Ebook Design and analysis of experiments (Volume 2: Advanced experimental design): Part 1

Design and Analysis of Experiments Volume Advanced Experimental Design KLAUS HINKELMANN Virginia Polytechnic Institute and State University Department of Statistics Blacksburg, VA OSCAR KEMPTHORNE Iowa State University Department of Statistics Ames, IA A JOHN WILEY & SONS, INC., PUBLICATION Copyright  2005 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008 Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services please contact our Customer Care Department within the U.S at 877-762-2974, outside the U.S at 317-572-3993 or fax 317-572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print, however, may not be available in electronic format Library of Congress Cataloging-in-Publication Data is available ISBN 0-471-55177-5 Printed in the United States of America 10 Contents Preface xix General Incomplete Block Design 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Introduction and Examples, General Remarks on the Analysis of Incomplete Block Designs, The Intrablock Analysis, 1.3.1 Notation and Model, 1.3.2 Normal and Reduced Normal Equations, 1.3.3 The C Matrix and Estimable Functions, 1.3.4 Solving the Reduced Normal Equations, 1.3.5 Estimable Functions of Treatment Effects, 10 1.3.6 Analyses of Variance, 12 Incomplete Designs with Variable Block Size, 13 Disconnected Incomplete Block Designs, 14 Randomization Analysis, 16 1.6.1 Derived Linear Model, 16 1.6.2 Randomization Analysis of ANOVA Tables, 18 Interblock Information in an Incomplete Block Design, 23 1.7.1 Introduction and Rationale, 23 1.7.2 Interblock Normal Equations, 23 1.7.3 Nonavailability of Interblock Information, 27 Combined Intra- and Interblock Analysis, 27 1.8.1 Combining Intra- and Interblock Information, 27 1.8.2 Linear Model, 27 1.8.3 Normal Equations, 28 1.8.4 Some Special Cases, 31 v vi CONTENTS 1.9 1.10 1.11 1.12 1.13 1.14 Relationships Among Intrablock, Interblock, and Combined Estimation, 31 1.9.1 General Case, 32 1.9.2 Case of Proper, Equireplicate Designs, 34 Estimation of Weights for the Combined Analysis, 36 1.10.1 Yates Procedure, 37 1.10.2 Properties of Combined Estimators, 38 Maximum-Likelihood Type Estimation, 39 1.11.1 Maximum-Likelihood Estimation, 39 1.11.2 Restricted Maximum-Likelihood Estimation, 40 Efficiency Factor of an Incomplete Block Design, 43 1.12.1 Average Variance for Treatment Comparisons for an IBD, 43 1.12.2 Definition of Efficiency Factor, 45 1.12.3 Upper Bound for the Efficiency Factor, 47 Optimal Designs, 48 1.13.1 Information Function, 48 1.13.2 Optimality Criteria, 49 1.13.3 Optimal Symmetric Designs, 50 1.13.4 Optimality and Research, 50 Computational Procedures, 52 1.14.1 Intrablock Analysis Using SAS PROC GLM, 52 1.14.2 Intrablock Analysis Using the Absorb Option in SAS PROC GLM, 58 1.14.3 Combined Intra- and Interblock Analysis Using the Yates Procedure, 61 1.14.4 Combined Intra- and Interblock Analysis Using SAS PROC MIXED, 63 1.14.5 Comparison of Estimation Procedures, 63 1.14.6 Testing of Hypotheses, 66 Balanced Incomplete Block Designs 2.1 2.2 2.3 2.4 2.5 Introduction, 71 Definition of the BIB Design, 71 Properties of BIB Designs, 72 Analysis of BIB Designs, 74 2.4.1 Intrablock Analysis, 74 2.4.2 Combined Analysis, 76 Estimation of ρ, 77 71 vii CONTENTS 2.6 2.7 2.8 Construction of Balanced Incomplete Block Designs 3.1 3.2 3.3 3.4 Significance Tests, 79 Some Special Arrangements, 89 2.7.1 Replication Groups Across Blocks, 89 2.7.2 Grouped Blocks, 91 2.7.3 α-Resolvable BIB Designs with Replication Groups Across Blocks, 96 Resistant and Susceptible BIB Designs, 98 2.8.1 Variance-Balanced Designs, 98 2.8.2 Definition