 rule, 390
23 in two blocks of size four, 389
24 in two blocks of eight, 390, 392
25 in eight blocks of four, 395
27 in 16 blocks of eight, 396
28−4 in two blocks of eight, 485
32 with A1B2 confounded, 404
Accelerated life tests, 33
Acid rain and birch seedlings, 32, 86
Addled goose eggs, 324
Adjustment variables, 495
Alfalfa meal and turkeys, 103
Aliases
in three-series, 490
in two-series, 475
Alpha designs, 376
tables of, 616
Alternate block, 391
Alternate fraction, 473, 490
Alternative hypotheses
fixed-effects F-test, 45
paired t-test, 21
two-sample t-test, 25
Amylase activity, 195, 213, 226, 228,
233
Analysis of covariance, 454
see also Covariates39
Analysis of variance, 44
balanced incomplete blocks, 361
completely randomized design, 46,
48
confounded designs, 400
expected mean squares, 257
factorial treatment structure, 179,
180
Latin Square, 328
lattice designs, 376
linear subspaces, 567
partially balanced incomplete
blocks, 372
random-effects, 257
Randomized complete block, 321
repeated measures, 440
residual effects, 342
split-plots, 424
split-split-plots, 430
weighted, 131
Youden Square, 369
ANOVA, see Analysis of variance
Anxiety, tension, and memory, 426
Artificial insemination in chickens, 127
Associate classes, 371
Assumptions, 111–143
and factorial treatment structure, 185
assessing, 114–123
constant variance, 118, 126, 136
fixed-effects models, 111
independence, 120, 133, 138
normality, 115, 124, 134, 272
random-effects models, 271
role of residuals, 112
Autocorrelation, 120
Axial points, 522, 523
Balanced incomplete block designs,648 Index
358–368
efficiency, 362
interblock information, 364
intrablock analysis, 360
model for, 360
randomization, 360
symmetric, 360
tables of, 609
unreduced, 360
Bartlett’s test, 118
Bayesian methods, 27, 28
Bioequivalence of drug delivery, 326,
329, 333, 337
Blinding, 6
Blocking, 315
complete, 316
confounding, 387
do not test blocks, 321
incomplete, 357–379, 387
initial, 373
reused in Latin Squares, 330
Split plot designs, 417
Bonferroni methods, 81–84
BSD, 91
for factorials, 205
Bootstrapping, 28
Box-Behnken designs, 525
Brown-Forsythe modified F, 133
Cadmium in soils, 17
Cake baking, 514, 516, 524, 526
Canonical analysis, 526
Canonical variables, 518, 521
Carbon monoxide emissions, 330, 331
Carcinogenic mixtures, 77
Cardiac arrhythmias, 11
Carton experiment three, 263,
266–268, 270, 271, 273
Causation, 2, 3
Center points, 513, 522
Central composite designs, 522
orthogonal blocking, 523
rotatable, 523
uniform precision, 524
Cheese tasting, 284, 287, 298
Chi-square distribution, 59, 161, 260
noncentral, 161, 575
table of, 626
Chick body weights, 173
Cloud seeding, 117, 125
Complete mixtures, 531
Completely randomized designs,
31–60
analysis of variance, 46, 48
degrees of freedom, 39, 41
expected mean squares, 52
factorial treatment structure,
165–196
model for, 37–39
parameter estimates for, 40, 41
parameters of, 37
randomization, 31
sample sizes, 31
sums of squares, 40
Components of variance, see Variance
components
Confidence intervals
and skewness, 135
for contrasts, 68
for intraclass correlation, 269
for means, 43
for ratios of variances, 269
for variance components, 267
Scheff´e method, 85
variance components and
nonnormality, 272
Williams’ method, 270
Confident directions, 100
Confounded designs, 387–410
analysis of, 397, 408
complete confounding, 400
double confounding, 402
fractional factorials, 485
guidelines, 394Index 649
partial confounding, 400
replication of, 399
three-series factorials, 403–409
two-series factorials, 388–403
two-series plans, 617
Confounding, 7
in split plots, 418
Confounding a 33 in nine blocks, 405,
407
Confounding a 35 in 27 blocks, 408
Connected designs, 358
Contour plots, 510
Contrasts, 65–75, 578
and empty cells, 234
for factorial treatment structure, 169,
203
