A first course in design and analysis of experiments

A First Course in Design and Analysis of Experiments A First Course in Design and Analysis of Experiments Gary W Oehlert University of Minnesota Cover design by Victoria Tomaselli Cover illustration by Peter Hamlin Minitab is a registered trademark of Minitab, Inc SAS is a registered trademark of SAS Institute, Inc S-Plus is a registered trademark of Mathsoft, Inc Design-Expert is a registered trademark of Stat-Ease, Inc Library of Congress Cataloging-in-Publication Data Oehlert, Gary W A first course in design and analysis of experiments / Gary W Oehlert p cm Includes bibligraphical references and index ISBN 0-7167-3510-5 Experimental Design I Title QA279.O34 2000 519.5—dc21 99-059934 Copyright c 2010 Gary W Oehlert All rights reserved This work is licensed under a “Creative Commons” license Briefly, you are free to copy, distribute, and transmit this work provided the following conditions are met: You must properly attribute the work You may not use this work for commercial purposes You may not alter, transform, or build upon this work A complete description of the license may be found at http://creativecommons.org/licenses/by-nc-nd/3.0/ For Becky who helped me all the way through and for Christie and Erica who put up with a lot while it was getting done Contents Preface Introduction 1.1 Why Experiment? 1.2 Components of an Experiment 1.3 Terms and Concepts 1.4 Outline 1.5 More About Experimental Units 1.6 More About Responses xvii 1 10 13 14 16 17 19 20 25 26 27 28 Completely Randomized Designs 3.1 Structure of a CRD 3.2 Preliminary Exploratory Analysis 3.3 Models and Parameters 31 31 33 34 Randomization and Design 2.1 Randomization Against Confounding 2.2 Randomizing Other Things 2.3 Performing a Randomization 2.4 Randomization for Inference 2.4.1 The paired t-test 2.4.2 Two-sample t-test 2.4.3 Randomization inference and standard inference 2.5 Further Reading and Extensions 2.6 Problems viii CONTENTS 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 Estimating Parameters Comparing Models: The Analysis of Variance Mechanics of ANOVA Why ANOVA Works Back to Model Comparison Side-by-Side Plots Dose-Response Modeling Further Reading and Extensions Problems Looking for Specific Differences—Contrasts 4.1 Contrast Basics 4.2 Inference for Contrasts 4.3 Orthogonal Contrasts 4.4 Polynomial Contrasts 4.5 Further Reading and Extensions 4.6 Problems 39 44 45 52 52 54 55 58 60 65 65 68 71 73 75 75 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 not control combined Type I error rates 98 5.4.10 Confident directions 100 CONTENTS 5.5 5.6 5.7 5.8 5.9 5.10 Comparison with Control or the Best 5.5.1 Comparison with a control 5.5.2 Comparison with the best Reality Check on Coverage Rates A Warning About Conditioning Some Controversy Further Reading and Extensions Problems ix Checking Assumptions 6.1 Assumptions 6.2 Transformations 6.3 Assessing Violations of Assumptions 6.3.1 Assessing nonnormality 6.3.2 Assessing nonconstant variance 6.3.3 Assessing dependence 6.4 Fixing Problems 6.4.1 Accommodating nonnormality 6.4.2 Accommodating nonconstant variance 6.4.3 Accommodating dependence 6.5 Effects of Incorrect Assumptions 6.5.1 Effects of nonnormality 6.5.2 Effects of nonconstant variance 6.5.3 Effects of dependence 6.6 Implications for Design 6.7 Further Reading and Extensions 6.8 Problems Power and Sample Size 7.1 Approaches to Sample Size Selection 7.2 Sample Size for Confidence Intervals 7.3 Power and Sample Size for ANOVA 7.4 Power and Sample Size for a Contrast 7.5 More about Units and Measurement Units 101 101 104 105 106 106 107 108 111 111 113 114 115 118 120 124 124 126 133 134 134 136 138 140 141 143 149 149 151 153 158 158 x CONTENTS 7.6 7.7 7.8 Allocation of Units for Two Special Cases 160 Further Reading and Extensions 161 Problems 162 Factorial Treatment Structure 8.1 Factorial Structure 8.2 Factorial Analysis: Main Effect and Interaction 8.3 Advantages of Factorials 8.4 Visualizing Interaction 8.5 Models with Parameters 8.6 The Analysis of Variance for Balanced Factorials 8.7 General Factorial Models 8.8 Assumptions and Transformations 8.9 Single Replicates 8.10 Pooling Terms into Error 8.11 Hierarchy 8.12 Problems 165 165 167 170 171 175 179 182 185 186 191 192 197 A Closer Look at Factorial Data 