of Resistant Designs, 99 2.8.3 Characterization of Resistant Designs, 100 2.8.4 Robustness and Connectedness, 103 Introduction, 104 Difference Methods, 104 3.2.1 Cyclic Development of Difference Sets, 104 3.2.2 Method of Symmetrically Repeated Differences, 107 3.2.3 Formulation in Terms of Galois Field Theory, 112 Other Methods, 113 3.3.1 Irreducible BIB Designs, 113 3.3.2 Complement of BIB Designs, 113 3.3.3 Residual BIB Designs, 114 3.3.4 Orthogonal Series, 114 Listing of Existing BIB Designs, 115 Partially Balanced Incomplete Block Designs 4.1 4.2 4.3 4.4 104 Introduction, 119 Preliminaries, 119 4.2.1 Association Scheme, 120 4.2.2 Association Matrices, 120 4.2.3 Solving the RNE, 121 4.2.4 Parameters of the Second Kind, 122 Definition and Properties of PBIB Designs, 123 4.3.1 Definition of PBIB Designs, 123 4.3.2 Relationships Among Parameters of a PBIB Design, 125 Association Schemes and Linear Associative Algebras, 127 4.4.1 Linear Associative Algebra of Association Matrices, 127 119 viii CONTENTS 4.5 4.6 4.7 4.4.2 Linear Associative Algebra of P Matrices, 128 4.4.3 Applications of the Algebras, 129 Analysis of PBIB Designs, 131 4.5.1 Intrablock Analysis, 131 4.5.2 Combined Analysis, 134 Classification of PBIB Designs, 137 4.6.1 Group-Divisible (GD) PBIB(2) Designs, 137 4.6.2 Triangular PBIB(2) Designs, 139 4.6.3 Latin Square Type PBIB(2) Designs, 140 4.6.4 Cyclic PBIB(2) Designs, 141 4.6.5 Rectangular PBIB(3) Designs, 142 4.6.6 Generalized Group-Divisible (GGD) PBIB(3) Designs, 143 4.6.7 Generalized Triangular PBIB(3) Designs, 144 4.6.8 Cubic PBIB(3) Designs, 146 4.6.9 Extended Group-Divisible (EGD) PBIB Designs, 147 4.6.10 Hypercubic PBIB Designs, 149 4.6.11 Right-Angular PBIB(4) Designs, 151 4.6.12 Cyclic PBIB Designs, 153 4.6.13 Some Remarks, 154 Estimation of ρ for PBIB(2) Designs, 155 4.7.1 Shah Estimator, 155 4.7.2 Application to PBIB(2) Designs, 156 Construction of Partially Balanced Incomplete Block Designs 5.1 5.2 5.3 5.4 Group-Divisible PBIB(2) Designs, 158 5.1.1 Duals of BIB Designs, 158 5.1.2 Method of Differences, 160 5.1.3 Finite Geometries, 162 5.1.4 Orthogonal Arrays, 164 Construction of Other PBIB(2) Designs, 165 5.2.1 Triangular PBIB(2) Designs, 165 5.2.2 Latin Square PBIB(2) Designs, 166 Cyclic PBIB Designs, 167 5.3.1 Construction of Cyclic Designs, 167 5.3.2 Analysis of Cyclic Designs, 169 Kronecker Product Designs, 172 5.4.1 Definition of Kronecker Product Designs, 172 158 ix CONTENTS 5.5 5.6 5.4.2 Properties of Kronecker Product Designs, 172 5.4.3 Usefulness of Kronecker Product Designs, 177 Extended Group-Divisible PBIB Designs, 178 5.5.1 EGD-PBIB Designs as Kronecker Product Designs, 178 5.5.2 Method of Balanced Arrays, 178 5.5.3 Direct Method, 180 5.5.4 Generalization of the Direct Method, 185 Hypercubic PBIB Designs, 187 More Block Designs and Blocking Structures 6.1 6.2 6.3 6.4 6.5 6.6 Introduction, 189 Alpha Designs, 190 6.2.1 Construction Method, 190 6.2.2 Available Software, 192 6.2.3 Alpha Designs with Unequal Block Sizes, 192 Generalized Cyclic Incomplete Block Designs, 193 Designs Based on the Successive Diagonalizing Method, 194 6.4.1 Designs for t = Kk, 194 6.4.2 Designs with t = n2 , 194 Comparing Treatments with a Control, 195 6.5.1 Supplemented Balance, 196 6.5.2 Efficiencies and Optimality Criteria, 197 6.5.3 Balanced Treatment Incomplete Block Designs, 199 6.5.4 Partially Balanced Treatment Incomplete Block Designs, 205 6.5.5 Optimal Designs, 211 Row–Column Designs, 213 6.6.1 Introduction, 213 6.6.2 Model and Normal Equations, 213 6.6.3 Analysis of Variance, 215 6.6.4 An Example, 216 6.6.5 Regular Row–Column Designs, 230 6.6.6 Doubly Incomplete Row–Column Designs, 230 6.6.7 Properties