in two-series factorials, 237
interaction, 170
main-effects, 169
orthogonal, 71–73, 578
polynomial, 67, 73–74, 213
table of orthogonal, 630
power, 158
Scheff´e method, 85
variances in mixed effects, 298–303
Control
of an experiment, 7
Control treatment, see Treatments,
control
Correlated errors, 138
Covariances of means, 302
Covariates, 453–466
affected by treatments, 460
and split plots, 466
centered, 460
CPU page faults, 187, 218
Crossover designs, 326, 441
Cyclic designs, 372
initial block, 373
tables of, 615
Data
advertising, 469
air cells, 251
alfalfa meal and turkeys, 353
alpine meadows, 62
amylase activity, 194
anticonvulsants, 251
bacteria in abused milk, 312
barley sprouting, 166
beer retained in mouth, 313
big sagebrush, 202
bioequivalence, 333
bird bones, 467
book ratings, 28
bread flours, 537
caffeine and adenine, 63
cake baking, 514, 525
car seats, 505
cardiac relaxants, 144
cisplatin, 224
cloud seeding, 117
CO emissions, 538
cockroaches, 29
coffee yields, 354
contaminated milk, 278
cracks in pavement, 345
cytokinin, 351
disk drive access, 380
disk drives, 347
fat acidity, 223
fillings, 310
free alpha amino nitrogen, 198
free amino acis in cheese, 313
fruit flies, 62
fruit punch, 530
gel strength, 224
gentleness, 144
graininess, 346
growth hormones, 354
gum water-binding, 250
highly unbalanced factorial, 230
ice creams, 249
icings, 508650 Index
impregnated cows, 29
interchanges, 448
irrigation, 444
Japanese beetles, 380
keyboarding pain, 456
laundry detergent, 449
leaf angles, 61
leaf springs, 497
leucine, 252
locations of good and bad chips, 123
long-distance quality, 446
mealy bugs, 317
melatonin, 143
memory errors, 426
mercury in soils, 470
milfoil, 415
milk chiller, 401
milk filtration, 381
milk production, 339
odor intensities, 414
oleoresin, 354
one-cell interaction, 211
orange pulp silage, 61
pacemaker delamination, 200
pacemaker substrate lengths, 353
page faults, 187
particleboard, 199
pediatricians, 252
pine oleoresin, 201
plates washed, 359
polypropylene concrete, 63
potato chips, 355
product scoring, 507
quack grass, 147
rat deaths, 193
rat liver iron, 172
rat liver weights, 60
resin lifetimes, 33
rocket errors, 352
rocket fuel, 537
ruffe, 451
runstitch times, 20
serial dilution, 144
serum lithium, 369
shear strength, 541
softness of clothes, 382
solder joints, 61
soybean herbicides, 350
soybean rotations, 350
speedometer casings, 504
State exams, 382
tensile strength, 278
thermocouples, 121
thickness of silicon, 506
tire wear, 276
total free amino acids, 178
tropical grasses, 201
two-series design, 246
vegetable oil, 276
Verapamil, 248
Visiplume, 30, 147
visual perception, 398
weed control in soybeans, 105
weight gain, 197
weight gain of calves, 275
welding strength, 480
wetland snowmelt, 29
wetland weeds, 432
white leghorns, 223
whole plant phosphorus, 25
work baskets, 349
yogurt odors, 415
yogurts, 248
Data snooping, 78, 85, 186
De-aliasing, 485
Defective integrated circuits on a
wafer, 123
Defining contrast, 389
Defining relation, 473
Degrees of freedom
approximate, 132, 262
factorial treatment structure, 177,
183
for error, 41Index 651
for treatments, 39
heuristics for, 51
interaction, 177, 183
main-effect, 177, 183
nested design, 281
Design of experiments, see
Experimental design
Design variables, 495
Deviations, 37
Dish detergent, 359, 361, 365
Doses, 55
Dunnett’s procedure, 101
Dunnett’s t distribution
table of, 635
Durbin-Watson statistic, 121, 142
Edge effects, 9, 159
Effect sparsity, 241
Effects
carryover, 339
confounded, 393, 407
covariate, 454
covariate-adjusted, 454
direct, 339
dispersion, 497
fixed, 254
interaction, 168, 176, 183
location, 497
main, 168, 175, 183
mixed, 285–288
nested design, 283
random, 253–275
residual, 339
simple, 205
standard errors of, 44
total, 238, 474
treatment, 38
Efficiency