9.1 Contrasts for Factorial Data 9.2 Modeling Interaction 9.2.1 Interaction plots 9.2.2 One-cell interaction 9.2.3 Quantitative factors 9.2.4 Tukey one-degree-of-freedom for nonadditivity 9.3 Further Reading and Extensions 9.4 Problems 203 203 209 209 210 212 217 220 222 225 225 226 227 230 233 234 10 Further Topics in Factorials 10.1 Unbalanced Data 10.1.1 Sums of squares in unbalanced data 10.1.2 Building models 10.1.3 Testing hypotheses 10.1.4 Empty cells 10.2 Multiple Comparisons Tables 645 Table D.11: Power curves for random-effects ANOVA, continued Numerator df = 99 95 P o w e r 3 10 30 100 300 1000 Ratio of EMS (times 10 for 01 level) Numerator df = 99 95 P o w e r 3 1000 300 100 30 10 Ratio of EMS (times 10 for 01 level) 646 Tables Table D.11: Power curves for random-effects ANOVA, continued Numerator df = 99 95 P o w e r 3 10 30 100 300 1000 Ratio of EMS (times 10 for 01 level) Numerator df = 99 95 P o w e r 3 1000 300 100 30 10 Ratio of EMS (times 10 for 01 level) 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 A1 B 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, Blocking, 315 complete, 316 confounding, 387 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, 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, 394 Index partial confounding, 400 replication of, 399 three-series factorials, 403–409 two-series factorials, 388–403 two-series plans, 617 Confounding, 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, 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 649 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, 508 650 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, 41 Index 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, 387 651 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, experimental, 6, 37 systematic, 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, 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, 293 652 Index Expected mean squares in the unrestricted model, 294 Expected mean squares unbalanced mixed-effects, 304 Experiment design, Experimental designs, 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, components of, 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, 257 Index 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, 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 15−10 , 487 Fold-over for a 2IV Fractional factorials, 471–499 aliases, 475, 490 653 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, 543 654 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, Levene’s test, 119 Leverage, 115 Lithium in blood, 369 Machine shop, 434 Mallows’ Cp , 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, 568 Index 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, 77–108 655 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, advantages and disadvantages, 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, 49 656 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, 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, 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, 360 Index 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, 526 657 second-order designs, 522 second-order models, 517 Responses, 2, 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, 134 658 Index Spanking, 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, 185 Index 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, control, placebo, 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, costs of, 159 crossed, 435 experimental, 2, 5, 6, 8, 158 independence of, measurement, 6, 8, 158 nested, 435 nested and crossed, 437 size of, 659 split plots, 418 split-split plots, 429 strip plots, 435 whole plots, 417 Validity, 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 ... saying main effects and/ or interactions are significant and to understand the relationships in the data I want them to learn that understanding what the data have to say is the goal ANOVA is a. .. SAS is far and away the market leader in statistical software You can practically every kind of analysis in SAS, but as a novice I spent many hours with the manuals trying to get SAS to any kind... trademark of Minitab, Inc SAS is a registered trademark of SAS Institute, Inc S-Plus is a registered trademark of Mathsoft, Inc Design- Expert is a registered trademark of Stat-Ease, Inc Library of