of Row–Column Designs, 232 6.6.8 Construction, 237 6.6.9 Resolvable Row–Column Designs, 238 189 x CONTENTS Two-Level Factorial Designs 7.1 7.2 7.3 7.4 7.5 7.6 Introduction, 241 Case of Two Factors, 241 7.2.1 Definition of Main Effects and Interaction, 241 7.2.2 Orthogonal Contrasts, 243 7.2.3 Parameterizations of Treatment Responses, 244 7.2.4 Alternative Representation of Treatment Combinations, Main Effects, and Interaction, 247 Case of Three Factors, 248 7.3.1 Definition of Main Effects and Interactions, 249 7.3.2 Parameterization of Treatment Responses, 252 7.3.3 The x -Representation, 252 General Case, 253 7.4.1 Definition of Main Effects and Interactions, 254 7.4.2 Parameterization of Treatment Responses, 256 7.4.3 Generalized Interactions, 258 Interpretation of Effects and Interactions, 260 Analysis of Factorial Experiments, 262 7.6.1 Linear Models, 262 7.6.2 Yates Algorithm, 263 7.6.3 Variances of Estimators, 265 7.6.4 Analysis of Variance, 265 7.6.5 Numerical Examples, 267 7.6.6 Use of Only One Replicate, 278 Confounding in 2n Factorial Designs 8.1 8.2 8.3 8.4 241 Introduction, 279 8.1.1 A Simple Example, 279 8.1.2 Comparison of Information, 280 8.1.3 Basic Analysis, 280 Systems of Confounding, 283 8.2.1 Blocks of Size 2n−1 , 283 8.2.2 Blocks of Size 2n−2 , 283 8.2.3 General Case, 285 Composition of Blocks for a Particular System of Confounding, 289 8.3.1 Intrablock Subgroup, 289 8.3.2 Remaining Blocks, 290 Detecting a System of Confounding, 291 279 xi CONTENTS 8.5 8.6 8.7 8.8 Using SAS for Constructing Systems of Confounding, 293 Analysis of Experiments with Confounding, 293 8.6.1 Estimation of Effects and Interactions, 293 8.6.2 Parameterization of Treatment Responses, 297 8.6.3 ANOVA Tables, 298 Interblock Information in Confounded Experiments, 303 Numerical Example Using SAS, 311 Partial Confounding in 2n Factorial Designs 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 Introduction, 312 Simple Case of Partial Confounding, 312 9.2.1 Basic Plan, 312 9.2.2 Analysis, 313 9.2.3 Use of Intra- and Interblock Information, 315 Partial Confounding as an Incomplete Block Design, 318 9.3.1 Two Models, 318 9.3.2 Normal Equations, 320 9.3.3 Block Contrasts, 322 Efficiency of Partial Confounding, 323 Partial Confounding in a 23 Experiment, 324 9.5.1 Blocks of Size 2, 324 9.5.2 Blocks of Size 4, 325 Partial Confounding in a 24 Experiment, 327 9.6.1 Blocks of Size 2, 327 9.6.2 Blocks of Size 4, 328 9.6.3 Blocks of Size 8, 328 General Case, 329 9.7.1 Intrablock Information, 330 9.7.2 The ANOVAs, 330 9.7.3 Interblock Information, 332 9.7.4 Combined Intra- and Interblock Information, 333 9.7.5 Estimation of Weights, 333 9.7.6 Efficiencies, 334 Double Confounding, 335 Confounding in Squares, 336 9.9.1 23 Factorial in Two × Squares, 337 9.9.2 24 Factorial in × Squares, 338 Numerical Examples Using SAS, 338 9.10.1 23 Factorial in Blocks of Size 2, 338 9.10.2 24 Factorial in Blocks of Size 4, 350 312 xii CONTENTS 10 Designs with Factors at Three Levels 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 Introduction, 359 Definition of Main Effects and Interactions, 359 10.2.1 The 32 Case, 359 10.2.2 General Case, 363 Parameterization in Terms of Main Effects and Interactions, 365 Analysis of 3n Experiments, 366 Confounding in a 3n Factorial, 368 10.5.1 The 33 Experiment in Blocks of Size 3, 369 10.5.2 Using SAS PROC FACTEX, 370 10.5.3 General Case, 374 Useful Systems of Confounding, 374 10.6.1 Two Factors, 376 10.6.2 Three Factors, 376 10.6.3 Treatment Comparisons, 376 10.6.4 Four Factors, 379 10.6.5 Five Factors, 379 10.6.6 Double Confounding, 379 Analysis of Confounded 3n Factorials, 380 10.7.1 Intrablock Information, 381 