Alpha design, 378
balanced incomplete block design,
362
confounded design, 387
cyclic design, 373
Latin Squares, 335
partially balanced incomplete block
design, 372
randomized complete block, 322
split-plots, 419
square lattice, 375
Eigenvalues, 521
Eigenvectors, 521
Embedded factorial, 477
Empty cells, 233
Entries
self-referencing, 651
Error
design to estimate, 5
experimental, 6, 37
systematic, 5
Error rates, 78–81
comparisonwise, 79, 98
conditional, 106
experimentwise, 79, 97
false discovery rate, 79, 96
simultaneous confidence intervals,
80, 90
strong familywise, 79, 92
Estimable functions, 75, 576
Estimates
see also individual designs39
of variance components, 264–266
unbiased, 39, 41, 272
Ethics, 4
Even/odd rule, 390
Exchangeability, 28
Expected mean squares, 258–260, 272,
274
completely randomized designs, 52
nested design, 281
random effects, 257
rules for, 293
Expected mean squares in the
restricted model, 293652 Index
Expected mean squares in the
unrestricted model, 294
Expected mean squares
unbalanced mixed-effects, 304
Experiment design, 4
Experimental designs, 4
alpha, 376
balanced incomplete block, 358–368
Box-Behnken, 525
central composite, 522
completely randomized, 31
confounding, 387–410
context, 543
cross-nested factors, 283
crossed factors, 280
crossover, 326
cyclic design, 372
factorial ratios, 531
factorial treatment structure, 165
fractional factorials, 471–499
generalized randomized complete
block, 344
goals, 543
Graeco-Latin Square, 343
hyper-Latin Square, 344
hypotheses, 543
Latin Square, 324–342
lattices, 374
main-effects, 483
mixtures, 529
nested factors, 280
objectives, 543
orthogonal-main-effects, 498
partially balanced incomplete block,
370
Plackett-Burman, 499
Randomized complete blocks, 324
randomized complete blocks, 316
repeated measures, 438–441
residual effects, 338
response surfaces, 509–535
row orthogonal, 369, 373
split block, 435
split plot, 417–428
split-split plot, 428–434
staggered nested, 306
strip plot, 435
with covariates, 453–466
Youden square, 368
Experimental error, see Error,
experimental
Experimental units, see Units,
experimental
Experiments
advantages of, 2
components of, 2
randomized, 13
Exploratory analysis, 33
Eyedrops, 357
F distribution, 59
noncentral, 153, 575
table of, 627
F-tests
p-value, 48
approximate, 260–264, 295
Brown-Forsythe modification, 133
completely randomized design, 48
factorial treatment structure, 181
for contrasts, 69
mixed-effects, 290
random-effects, 258–260
Scheff´e method, 85
Factorial contrasts, 205
Factorial ratios designs, 531
Factorial treatment structure, 165–196
advantages of, 170
analysis of variance, 179, 180
balanced, 166
confounding, 387–410
contrasts, 169, 203
degrees of freedom, 177, 183
empty cells, 233
expected mean squares, 257Index 653
F-test, 181
fractional factorials, 471–499
hierarchical models, 192
interaction effects, 168, 176, 183
main effects, 168, 175, 183
mixed effects, 285
models and weighting, 193
models for, 175
models of interaction, 209
noncentrality parameter, 235
pairwise comparisons, 204
parameter estimates for, 177
pooling terms, 191
power, 235
random effects, 255
single replicates, 186, 240, 397
sums of squares, 180, 184
transformations and interactions,
185
unbalanced data, 225–234
unweighted tests, 230
Factors, 7
coded, 513
continuous, 509
crossed, 280
grouping, 438
nested, 279–283
noise, 515
random, 255
split-plot, 418
split-split plot, 429
trial, 438
whole-plot, 417
Finding mean squares for an
approximate test, 262
First-order designs, 512
Fish
flopping, 340
Fold-over, 487
Fold-over for a 215−10
IV , 487
Fractional factorials, 471–499
aliases, 475, 490
analysis, 479
confounding, 485
de-aliasing, 485
fold-over, 487
in quality experiments, 493
minimum aberration, 483
motivation for, 471
pitfalls, 492
projection, 482
resolution, 482, 491