10.7.2 The ANOVAs, 381 10.7.3 Tests of Hypotheses, 384 10.7.4 Interblock Information, 385 10.7.5 Combined Intra- and Interblock Information, 386 10.7.6 Estimation of Weights, 386 Numerical Example, 387 10.8.1 Intrablock Analysis, 387 10.8.2 Combined Analysis, 387 11 General Symmetrical Factorial Design 11.1 11.2 11.3 11.4 11.5 11.6 359 Introduction, 393 Representation of Effects and Interactions, 395 Generalized Interactions, 396 Systems of Confounding, 398 Intrablock Subgroup, 400 Enumerating Systems of Confounding, 402 393 Tai lieu Luan van Luan an Do an 316 PARTIAL CONFOUNDING IN 2n FACTORIAL DESIGNS know from our previous discussion that II,III ) = var(B I,III ) = var(AB  I,II ) = σe2 var(A 2q and, as follows from Section 8.6, II ) = var(AB  III ) = (σe2 + 2σβ2 ) I ) = var(B var(A q Hence the combined estimate of A, for example, is the weighted average of I , that is, letting w = 1/σe2 and w = 1/(σe2 + 2σ ), II, III and A A β     = 2qwAII, III + qw AI A  2qw + qw (9.2) with similar expressions for B and AB (For a proof of this result we refer to Section 11.6.) Since w and w or ρ = w/w are usually not known, the quantities have to be estimated and then substituted into (9.2) We know that  σe2 = MS(Residual) and we also know that (see Section 8.6) σe2
+ 2σβ2 = MS(Remainder) There is, however, another way of estimating σe2 + 2σβ2 This is accomplished through the B|T-ANOVA as given in Table 9.3 by utilizing all 3q d.f for SS(Xβ ∗ |I, Xρ , X τ ) and not just the 3(q − 1) d.f for SS(Remainder) We comment briefly on how SS(Xβ ∗ |I, Xρ , Xτ ) is obtained in a way other than the usual as indicated in Section 1.3 As mentioned above, SS(Remainder) from the T|B-ANOVA is part of SS(Xβ ∗ |I, Xρ , X τ ) because it is free of treatment effects Since this SS accounts for 3(q − 1) d.f., there remain d.f to be accounted for These are obtained by realizing that for each effect and interaction we have two estimates, namely one from replicates in which the effect is not confounded and one from replicates in II,III and A I , B I,III and which the effect is confounded Specifically, we have A II , and AB  I,II and AB  III Obviously, the comparison of these two types of estiB mates is a function of block effects and error only, and hence the associated sum II,III is obtained of squares belongs to SS(Xβ ∗ |I, Xρ , X τ ) Since, for example, A  from 2q replicates and AI from q replicates and these estimates are uncorrelated, we have   II,III − A II,III − A I ) = [AII,III − AI ] = 2q [A I ]2 SS(A 1/2q + 1/q Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn (9.3) Tai lieu Luan van Luan an Do an 317 SIMPLE CASE OF PARTIAL CONFOUNDING Table 9.3 Structure of B|T-ANOVA for Partially Confounded Design Source d.f Xρ |I 3q − Xτ |I, X ρ Xβ ∗ |I, X ρ , X τ 3q I II,III vs A A II vs B SS From Table 9.1
3q (y ··k
− y ···· )2 k
III  I,II vs A AB Remainder 3(q − 1) 2q  I AII,III − A 2q  II BI,III − B 2q   III AB I,II − AB From Table 9.2a Residual 3(2q − 1) From Table 9.2b Total 12q − From Table 9.1 I, B III E(MS) σe2 + σβ2 σe2 + σβ2 σe2 + σβ2 σe2 + 2σβ2 σe2 Similarly 2q  II ]2 [BI,III − B  I,II − AB  III ) = 2q [AB  III ]2  I,II − AB SS(AB II ) = I,III − B SS(B (9.4) (9.5) II,III − A I , B I,III − B III , and AB  I,II − AB  III are orthogSince the comparisons A onal to each other, the SS (9.3), (9.4), and (9.5) are orthogonal Also, the d.f associated with (9.3), (9.4), and (9.5) are not accounted for in SS(Remainder) since the comparisons leading to SS(Remainder) involve only comparisons among blocks from those replicates in which the respective effects are confounded, as is evident from Table 9.2a Returning now to the estimation of σe2 + 2σβ2 , to be used in (9.2), we evaluate as usual E[MS(Xβ ∗ |I, Xρ , Xτ )] assuming that the βij∗ are i.i.d random variables with mean zero and variance σβ2 We this by obtaining the expected value of each component of SS(Xβ ∗ |I, Xρ , Xτ ) as given in Table 9.3 We already know that E[SS(Remainder)] = 3(q − 1)(σe2 + 2σβ2 ) Now II,III − A I )] = 2q var(A I ) II,III − A E[SS(A = σe2 + σβ2 Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn (9.6) (9.7) Tai lieu Luan van Luan an Do an 318 PARTIAL CONFOUNDING IN 2n FACTORIAL DESIGNS since II,III ) = var(A and I ) = var(A σ 2q e (σ + 2σβ2 ) q e I,III − B II ) and The same result (9.7) is, of course, obtained also for SS(B   SS(AB I,II − AIII ) Hence,     E[(MS(Xβ ∗ |I, X ρ , Xτ )] = σe2 + σβ2 + 3(q − 1)(σe2 + 2σβ2 ) 3q 3q − σβ 3q (9.8) [3q MS(X β ∗ |I, Xρ , Xτ ) − MS(Residual)] 3q − (9.9) = σe2 + An estimator for σe2 + 2σβ2 is then σe2
+ 2σβ2 = We note here that if q is large, MS(Remainder) may actually be a quite satisfactory estimator for σe2 + 2σβ2 9.3 PARTIAL CONFOUNDING AS AN INCOMPLETE BLOCK DESIGN In the previous section we have presented a simple example of partial confounding together with the analysis based entirely on reasoning suggested by the factorial structure of the treatments and the corresponding allocation of the treatments to blocks The design given in Figure 9.1, replicated q times, is, of course, an example of an incomplete block design, in fact a resolvable BIB design in this case as we pointed out earlier As such, data from this design can be analyzed according to the general principles discussed in Chapter 1, or, more specifically, as in Chapter We shall this now and show that the resulting analysis agrees with that given in the previous section 9.3.1 Two Models Without loss of generality and for purpose of ease of notation only, we assume that the arrangement of treatments in blocks is the same for each repetition of the basic design of Figure 9.1 Using the model for an incomplete block design, we have y = µI + Xβ β + Xτ τ + e Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn (9.10) Tai lieu Luan van Luan an Do an 319 PARTIAL CONFOUNDING AS AN INCOMPLETE BLOCK DESIGN with y = (yij k
m ), i = 1, 2, , q indicating the repetition, j = 1, 2, indicating the replication within repetition (as labeled in Fig 9.1), k = 1, denoting the block within replication, and
, m = 0, denoting the levels of factors A and B The observation vector y consists then of q segments (yi1110 , yi1111 , yi1200 , yi1201 , yi2101 , yi2111 , yi2200 , yi2210 , yi3100 , yi3111 , yi3210 , yi3201 ) (i = 1, 2, , q) following Figure 9.1 Further, it can be deduced easily that   (9.11) X β = I 6q ×  0  1   0  0  0 X τ = Iq ×  1  0   1  0  0 0 0 0 0 0 0 1 0 0 0  1  0   0  0  1  0  0   0  1  0 and τ  = (τ00 , τ10 , τ01 , τ11 ) Alternatively, we may write (9.1) as y = µI + X β β + X τ ∗ τ ∗ + e (9.12) where  Xτ ∗ 1  1  1  1  1  = Iq ×  1  1  1  1  1 1 −1 −1 −1 −1 −1 1 −1 −1 −1 1 −1 −1 −1 −1  −1 1  1  −1  −1  1   1  −1  1  1  −1 −1 Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn (9.13) Tai lieu Luan van Luan an Do an 320 PARTIAL CONFOUNDING IN 2n FACTORIAL DESIGNS    and τ ∗ = M, 12 A, 12 B, 12 AB The “factorial incidence matrix” N F = Xτ ∗ X β is then of the form   2 2 2 2 −2 0 0  N F = Iq ×  0 −2 0 0 0 −2 or   1 1 1 1 −1 0 0  (9.14) N F = 2Iq ×  0 −1 0 0 0 −1 9.3.2 Normal Equations We consider now the RNE for τ ∗ , that is, (X τ ∗ Xτ ∗ − N F (X β Xβ )−1 N F )τ ∗ = Xτ ∗ y − N F (Xβ Xβ )−1 Xβ y or   12qI − 12 N F N F τ ∗ = T ∗ − 12 N F B (9.15) (9.16) where   12q M     6q A   ∗ T = ,   6q B    6q AB   B111  B   112    B=      Bq32 are the effect totals and block totals, respectively  0 0  N F N F = 4q  0 0 With  0  0 