sequences of, 489
three-series, 489
two-series, 472
two-series plans, 617
Free amino acids in cheese, 91, 94, 96,
97
Free height of leaf springs, 496
Fruit punch, 529
Functional magnetic resonance
imaging, 80
Generalized interactions, 394, 407,
475, 490
Generalized linear models, 142
Generating array, 376
Generator, 473
Goals, 543
Graeco-Latin Squares, 343
Greenhouse-Geisser adjustment, 442
Gum arabic, 284, 287
Haphazard, 13, 14
Hartley’s test, 118
Harvey Wallbangers, 531, 534
Hasse diagrams, 289–298
and expected mean squares, 293
and test denominators, 290
construction of, 296
Huynh-Feldt adjustment, 442
Huynh-Feldt condition, 439
Hyper-Latin Squares, 344
Hypotheses, 543654 Index
Index plot, 120, 122
Inefficiency
of incomplete blocks, 357, 387
Inner noise, 494
Interaction
column-model, 221
dose-response, 212
Johnson-Graybill, 222
one-cell, 210
polynomial, 212
row-model, 221
slopes-model, 221
Tukey one-degree-of-freedom, 217,
220
Interaction plot, 171–174, 209
Intermediate array, 376
Interpolation, 56
Intraclass correlation, 254
confidence interval for, 269
Keyboarding pain, 453, 456, 461, 464
Kurtosis, 134
Lack of fit, 513, 516
Land’s method, 114, 126
Latin Squares, 324–342
analysis of variance, 328
estimated effects, 332
incomplete, 368
model for, 327, 331
orthogonal, 343, 374
randomization, 327
relative efficiency of, 335
replicated, 330
standard, 327
tables of, 607
Lattice designs
balanced, 375
cubic, 374, 375
efficiency of, 375
rectangular, 374, 375
simple, 374
square, 374
triple, 374
Lattice Squares, 378
Leaflet angles, 74
Least squares, 45, 58, 566
Lenth’s PSE, 241, 479
Levels, 7
Levene’s test, 119
Leverage, 115
Lithium in blood, 369
Machine shop, 434
Mallows’ C
p, 59
Masking, 118
Mauchly test, 439
Mealybugs on cycads, 316, 321, 323
Means
and transformations, 113
covariate-adjusted, 456
variances in mixed effects, 298–303
Measurement units, see Units,
measurement
Milk chiller, 401
Milk yield, 340
Minimum aberration design, 483
Mixture designs, 529
constrained, 532, 535
factorial ratios, 531
first-order model, 533
pseudocomponents, 532
second-order model, 533
simplex centroid, 530
simplex lattice, 530
third-order model, 533
Models, 34
additive, 171, 322, 328, 343, 360
analysis of covariance, 454
assumptions for mixed-effects, 286
balanced incomplete block design,
360
canonical form, 533
comparison of, 44, 226, 455, 568Index 655
completely randomized design,
37–39
cross-nested factors, 284
dose-response, 55–58, 212
factorial treatment structure, 175
first-order, 511, 533
fixed-effects, 254
for errors, 36
for interaction, 209
for means, 36
for mixtures, 533
full, 37
hierarchical, 192, 213, 227, 255
Latin Squares, 327
lattice of, 565
linear subspaces, 563
overparameterized, 38
parallel-lines, 455
parameters of, 34
polynomial, 55–58, 212, 511, 517,
530, 533
randomized complete block, 319
reduced, 37
repeated measures, 440
replicated Latin Squares, 331
restricted assumptions, 286, 288
second-order, 517, 533
separate means, 37
separate-intercepts, 455
separate-lines, 464
separate-slopes, 464
single-line, 455
single-mean, 37
split-plot, 421–423
split-split-plot, 429
steps for building, 285, 288
strip plot, 436
third-order, 533
Tukey one-degree-of-freedom, 217,
220
unrestricted assumptions, 286, 288
Multiple comparisons, 77–108
see also Simultaneous inference77
see also Pairwise comparisons77
with best, 104
Nesting, 279–283
Noncentrality parameter, 154
in factorial treatment structure, 235
in mixed effects, 293
Nonstarter bacteria in cheddar cheese,
178, 181
Normal distribution, 36, 572
table of, 624
Normal probability plot, 115, 118
Normal scores, 115
Null hypotheses
and transformations, 113
family of, 78
fixed-effects F-test, 45
interactions, 181
main-effects, 181
overall, 78
paired t-test, 21
random-effects, 255
randomization test, 22, 26