and  B···     B·11 − B·12   NF B =  B − B  ·22   ·21 B·31 − B·32 Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn (9.17) Tai lieu Luan van Luan an Do an 321 PARTIAL CONFOUNDING AS AN INCOMPLETE BLOCK DESIGN Eq (9.16) reduces to        − (B·11 − B·12 ) II,III 6q A 4q A        − (B·21 − B·22 )  = 4q B I,III    = 6q B 8q · B       1   6q AB − (B − B ) 4q AB 8q · AB ·31 ·32 I,II  8q · 12 A and hence to     II,III A A       B  =  B    I,III    AB AB I,II I , B·21 − B·22 = 2q B II , B·31 − B·32 = 2q AB  III , A = since B·11 − B·12 = 2q A  1        (AI + 2AII,III ), B = (BII + 2BI,III ), AB = (AB III + 2AB I,II ) Turning to the RNE for β, we consider (X β X β − N F (X τ ∗ Xτ ∗ )−1 N F )β = X β y − N F (Xτ ∗ Xτ ∗ )−1 Xτ ∗ y (9.18) or  2I 6q −  1 N F N F β = B − N T ∗ 12q 12q F (9.19) with B and T ∗ as defined in (9.17) With N F N F  0  1 = 4(Iq Iq ) ×  1 1 1 1 1 1 1 1 1 1  1  1  1 0 = 4(Iq Iq ) × H say we see that the coefficient matrix for β, that is,   6qI − Iq Iq × H 3q is of rank 6q − Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn (9.20) Tai lieu Luan van Luan an Do an 322 PARTIAL CONFOUNDING IN 2n FACTORIAL DESIGNS 9.3.3 Block Contrasts From the form of (9.20) we can write out 6q − identifiable functions of β, namely 3q − of the form S1 : {2(βij + βij − βq31 − βq32 ); i = 1, 2, , q; j = 1, 2, 3; (ij ) = (q3)} 3(q − 1) of the form S2 : {2(βij − βij − βqj + βqj ); i = 1, 2, , q − 1; j = 1, 2, 3} and of the form S3 : ! q 4
(βij − βij ); j = 1, 2, 3 i=1 These three sets of identifiable functions are orthogonal to each other in the sense that any function in Sν is orthogonal to any function in Sν  (ν, ν  = 1, 2, 3; ν = ν  ) The corresponding RHSs are obtained by writing the RHS of (9.19) in the form   +A  2M     2M − A    2M     +B B − Iq ×   −B   2M   2M     + AB  − AB  2M We then find the following RHSs for " # S1 : Bij · − Bq3· ; i = 1, , q; j = 1, 2, 3; (ij ) = (3q) " # S2 : Bij − Bij − Bqj + Bqj ; i = 1, , q − 1; j = 1, 2, " #  B·21 − B·22 − 2q B,  B·31 − B·32 − 2q AB  S3 : B·11 − B·12 − 2q A, Within the context of the factorial calculus we can rewrite these for S2 and S3 as " i − A q ) (for j = 1) S2 : 2(A i − B q ) (for j = 2) 2(B #  i − AB  q ) (for j = 3); i = 1, , q − 2(AB $ S3 : % 4q  II,III ), 4q (B I,III ), 4q (AB  I,II ) II − B  III − AB (AI − A 3 iI , Bi21 − Bi22 = 2B iII , Bi21 − Bi32 = 2AB  iIII since Bi11 − Bi12 = 2A Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn Tai lieu Luan van Luan an Do an 323 EFFICIENCY OF PARTIAL CONFOUNDING We recognize that the comparisons S1 represent comparisons among replicates and the associated sum of squares is SS(Replicate) in Table 9.1 The comparisons of S2 and S3 are comparisons of block effects within replicates and hence belong to (Xβ ∗ |I, X ρ , Xτ ) The sum of squares associated with S2 is SS(Remainder) of Table 9.2, each set for j = 1, 2, leading to a sum of squares with q − d.f Finally, the three functions of S3 give rise to the remaining three single d.f sums of squares belonging to (X β ∗ |I, Xρ , Xτ ) as given in Table 9.3 We have thus shown that a system of partial confounding can be analyzed using the general principles of incomplete block designs Although we have only considered the intrablock analysis of a particular case, this line of argument can, of course, be carried over to the combined analysis and to the general case It is hoped, however, that the reader realizes that the analysis can be described and performed much easier via the concepts of factorial experiments as the factorial structure of the treatments determines the structure of the incomplete block design 9.4 EFFICIENCY OF PARTIAL CONFOUNDING In general, reduction of block size will lead to a reduction of experimental error variance On the other hand, this forces the experimenter to use a system of partial confounding, which may offset this gain Therefore, if the experimenter has a choice of using blocks either of size or of size 2, he or she needs a criterion on which to base the choice For the simple case discussed so far, this is obtained by comparing the information provided by both designs (using the same number of replicates) as given in Table 9.4 The information of the partially confounded scheme relative to that of the scheme with no confounding is 2q/σ22 3q/σ42 = σ42 σ22 (9.21) where the subscript on σ denotes the number of units per block If σ42 is greater than 3/2σ22 (or σ22 less than 2/3σ42 ), the information is greater with the partially confounded design Equivalently, we say then that the efficiency of the partially confounded design relative to the unconfounded design, as given by (9.21), is Table 9.4 Information on Effects with Equal Confounding No Confounding A B AB        3q σ42 Partial Confounding of Section 9.2 2q σ22 Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn Tai lieu Luan van Luan an Do an 324 PARTIAL CONFOUNDING IN 2n FACTORIAL DESIGNS Table 9.5 Information on Effects with Unequal Confounding A B AB No Confounding  %    4q  σ   Partial Confounding 3q σ22 2q σ22 larger than In general, σ42 will be greater than σ22 , but whether it will be sufficiently greater to give the advantage to the partially confounded design depends on the experimental material In many cases this may prove to be a difficult question We have considered a scheme of partial confounding that results in equal information on main effects and the 2-factor interaction In some cases it may be more appropriate to obtain greater information on main effects This would entail greater representation of replicates of type III, to an extent depending on the relative amounts of information required Suppose that a basic repetition consists of one replicate each of types I and II and replicates of type III and that there are q such repetitions, that is, 4q replicates all together The amounts of information to be compared then are as given in Table 9.5 Since the emphasis is on main effects, it seems reasonable to use the partial confounding scheme if σ22 < 34 σ42 The design utilizing partial confounding that we have just mentioned is obviously only one of many choices The number of choices becomes even larger as the number of factors increases Some such cases will be discussed in the following sections 9.5 PARTIAL CONFOUNDING IN A 23 EXPERIMENT As we have mentioned earlier, the arrangements utilizing partial confounding that are best for any given situation depend on the information the experimenter wishes to obtain Suppose, for example, with an experiment of three factors A, B, C, each at two levels, the experimenter desires maximum possible accuracy on main effects and equal information on the 2-factor and 3-factor interactions 9.5.1 Blocks of Size A suitable system of partial confounding will consist of a number of repetitions, q say, of the following types of replicates: Type I: Confound AB, AC, BC Type II: Same as type I Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn Tai lieu Luan van Luan an Do an 325 PARTIAL CONFOUNDING IN A 23 EXPERIMENT Type III: Confound A, BC, ABC Type IV: Confound B, AC, ABC Type V: Confound C, AB, ABC This design requires 5q replicates, that is, 20q blocks, which may not always be feasible The information that this design yields as compared to the unconfounded design is given in Table 9.6 For an example using SAS PROC FACTEX see Table 9.14, Section 9.10.1 9.5.2 Blocks of Size In this case