two-sample t-test, 25
unbalanced factorials, 230, 244
Objectives, 543
Observational study, 2
advantages and disadvantages, 3
Occam’s razor, 45
Off-line quality control, 493
One-at-a-time designs, 170
One-cell interaction, 210, 211
Optimal design, 344, 379, 535
Orthogonal-main-effects designs, 498
Outer noise, 494
Outliers, 116, 124, 136, 141
Overall mean, see Parameters, overall
mean
p-values, 48
calibrated, 49656 Index
F-test, 48
paired t-test, 21
randomization test, 22, 24, 26, 27
Pacemaker substrates, 240, 241
Pairwise comparisons, 66, 87–101
BSD, 91, 205
confident directions, 100
DSD, 101, 107
Duncan’s multiple range, 99
Dunnett’s procedure, 101
for factorial treatment structure, 204
LSD, 98, 107
MCB, 104
predictive methods, 100
protected LSD, 97
REGWR, 94, 107
SNK, 96, 107
step-down methods, 92
Tukey HSD, 90, 107, 205
Tukey-Kramer, 91, 108
with best, 104
with control, 101
Parameters
interaction effects, 176, 183
main effects, 175, 183
noncentrality, 154
of CRD, 37
of factorials, 175, 183
overall mean, 38, 175, 183
restrictions on, 38
Partially balanced incomplete blocks,
370
associate classes, 371
randomization, 371
Particle sampling, 287
Permutation tests, 27
Perspective plots, 510
Placebo, 7
Plackett-Burman designs, 499
Planning an experiment, 544
Polynomials
see also Models, polynomial55
see also Contrasts, polynomial55
Power, 150
curves, 154, 273
factorial effects, 235
for a contrast, 158
random effects, 272
software, 156
Power curves
fixed-effects, 639
random-effects, 643
Practical significance, 49
Precision, 5
Prediction, 59
Principal block, 391, 404
Principal fraction, 473, 490
Profile plot, 171–174
Projection
of fractional factorials, 482
onto linear subspace, 570
orthogonal, 571
Proportional balance, 244
Proportions, 529
Protein/amino acid effects on growing
rats, 318
Pseudo-standard error, 241
Pseudocomponents, 532
Pseudorandom numbers, 19
Pure error, 513
Pure interactive response, 171
Quarter fraction of a 25 design, 473
Random digits
table of, 622
Random effects, see Effects, random
Randomization, 6, 13–28
completely randomized design, 31
determines design, 16
inference, 19–27
lack of, 15
Latin Squares, 327
of balanced incomplete block
design, 360Index 657
partially balanced incomplete
blocks, 371
performing, 17–19
repeated measures, 438
restricted, 315, 418, 430
to determine design, 318
Randomization tests, 126
and standard inference, 26
subsampled distribution, 24
Randomized complete blocks, 316–324
generalized, 344
model for, 319
relative efficiency of, 322
unbalanced, 324
Rank-based methods, 124, 141
Rankits, 115
Rat liver iron, 172
Rat liver weights, 69
Regression, 56, 60, 455
Repeated-measures designs, 438–441
model for, 440
randomization, 438
univariate analysis, 439
Residual plot, 119, 120
Residuals, 45, 112, 573
externally Studentized, 115
internally Studentized, 114
raw, 114
Resin lifetimes, 32, 34, 42, 50, 57, 119,
130, 133
Resistance, 136
Resolution, 482, 491
Resolvable designs, 358
Response surface designs, 509–535
Box-Behnken, 525
canonical analysis, 526
canonical variables, 518, 521
central composite, 522
first-order analysis, 514
first-order designs, 512
first-order models, 511
second-order analysis, 526
second-order designs, 522
second-order models, 517
Responses, 2, 6
audit, 11
multivariate, 439
predictive, 11, 453
primary, 10
surrogate, 10
Ridge surface, 518, 521
Robust methods, 124, 136, 141
Robustness of validity, 112, 136
Rotatable designs, 523
Saddle point, 518, 519, 521
Sample size
choosing, 149–161
effective, 140, 363
fixed-effects power, 153
for a contrast, 158
for comparison with control, 160
for comparisons with control, 103
for confidence intervals, 151
for random effects, 273
Satterthwaite approximation, 262, 274
Scheff´e method, 85–86, 579
Second-order designs, 522
Seed maturation on cut stems, 243
Seed viability, 215
Sensory characteristics of cottage
cheeses, 83
Serial dependence, 120
Side-by-side plots, 54
Signficance level, 48
Significant differences, 88
Simplex, 529
Simultaneous inference, 77–108
Bonferroni, 81, 117
false discovery rate, 82
for factorial treatment structure, 234
Holm procedure, 82
Scheff´e method, 85
Skewness, 134658 Index
Spanking, 2
Spatial association, 122
Split plot examples, 421–423
Split-block designs, 435
Split-plot designs, 417–428
analysis of, 420
analysis of variance, 424–428
and covariates, 466
blocked, 420, 423
generalized, 434
models for, 420
randomization of, 418
Split-split plot examples, 429
Split-split-plot designs, 428–434
analysis of variance, 430–431
randomization of, 430
Staggered nested designs, 306
Standard order, 237
Stationary point, 519
Steepest ascent, 512, 515
Step-down methods, 92
Strip-plot designs, 435
Structures, see Models
Student-Newman-Keuls procedure, 96
Studentized range, 89
table of, 632
Subsampling, 256
Sums of squares
balanced incomplete block design,
363
completely randomized designs, 40
error, 46
factorial treatment structure, 180,
184
for contrasts, 69
for residuals, 45
fully adjusted, 232
linear, 56
model, 575
nested design, 282
polynomial, 56
quadratic, 56
residual, 53, 574
sequential, 56, 227
total, 46
treatment, 46
Type I, 227
Type II, 228
Type III, 232
t distribution, 21
table of, 625
t-tests
for contrasts, 69
paired, 20–25
Scheff´e method, 85
two-sample, 25–26
Welch, 132
Tables
Bonferroni t distribution, 631
chi-square distribution, 626
Dunnett’s t distribution, 635
F distribution, 627
fixed-effects power, 639
normal distribution, 624
orthogonal polynomial contrasts,
630
random digits, 622
random-effects power, 643
Studentized range, 632
t distribution, 625
Taguchi methods, 493
Temperature differences, 121
Test denominators in the restricted
model, 291
Test denominators in the unrestricted
model, 292
Three-series factorials
confounding, 403–409
fractioning, 489
Total effect, 238
Transformable nonadditivity, 217, 220
Transformations, 113–114, 141
and interactions, 185Index 659
Box-Cox, 128–131, 495
logarithmic, 124, 125
power, 124
power family, 128
variance stabilizing, 126, 127
Treatment design, 4, 471
Treatment effects, see Effects,
treatment
Treatments, 2, 6
control, 7
placebo, 7
quantitative, 55
split-plot, 418
whole-plot, 417
Treatments in a 28−4 design, 478
Tukey HSD, 90
Tukey one-degree-of-freedom, 217,
220, 322
Two-degree-of-freedom bundles, 403
Two-series factorials, 236–243
confounding, 388–403
contrasts, 237
fractioning, 472
single replicates, 240
Unbalanced data, 225–234
empty cells, 233
Unbalanced data puzzle, 229, 232
Unbalanced expected mean squares,
304
Underline diagram, 88
Unequal weights, 193
Units
allocation of, 9
costs of, 159
crossed, 435
experimental, 2, 5, 6, 8, 158
independence of, 9
measurement, 6, 8, 158
nested, 435
nested and crossed, 437
size of, 9
split plots, 418
split-split plots, 429
strip plots, 435
whole plots, 417
Validity, 5
versus precision, 5, 315
Variance
estimate of, 41
in quality experiment, 494
negative estimate of, 266
of contrasts, 68
of means, 300
pooled estimate of, 25
Variance components, 254
confidence intervals for, 267
estimates in nested designs, 282
estimates of, 264–266, 275
standard error of estimates, 267
Williams’ method, 270
Variance reduction designs, see
Experimental designs
Variogram, 122
Viruses and variances, 36
Visual perception, 397
VOR in ataxia patients, 150, 152, 155,
156, 158
Washout period, 10, 338
Weed biomass in wetlands, 432
Weed control in soybeans, 104
Welch t-test, 132
Welding strength, 479, 484
Whole plots, 417
Williams’ method, 270
Yates’ algorithm, 239
Youden squares, 368
Zero-sum restrictions, 38, 176, 183,
564
Zinc retention, 174
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