the experimenter will use a number of repetitions of the following basic pattern (with each column representing a block): I (1) ab c abc II (1) ac b abc a b ac bc b c ab bc III (1) bc a abc b c ab ac IV (1) ab ac bc a b c abc confounding AB, AC, BC, ABC in replicates of types I, II, III, IV, respectively Suppose we use q repetitions, the positions of replicates, and blocks within replicates and treatment combinations within blocks being randomized The information from this design and the corresponding unconfounded design is given in Table 9.7 It follows then that the partially confounded design will yield more information on interactions than the unconfounded design and substantially more information on main effects if σ42 is less than 34 σ82 Table 9.6 Information Given by Design for 23 System in Blocks of Size No Confounding Information A B AB C AC BC ABC Estimate From               10q  σ82             All replicates Partial Confounding of Section 9.5.1 Information Estimate from Replicate of Types 8q/σ22 I, II, IV, V 8q/σ22 4q/σ22 8q/σ22 4q/σ22 4q/σ22 4q/σ22 I, II, III, V Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn III, IV I, II, III, IV III, V IV, V I, II Tai lieu Luan van Luan an Do an 326 PARTIAL CONFOUNDING IN 2n FACTORIAL DESIGNS Table 9.7 Information Given by Design for 23 System in Blocks of No Confounding Information Partial Confounding of Section 9.5.2 Estimate From Estimate from Replicate of Types Information A 8q/σ42 All B 8q/σ42 All AB 6q/σ42 II, III, V 8q/σ42 All 6q/σ42 6q/σ42 6q/σ42 I, III, V C 8q σ82 All replicates AC BC ABC Table 9.8 b I, II, III Structure of T|B-ANOVA Source of Variation a I, II, V d.f Replicates Blocks/replicates A B AB C AC BC ABC Residual 4q − 4q 1 1 1 24q − Total 32q − Blocks/replicates AB AC BC ABC AB × repsI AC × repsII BC × repsIII ABC × repsIV 4q 1 1 q −1 q −1 q −1 q −1 Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn Tai lieu Luan van Luan an Do an 327 PARTIAL CONFOUNDING IN A 24 EXPERIMENT Table 9.8 (Continued ) Source of Variation c d.f Residual A × reps (all) B × reps (all) AB × reps (II, III, IV) C × reps (all) AC × reps (I, III, IV) BC × reps (I, II, IV) ABC × reps (I, II, III) 24q − 4q − 4q − 3q − 4q − 3q − 3q − 3q − The partition of the degrees of freedom in the analysis of variance is given in Table 9.8a with a breakdown of the degrees of freedom for blocks in Table 9.8b and residual in Table 9.8c The computation of the various sums of squares follows the general procedure as indicated in Tables 9.1 and 9.2 using the observations from the respective replicates according to Tables 9.7 and 9.8 9.6 PARTIAL CONFOUNDING IN A 24 EXPERIMENT 9.6.1 Blocks of Size In this situation each replicate will consist of eight blocks With s types of replicates, each utilizing a different system of confounding, there will be 8s − d.f for blocks, 15 d.f for treatments if no effect is completely confounded, and therefore 8s − 15 d.f for residual In order to have an error based on a reasonable number of d.f., it seems that we need at least s = replicates It follows from Table 8.3 that under these circumstances the best system of confounding appears to be as follows: Replicate I II III IV Effects and Interactions Confounded A, B, C, D, BC, AC, AB, AB, ABC, ABC, ABC, ABD, BD, AD, AD, AC, ABD, ABD, ACD, ACD, CD, CD, BD, BC, ACD BCD BCD BCD These four replicates give 34 relative information on main effects (in the sense that all main effects are estimated from three out of four replicates), 12 relative information on 2-factor interactions, 14 relative information on 3-factor interactions, and full information on the 4-factor interaction Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn