Applied Optimal Designs Edited by Martijn P F Berger Department of Methodology and Statistics, University of Maastricht, The Netherlands Weng Kee Wong Department of Biostatistics, UCLA, Los Angeles, USA Copyright # 2005 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved 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 under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons 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Cataloging-in-Publication Data Applied optimal designs/edited by Martijn P F Berger, Weng Kee Wong p cm Includes bibliographical references and index ISBN 0-470-85697-1 (alk paper) Optimal designs (Statistics) Experimental design I Berger, Martijn P F II Wong, Weng Kee QA279.A67 2005 519.50 7–dc22 2004058017 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-85697-1 Typeset in 10/12pt Times by Thomson Press (India) Limited, New Delhi Printed and bound in Great Britain by TJ International Ltd., Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production Contents List of Contributors Editors’ Foreword Optimal Design in Educational Testing xi xv Steven Buyske 1.1 Introduction 1.1.1 Paper-and-pencil or computerized adaptive testing 1.1.2 Dichotomous response 1.1.3 Polytomous response 1.1.4 Information functions 1.1.5 Design problems 1.2 Test Design 1.2.1 Fixed-form test design 1.2.2 Test design for CAT 1.3 Sampling Design 1.3.1 Paper-and-pencil calibration 1.3.2 CAT calibration 1.4 Future Directions Acknowledgements References Optimal On-line Calibration of Testlets 2 7 11 12 12 14 15 16 16 21 Douglas H Jones and Mikhail S Nediak 2.1 2.2 2.3 Introduction Background 2.2.1 Item response functions 2.2.2 D-optimal design criterion Solution for Optimal Designs 2.3.1 Mathematical programming model 2.3.2 Unconstrained conjugate-gradient method 2.3.3 Constrained conjugate-gradient method 21 23 23 24 25 25 27 28 vi CONTENTS 2.3.4 Gradient of log det MðB; H; xÞ 2.3.5 MCMC sequential estimation of item parameters 2.3.6 Note on performance measures 2.4 Simulation Results 2.5 Discussion Appendix A Derivation of the Gradient of log det MðB; H; xÞ Appendix B Projection on the Null Space of the Constraint Matrix Acknowledgements References 28 29 30 31 35 38 39 41 41 On the Empirical Relevance of Optimal Designs for the Measurement of Preferences 45 Heiko Großmann, Heinz Holling, Michaela Brocke, Ulrike Graßhoff and Rainer Schwabe 3.1 3.2 3.3 3.4 3.5 Introduction Conjoint Analysis Paired Comparison Models in Conjoint Analysis Design Issues Experiments 3.5.1 Experiment 3.5.2 Experiment 3.6 Discussion Acknowledgements References Designing Optimal Two-stage Epidemiological Studies 45 48 49 53 54 55 58 61 63 63 67 Marie Reilly and Agus Salim 4.1 4.2 4.3 4.4 4.5 Introduction Illustrative Examples 4.2.1 Example 4.2.2 Example 4.2.3 Example Meanscore 4.3.1 Example of meanscore Optimal Design and Meanscore 4.4.1 Optimal design derivation for fixed second stage sample size 4.4.2 Optimal design derivation for fixed budget 4.4.3 Optimal design derivation for fixed precision 4.4.4 Computational issues Deriving Optimal Designs in Practice 4.5.1 Data needed to compute optimal designs 4.5.2 Examples of optimal design 4.5.3 The optimal sampling package 4.5.4 Sensitivity of design to sampling variation in pilot data 67 69 69 70 71 72 76 77 77 78 79 80 81 81 82 85 85 CONTENTS 4.6 4.7 Summary Appendix Brief Description of Software Used 4.7.1 R language 4.7.2 S-PLUS 4.7.3 STATA 4.8 Appendix The Optimal Sampling Package 4.8.1 Illustrative data sets 4.9 Appendix Using the Optimal Package in R 4.9.1 Syntax and features of optimal sampling command ‘budget’ in R 4.9.2 Example 4.10 Appendix Using the Optimal Package in S-Plus 4.11 Appendix Using the Optimal Package in STATA 4.11.1 Syntax and features of ‘optbud’ function in STATA 4.11.2 Analysis with categorical variables 4.11.3 Illustrative example References Response-Driven Designs in Drug Development vii 88 89 89 90 90 90 92 92 93 94 97 97 98 99 99 101 103 Valerii V Fedorov and Sergei L Leonov 5.1 5.2 Introduction Motivating Example: Quantal Models for Dose Response 5.2.1 Optimality criteria 5.3 Continuous Models 5.3.1 Example 3.1 5.3.2 Example 3.2 5.4 Variance Depending on Unknown Parameters and Multi-response Models 5.4.1 Example 4.1 5.4.2 Optimal designs as a reference point 5.4.3 Remark 4.1 5.5 Optimal Designs with Cost Constraints 5.5.1 Example 5.1 5.5.2 Example 5.2 Pharmacokinetic model, serial sampling 5.5.3 Remark 5.1 5.6 Adaptive Designs 5.6.1 Example 6.1 5.7 Discussion Acknowledgements References Design of Experiments for Microbiological Models 103 104 105 108 108 109 110 114 116 117 117 120 121 124 127 129 131 133 133 137 Holger Dette, Viatcheslav B Melas and Nikolay Strigul 6.1 6.2 Introduction Experimental Design for Nonlinear Models 6.2.1 Example 2.1 The exponential regression model 137 138 140 viii CONTENTS 6.2.2 Example 2.2 Three-parameter logistic distribution 6.2.3 Example 2.3 The Monod differential equation 6.2.4 Example 2.4 6.3 Applications of Optimal Experimental Design in Microbiology 6.3.1 The Monod model 6.3.2 Application of optimal experimental design in microbiological models 6.4 Bayesian Methods for Regression Models 6.5 Conclusions Acknowledgements References Selected Issues in the Design of Studies of Interrater Agreement 140 141 143 148 149 160 170 173 174 175 181 Allan Donner and Mekibib Altaye 7.1 7.2 Introduction The Choice between a Continuous or Dichotomous Variable 7.2.1 Continuous outcome variable 7.2.2 Dichotomous Outcome Variable 7.3 The Choice between a Polychotomous or Dichotomous Outcome Variable 7.4 Incorporation of Cost Considerations 7.5 Final Comments Appendix Acknowledgement References 181 182 183 184 189 191 193 194 194 195 Restricted Optimal Design in the Measurement of Cerebral Blood Flow Using the Kety–Schmidt Technique 197 J.N.S Matthews and P.W James 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Introduction The Kety–Schmidt Method The Statistical Model and Optimality Criteria Locally Optimal Designs 8.4.1 DS -optimal designs ^Þ 8.4.2 Designs minimising varðD Bayesian Designs and Prior Distributions 8.5.1 Bayesian criteria 8.5.2 Prior distribution Optimal Bayesian Designs 8.6.1 Numerical methods 8.6.2 DS -optimal designs ^Þ 8.6.3 Optimal designs for varðD Practical Designs 8.7.1 Reservations about the optimal designs 197 198 199 202 202 203 205 205 206 208 208 209 210 211 211 CONTENTS 8.7.2 Discrete designs 8.8 Concluding Remarks References ix 212 216 218 Optimal Experimental Design for Parameter Estimation and Contaminant Plume Characterization in Groundwater Modelling 219 James McPhee and William W-G Yeh 9.1 9.2 Introduction Groundwater Flow and Mass Transport in Porous Media: Modelling Issues 9.2.1 Governing equations 9.2.2 Parameter estimation 9.3 Problem Formulation 9.3.1 Experimental design for parameter estimation 9.3.2 Monitoring network design for plume characterization 9.4 Solution Algorithms 9.5 Case Studies 9.5.1 Experimental design for parameter estimation 9.5.2 Experimental design for contaminant plume detection 9.6 Summary and Conclusions Acknowledgements References 10 The Optimal Design of Blocked Experiments in Industry 219 220 220 222 224 224 226 230 231 231 238 241 243 243 247 Peter Goos, Lieven Tack and Martina Vandebroek 10.1 10.2 10.3 10.4 Introduction The Pastry Dough Mixing Experiment The Problem Fixed Block Effects Model 10.4.1 Model and estimation 10.4.2 The use of standard designs 10.4.3 Optimal design 10.4.4 Some theoretical results 10.4.5 Computational results 10.5 Random Block Effects Model 10.5.1 Model and estimation 10.5.2 Theoretical results 10.5.3 Computational results 10.6 The Pastry Dough Mixing Experiment Revisited 10.7 Time Trends and Cost Considerations 10.7.1 Time trend effects 10.7.2 Cost considerations 247 248 249 251 251 252 254 254 256 257 257 258 262 262 265 265 266 x CONTENTS 10.7.3 The trade-off between trend resistance and cost-efficiency 10.8 Optimal Run Orders for Blocked Experiments 10.8.1 Model and estimation 10.8.2 Computational results 10.9 A Time Trend in the Pastry Dough Mixing Experiment 10.10 Summary Acknowledgement Appendix: Design Construction Algorithms References Index 267 269 269 271 273 275 275 275 277 281 List of Contributors Mekibib Altaye Center for Epidemiology and Biostatistics Cincinnati Children’s Hospital and The University of Cincinnati College of Medicine Cincinnati, Ohio USA Allan Donner Department of Epidemiology and Biostatistics Faculty of Medicine and Dentistry University of Western Ontario and Robarts Clinical Trials Robarts Research Institute London Ontario Canada Michaela Brocke Westfaălische Wilhelms-Universitaăt Muănster Psychologisches Institut IV Fliednerstr 21 D-48149 Muănster Germany Valerii Fedorov GlaxoSmithKline 1250 So Collegeville Road PO Box 5089, UP 4315 Collegeville PA 19426-0989 USA Steven Buyske Rutgers University Department of Statistics 110 Frelinghuysen Rd Pitscataway NJ 08854-8019 USA Peter Goos Department of Mathematics, Statistics & Actuarial Sciences Faculty of Applied Economics University of Antwerp Prinsstraat 13 2000 Antwerpen Belgium Holger Dette Ruhr-Universitaăt Bochum Fakultaăt und Institut fuăr Mathematik 44780 Bochum Germany Ulrike Graòhoff Otto-von-Guericke-Universitaăt Magdeburg Insitut fuăr Mathematische Stochastik Postfach 4120 D-39016 Magdeburg Germany VARIANCE DEPENDING ON UNKNOWN PARAMETERS 1600 ψ ψ1 (standard) ψ2 (new term) 12 10 ψ(x,ξ ,θ) 1200 η(x,θ) 1400 1000 800 2 115 log(x ) 6 log(x ) 1000 10 600 * ψ(x,ξ ,θ) d(x,ξ,θ) 12 800 400 ξ2 (uniform) * ξ (optimal) 200 0 2 log(x ) 0 log(x ) Figure 5.2 Plots for Example 4.1 Upper left panel: response fuction ðx; Þ Upper right panel: normalized variance ðx; ; Þ for twofold serial dilution design 2 Lower right: normalized variance ðx; ; Þ for optimal design à Lower left: unnormalized variance d1 ðx; ; Þ (triangles-optimal design, circles-serial dilution) Function ðx; ; Þ is defined in (5.20), function d1 ðx; ; Þ is defined in (5.21) Dashed line in the right panels is ẳ d1 x; ; ị=Sx; ị, see (5.20) Dotted line in the right panels is ẳ d2 x; ; ị=ẵ2S2 x; ị prediction d1 x; ; Þ defined in (5.21), is given on the lower left panel Note that the weights of the support points are not equal In our example p ¼ f0:28; 0:22; 0:22; 0:28g It is worthy to note that the optimal design in our example is supported at just four points, which is less than the number of estimated parameters m ¼ This happens because the information matrix ðx; Þ of a single observation at point x is defined as the sum of the two terms; see (5.11) So even in the case of a single response, k ¼ 1, the rank of the information matrix ðx; Þ may be greater than one On the other hand, in the single-response regression models where the variance function does not depend on unknown parameters, the second term on the righthand side of (5.11) is missing Therefore, in that case for the information matrix of any design ~ to be non-degenerate, it is necessary to have at least m support points in the design 116 5.4.2 RESPONSE-DRIVEN DESIGNS IN DRUG DEVELOPMENT Optimal designs as a reference point It is worthwhile to remark that the direct use of locally optimal designs is not always realistic As mentioned in Section 5.2, if the logistic model (5.4) is used in the dose response experiment, then optimal design theory suggests putting half of the patients on each of the two doses, ED17:6 and ED82:4 Note that dose ED17:6 may be too low and not efficacious while dose ED82:4 may be quite toxic and not well tolerated by many patients Recall also that, though the optimal designs are unique in z-space, one needs preliminary estimates of parameters  to find optimal doses in the original x-space Hence, appealing to D-optimality of doses in a real-life study is very likely not to be a strong argument Nevertheless, the importance of locally optimal designs is in that they provide a reference point for the sensitivity analysis and allow us to calculate relative efficiency of alternative designs For instance, in the study described in Example 4.1, the scientists were traditionally using the twofold serial dilution design 2 , as shown in Figure 5.2 In practice, serial dilution designs can be easily implemented with an arbitrary dilution factor So the main practical goal was not in constructing optimal designs per se, but to answer the following questions:  How far is the twofold dilution design from the optimal one?  Are there ‘good’ serial dilution designs a with factors a other than 2? Since there existed a special interest in the estimation of parameter 1 ¼ ED50, we compared relative efficiency of various serial dilution designs with respect to D-optimality and ED50 -optimality: jMða ; Þj Eff D ða Þ ¼ jMðà ; Þj !1=m ; Eff ED50 ða Þ ẳ ẵM  ; ị11 ; ẵM a ; ị11 where by ẵA11 we denote the first diagonal element of matrix A Serial dilution designs are invariant with respect to model parameters, so they serve as a robust alternative to optimal designs If it could be shown that the efficiency of serial dilutions was close to one, it would provide a solid support for the use of such designs in practice We fitted the data for over 20 different assays with twofold serial dilution design 2 (concentrations from 500 to 0.98 ng/ml), constructed locally optimal designs, and calculated the relative efficiency of dilution designs The results could be split into two groups: (G1) In most cases design 2 covered the whole range of the logistic curve, and the loss of efficiency was within 10–15% This means that 10–15% more observations are required to get the same efficiency as in the optimal design OPTIMAL DESIGNS WITH COST CONSTRAINTS 117 (G2) However, in some examples either the left or right end of the curve was not properly covered (as the right end in Example 4.1, see Figure 5.2, top left), and the efficiency of designs 2 was less than 0.6, both with respect to D-criterion and variance of ^1 Therefore we suggested trying designs 2:5 or 3, to start with higher concentrations, and to have 10 different concentrations on the same 96-well plate as before, to preserve the logistics of the assays For example, one of the options with design 2:5 is to start from 2000 ng/ml, and go down to 800; 320; ; 0:52 ng=ml The comparison of designs 2:5 with the original twofold designs showed that for the cases in group (G1), designs 2:5 performed similarly to designs 2 , with essentially the same efficiency On the other hand, for cases in group (G2) designs 2:5 were significantly superior, with ratio Effð2:5 Þ=Effð2 Þ often greater than 1.5 for both optimality criteria Moreover, designs 2:5 were relatively close to the optimal ones, with the loss of efficiency not more than 15% Thus, it was suggested switching to 2.5-fold serial dilution designs for future studies For more discussion on robustness issues, see Sitter (1992) 5.4.3 Remark 4.1 In the study discussed above the estimated values of slope parameter 2 were always within the interval [1.2–1.5], so the logistic curve was not very steep This was one of the reasons for the efficiency of 2.5- or 3-fold designs When parameter 2 is significantly bigger than (say and above), then the quality of designs with large dilution factors deteriorates, especially with respect to the estimation of 1 It happens because when the logistic curve is rather steep, then designs a with larger a will more likely miss points in the middle of the curve, near values x ¼ ED50 , which in turn will greatly reduce the precision of the parameter estimation 5.5 Optimal Designs with Cost Constraints Traditionally when normalized designs are discussed, the normalization factor is equal to the number of experiments N; see Section 5.2 Now let each measurement at point xi be associated with a cost cðxi Þ, and there exists a restriction on the total cost, n X iẳ1 ni cxi ị C: 5:23ị 118 RESPONSE-DRIVEN DESIGNS IN DRUG DEVELOPMENT In this case it is quite natural to normalize the information matrix by the total cost C and introduce MC ; ị ẳ MN ị X ẳ wi ~xi ; ị; C i with wi ẳ ni cxi Þ ðx; Þ ; ~ðx; Þ ¼ : C cðxÞ ð5:24Þ Note that the considered case should not be confused with the case when additionally to (5.24) one also imposes that Ỉi ni N The latter design problem is more complicated and must be addressed as discussed in Cook and Fedorov (1995) Once cost function cðxÞ is defined, one can introduce a cost-based design C ¼ fwi ; xi g and use the well-elaborated techniques of constructing continuous designs for various criteria of optimality  ẳ arg ẫẵMC1 ; ị;  where  ẳ fwi ; xi g: As usual, to obtain frequencies ni , values ~ ni ¼ wi C=cðxi Þ have to be rounded to the nearest integers ni subject to ặi ni cxi ị C We think that the introduction of cost constraints and the alternative normalization (5.24) makes the construction of optimal designs for models with multiple responses more sound It allows for a meaningful comparison of ‘points’ x with distinct number of responses To illustrate the potential of this approach, we start this section with an example of the two-dimensional response function ðx; Þ ẳ ẵ1 x; ị; 2 x; ịT , with the variance matrix   S11 ðx; Þ S12 ðx; Þ : Sx; ị ẳ S12 x; ị S22 x; ị Let a single measurement of function i ðx; Þ cost ci xị, i ẳ 1; Additionally, we impose a cost cv ðxÞ on any single or pair of measurements The rationale behind this model comes from considering a hypothetical visit of a patient to the clinic to participate in a clinical trial It is assumed that each visit costs cv ðxÞ, where x denotes a patient (or more appropriately, some patient’s characteristics) There are three options for each patient: Take test t1 which by itself costs c1 ðxÞ; the total cost of this option is C1 xị ẳ cv xị ỵ c1 xị Take test t2 which costs c2 xị; the total cost is C2 xị ẳ cv xị ỵ c2 xị Take both tests t1 and t2 ; in this case the cost is C3 xị ẳ cv xị ỵ c1 xịỵ c2 xị OPTIMAL DESIGNS WITH COST CONSTRAINTS 119 Another interpretation could be to measure pharmacokinetic profile (blood concentration) at one or two time points To consider this example within the traditional framework, introduce binary variables x1 and x2 , xi ¼ f0 or 1g, i ¼ 1; Let X ¼ ðx; x1 ; x2 Þ where x belongs to a ‘traditional’ design region X, and pair ðx1 ; x2 Þ belongs to X12 ẳ fx1 ; x2 ị : xi ẳ or 1; maxx1 ; x2 ị ẳ 1g: Define X; ị ẳ Ix1 ;x2 x; ị; SX; ị ẳ Ix1 ;x2 Sðx; Þ Ix1 ;x2 ; ð5:25Þ where  Ix1 ;x2 ¼  : x2 x1 ~, Now introduce the ‘extended’ design region X ~ ¼ X X12 ẳ Z1 X [ Z2 [ 5:26ị Z3 ; where Z1 ẳ fx; x1 ; x2 ị : x X; x1 ¼ 1; x2 ¼ 0g; Z2 ¼ fðx; x1 ; x2 Þ : x X; x1 ¼ 0; x2 ¼ 1g; Z3 ¼ fðx; x1 ; x2 ị : x X; x1 ẳ x2 ¼ 1g; and Z1 \ Z2 \ Z3 ¼ ;: The normalized information matrix MC ð; Þ and design  are defined as MC ; ị ẳ n X iẳ1 wi ~Xi ; ị; n X wi ẳ 1;  ¼ fXi ; wi g; i¼1 where ~ðX; Þ ¼ X; ị=Ci xị if X Zi , i ẳ 1; 2; 3; with ðX; Þ defined in (5.11) and ðX; Þ, SðX; Þ introduced in (5.25) Note that the generalization to k > is straightforward; one has to introduce k binary variables xi and matrix Ix1 ; ;xk ¼ diagðxi Þ The number of subregions Zi in this case is equal to 2k À The presentation (5.26) is used in the example below to demonstrate the performance of various designs with respect to the sensitivity function ðX; ; Þ, X Zi 120 5.5.1 RESPONSE-DRIVEN DESIGNS IN DRUG DEVELOPMENT Example 5.1 We take 1 ðx; Þ ẳ ỵ x ỵ x2 ỵ x3 ẳ F1T xị ; 2 x; ị ẳ ỵ x ỵ xỵ ẳ F2T xị ; where F1 xị ẳ 1; x; x2 ; x3 ; 0; 0ịT ; F2 xị ẳ 1; 0; 0; 0; x; xỵ ịT ; x  ẳ ẵ1; 1, and xỵ ¼ fx if x ! 0, and otherwiseg Cost functions are selected as constants cv ; c1 ; c2 ½0; 1Š and not depend on x Similarly, the variance matrix Sðx; Þ is constant,   S11 S12 : Sx; ị ẳ S12 S22 In our computations, we take S11 ¼ S22 ¼ 1, S12 ¼ ,  1, thus changing the value of S12 only Note that  ¼ ð 1 ; ; ; ; S11 ; S12 ; S22 ÞT The functions 1 , 2 are linear with respect to unknown parameters Thus optimal designs not depend on their values On the contrary, in this example optimal designs depend on the values of the variance parameters Sij , i.e we construct locally optimal designs with respect to their values We considered a rather simple example to illustrate the approach Nevertheless, it allows us to demonstrate how the change in cost functions and variance parameter  affects the selection of design points For the first run, we choose cv ¼ 1, c1 ¼ c2 ¼ 0,  ¼ 0; see Figure 5.3 which shows the sensitivity function X; ; ị ẳ trẵ~ X; ịMC1  ; ị for X Zj, j ¼ 1; 2; Not surprisingly, in this case the selected design points are in subregion Z3 Indeed, since individual measurements cost nothing, it is beneficial to take two measurements instead of a single one to gain more information and to decrease the variability of parameter estimates The weights of the support points are shown in the plot which illustrates the generalized equivalence theorem: the sensitivity function hits the reference line m ¼ at the support points of D-optimal design; recall that dimðÞ ¼ The next case deals with positive correlation  ¼ 0:3 and cv ¼ 1, c1 ¼ c2 ¼ 0:5; see Figure 5.4 Now there are just four support points in the design: two of them are at the boundaries of subregion Z3 with weights w1 ¼ w2 ¼ 0:33, and the other two are in the middle of subregion Z1 , x3;4 ẳ ặ0:45 with weights w3 ẳ w4 ¼ 0:17 Unlike the traditional normalization by the total sample size, in case of cost constraints the randomization ratio n1 : n2 : is not defined simply by the ratio of weights w1 : w2 : Recall that with cost constraints values ni are given by ni ẳ wi C=cXi ị Therefore, in the last example n1 w1 cðX3 Þ 0:33  1:5 % 1:45; ẳ ẳ 0:17 n3 w3 cX1 ị OPTIMAL DESIGNS WITH COST CONSTRAINTS 121 1st function only: cost = 1.0 + 0.0 ψ(X), X ∈ Z1 -1 2nd function only: cost = 1.0 + 0.0 ψ(X), X ∈ Z2 -1 Both functions: cost = 1.0 + 0.0 + 0.0 ψ(X), X ∈ Z3 9• 0.321 • 0.162 • • 0.043 0.164 0.5 0.325 • -1 –0.5 x Figure 5.3 Sensitivity function ðX; ; Þ for three subregions Z1 (top), Z2 (middle) and Z3 (bottom), and D-optimal design for cv ¼ 1, c1 ¼ c2 ¼ 0;  ¼ for Example 5.1 All optimal points, marked by dots, are in subregion Z3 The value of sensitivity function is equal to m ¼ at the optimal points and therefore n1 ¼ n2 % 0:29 Á N, n3 ¼ n4 % 0:21 Á N, where N is the total sample size In the next example we demonstrate how to incorporate cost constraints in the random coefficients regression model 5.5.2 Example 5.2 Pharmacokinetic model, serial sampling At the early stages of clinical trials, patients often undergo serial blood sampling to determine activity level of various factors Compartmental models are an important tool in mathematical modelling of how drugs circulate through the body over time, even though the compartments usually have no physiological equivalents; cf Gibaldi and Perrier (1982) It is often assumed that the rate of transfer between compartments and the rate of elimination from compartments follow first-order (linear) kinetics which leads to systems of linear differential equations For 122 RESPONSE-DRIVEN DESIGNS IN DRUG DEVELOPMENT 1st function only: cost = 1.0 + 0.5 • ψ(X), X∈ Z1 • 0.173 0.174 -1 2nd function only: cost = 1.0 + 0.5 ψ(X), X∈ Z2 -1 Both functions: cost = 1.0 + 0.5 + 0.5 ψ(X), X∈ Z3 9• 0.331 0.33 • -1 –0.5 0.5 x Figure 5.4 Sensitivity functions and optimal design for cv ¼ 1, c1 ¼ c2 ¼ 0:5,  ¼ 0:3 for Example 5.1 Two optimal points are left in subregion Z3 , the other two are in subregion Z1 example, if a dose D is administered orally at time x ¼ 0, a two-compartment model with first-order absorption and elimination from central compartment can be used, > < g_ xị ẳ Ka g0 xị g_ xị ẳ Ka g0 xị K12 ỵ K10 ịg1 xị ỵ K21 g2 xị > : g_ xị ẳ K12 g1 xị K21 g2 xị 5:27ị with initial condition g0 0ị ẳ D, g1 0ị ẳ g2 0ị ẳ Here Ka is an absorption rate constant, K10 is an elimination rate constant, and K12 and K21 are transfer rate constants between compartments The central, or compartment 1, corresponds to plasma (serum); the peripheral, or compartment 2, is associated with tissue Measurements fyg are taken on the central compartment, Eyj ; xị ẳ x; Þ ¼ g1 ðxÞ=V1 ; Varðyj ; xÞ ¼ 2 ; OPTIMAL DESIGNS WITH COST CONSTRAINTS 123 where V1 is the apparent volume of distribution of the central compartment, and by we denote the vector of unknown parameters, ¼ ðV1 ; Ka ; K10 ; K12 ; K21 Þ of dimension m Solving system (5.27) leads to ! Ka D ðK21 À Ka ÞeÀKa x ðK21 À u1 ÞeÀu1 x K21 u2 ịeu2 x x; ị ẳ ỵ ỵ ; V1 ðu1 À Ka Þðu2 À Ka Þ ðKa À u1 Þðu2 À u1 Þ ðKa À u2 Þðu1 u2 ị 5:28ị with u1;2 q ! ẳ 0:5 K12 ỵ K21 ỵ K10 ặ K12 ỵ K21 ỵ K10 Þ2 À 4K21 K10 ; see Gibaldi and Perrier (1982, Appendix B), or Seber and Wild (1989, Example 8.5) We assume that given , all measurements are independent, and that parameters j of patient j are independently sampled from the normal distribution with E j ị ẳ ; Var j Þ ¼ Ã; ð5:29Þ where and à are usually referred to as ‘population’, or ‘global’ parameters In practice, different number of samples kj can be obtained for different j’s (patients) Therefore, it seems reasonable to introduce costs similar to Example 5.1 In particular, one of the options is to impose a cost cv for each visit, and a cost cs for each of the individual samples Then if k samples are taken, the total cost per patient will be described as Ck ẳ cv ỵ kcs ; k ẳ 1; ; q; with the restriction (5.23) on the total cost Let Xj ¼ ðx1j ; x2j ; ; xkj j Þ and Yj ¼ ðy1j ; y2j ; ; ykj j Þ be sampling times and measured concentrations, respectively, for patient j If Xj ; ị ẳ ẵx1j ; Þ; ; ðxkj j ; ފT and FðXj ; Þ is a ðkj  m Þ matrix of partial derivatives, ! @ ðXj ; Þ  FðXj ; Þ ¼  0; @  ¼ then using (5.29), one gets the following approximation of the variancecovariance matrix S, SXj ; ị ẳ VarYj jXj Þ ¼ FðXj ; Þà F T ðXj ; ị ỵ 2 Ikj ; where  ẳ ; Ã; 2 Þ is a combined vector of model parameters ð5:30Þ 124 RESPONSE-DRIVEN DESIGNS IN DRUG DEVELOPMENT To formally introduce the design region X, we note that there is a natural ordering corresponding to the timing of the samples The design subregion for a patient depends upon the number and timing of samples taken from that patient For example, if a single sample is drawn, the design subregion Z1 will consist of a single point x, x T For a patient having two samples taken, the design subregion Z2 will consist of vectors X ¼ ðx1 ; x2 Þ Moreover, the two consecutive samples have to be separated in time by at least some positive Á, therefore the design subregion Z2 is defined as Z2 ¼ fX ¼ ðx1 ; x2 Þ : x1 ; x1 þ Á x2 Tg; etc If q samples are taken from a patient, then Zq ¼ fX ¼ ðx1 ; ; xq ị : Finally, X ẳ Z1 defined as S Z2 S x1 ; x1 þ Á x2 ; ; xqÀ1 þ Á xq Tg: Zq , and the normalized information matrix can be MC ; ị ẳ n X iẳ1 wi ~Xi ị; n X wi ẳ 1; iẳ1 where ~X; ị ẳ X; ị=Ck , if dimXị ¼ k and X Zk ; the information matrix ðX; Þ is defined in (5.11), with SðX; Þ introduced in (5.30) 5.5.3 Remark 5.1 If one uses the Taylor expansion for ðx; Þ up to the second-order terms, !T @x; ị x; ị ẳ x; ị ỵ ẵ ỵ ẵ T H ịẵ ỵ @  where ! @ ðx; Þ  Hð Þ ¼ ; @  @  ¼ 0 then it follows (5.29) that the expectation of ðx; j Þ with respect to the distribution of parameters j can be approximated as E ðx; j Þ % ðx; Þ ỵ trẵH ị: 5:31ị OPTIMAL DESIGNS WITH COST CONSTRAINTS 125 If the second term on the right-hand side of (5.31) is not ‘small’ compared to ðx; Þ, then to estimate global parameters , some adjustments need to be made in the estimation procedures (such as iteratively reweighted algorithms (5.14)– (5.17)), to incorporate the additional term in the mean function However, if the second term is relatively small, then disregarding this term should not significantly affect the estimation algorithm This is exactly the case in our numerical example below As mentioned at the beginning of the section, the introduction of costs allows for a meaningful comparison of patients with distinct number of blood samples taken For this example, we used the data from the study where patients received oral drug and had one or two blood samples drawn The pharmacokinetic model was parameterized in terms of oral clearance CL, distributional clearance Q, and volumes of distribution V1 , V2 of central and peripheral compartments, respectively This leads to the following expressions for the compartment rate constants in (5.27): K10 ¼ CL=V1 ; K12 ¼ Q=V1 ; K21 ¼ Q=V2 : The absorption rate constant was fixed to 1:70 hÀ1 , and the parameters of response function ¼ ðCL; Q; V1 ; V2 Þ, covariance parameters à and residual variance 2 were estimated from the data Those estimates were used to construct locally optimal designs For the examples below, we take the diagonal covariance matrix à ¼ diagði Þ, so that the combined vector of unknown parameters is  ¼ ðCL; Q; V1 ; V2 ; CL ; Q ; V1 ; V2 ; 2 ÞT ; dimðÞ ¼ 9: We assume that the drug is administered at x ¼ 0; the sampling, one or two samples per patient, is allowed every hour from x ¼ to x ¼ 20, with samples separated by at least Á ¼ hours The design region X is presented in Figure 5.5, top panel D-optimal design and sensitivity function under the assumption of equal costs for single samples and pairs, i.e C1 ¼ C2 ¼ 1, are shown in Figure 5.5, bottom panel Similar to the first case in Example 5.1, it is not surprising that the pairs of samples are selected, X1 ẳ 1; 5ị; X2 ẳ 3; 11ị; X3 ẳ 10; 20ị with weights 0.39, 0.27 and 0.34, respectively Note that for subregion Z2 , the sensitivity function hits the reference plane ¼ at the D-optimal points In contrast, the sensitivity function for subregion Z1 is well below reference line m ¼ and is not presented here If the cost of drawing two blood samples is increased to C2 ¼ 10, then four single points appear in the D-optimal design, see Figure 5.6: X1 ¼ f1; 3; 8; 20g with weights w1 ¼ f0:2; 0:21; 0:2; 0:19g: 126 RESPONSE-DRIVEN DESIGNS IN DRUG DEVELOPMENT Design region 20 Z2 x2 15 10 Z1 10 x1 12 14 16 18 20 10 ψ(X), X∈ Z2 20 10 0 x2 10 12 14 16 18 20 x1 Figure 5.5 Design region and sensitivity function for Example 5.2 Top panel: design region, singles Z1 (triangles) and pairs Z2 (crosses) Bottom panel: sensitivity function for equal costs Note that ¼ at the support points of optimal design Still, there are two optimal pairs of samples, X5 ẳ 1; 6ị and X6 ¼ ð14; 18Þ with weights w5 ¼ 0:09 and w6 ¼ 0:11 As in Example 5.1, to get actual randomization ratio, one needs to take cost function cðXÞ into account For example, n1 0:2  10 % 22:2; etc: ¼ n5 0:09  and finally, n1 ¼ n3 ¼ 0:244  N; n2 ¼ 0:256  N; n4 ¼ 0:232  N; n5 ¼ 0:013  N; n6 ¼ 0:011  N: ADAPTIVE DESIGNS 127 C1=1 (singles), C2=10 (pairs) 10 ψ(X), X∈ Z 4 10 12 14 16 18 20 10 ψ(X), X∈ Z 20 10 0 x2 10 12 14 16 18 20 x1 Figure 5.6 Sensitivity functions for Example 5.2 with unequal costs: costsinglesị ẳ 1, costpairsị ẳ 10 Singles Z1 (top panel) and pairs Z2 (bottom panel) The interpretation of the optimal design is as follows:  Take one sample at times f1; 3; 8; 20g using 24.4, 25.6, 24.4 and 23.2% of the patients, respectively  Take two samples at times (1, 6) and (14, 18) using 1.3 and 1.1% of the patients, respectively For other methods of generating optimal designs with constraints in random effects models, see Mentre´ et al (1997), where the authors allow for repeated measurements at the same design point This, however, does not seem appropriate in our applications of interest 5.6 Adaptive Designs As mentioned earlier, for nonlinear models the information matrix depends, in general, on the values of unknown parameters which leads to the concept of locally 128 RESPONSE-DRIVEN DESIGNS IN DRUG DEVELOPMENT optimal designs The misspecification of the preliminary estimates may result in the poor performance of locally optimal designs Therefore the use of an adaptive, or sequential, approach in this situation can be beneficial; see Box and Hunter (1965), Fedorov (1972, Ch 4), Wu (1985a,b), Atkinson and Donev (1992, Ch 11) and Zacks (1996) In the rest of this section, we shall illustrate the adaptive approach in the context of dose response experiments The adaptive designs are performed in stages in order to correct dose levels at each stage and approach the optimal doses as the number of stages increases The adaptive approach relies on the reiteration of two steps: estimation and selection of new doses We first briefly outline the adaptive algorithm and then give a more formal description: (A.0) Initial non-singular design (doses) is chosen and preliminary parameter estimates are obtained (A.1) Additional dose(s) is selected from the available range of doses which provides the biggest improvement of the design with respect to selected criterion of optimality and current parameter estimates (A.2) The estimates of unknown parameters are refined given additional observation(s) Steps (A.1) and (A.2) are repeated given available resources (for instance, maximal number of patients to be enrolled in a study) For a formal introduction of adaptive designs, it is convenient to revert to unnormalized information matrix MN ị ẳ MN ; ị introduced in (5.5) In this section by  ¼ fxi ; ni g we denote an unnormalized design, if not stated otherwise Step A preliminary experiment is performed according to some initial nonsingular design 0 with N0 observations, i.e Ey0i ẳ x0i ; ị, i ¼ 1; 2; ; N0 Then an initial parameter estimate 0 is obtained from model fitting Step Let 1 ¼ 0 [ x1 , where x1 ẳ arg ẵM1 N0 ỵ1  ; 0;x ފ; x É is a chosen criterion of optimality, and 0;x ¼ 0 [ x, which means that one additional observation at dose level x is added to design 0 The minimization is performed with respect to all admissible doses Next, a new experiment at dose level x1 is carried out, Ey1 ẳ x1 ; ị, and an updated parameter estimate 1 is obtained from model fitting Step s ỵ Let sỵ1 ẳ s [ xsỵ1 , where xsỵ1 ẳ arg ẫẵMN10 ỵsỵ1 s ; s;x ị; x and s;x ẳ s [ x Then an updated parameter estimate sỵ1 is obtained, etc ... distribution Optimal Bayesian Designs 8.6.1 Numerical methods 8.6.2 DS -optimal designs ^Þ 8.6.3 Optimal designs for varðD Practical Designs 8.7.1 Reservations about the optimal designs 197 198... Kety–Schmidt Method The Statistical Model and Optimality Criteria Locally Optimal Designs 8.4.1 DS -optimal designs ^Þ 8.4.2 Designs minimising varðD Bayesian Designs and Prior Distributions 8.5.1 Bayesian... precision 4.4.4 Computational issues Deriving Optimal Designs in Practice 4.5.1 Data needed to compute optimal designs 4.5.2 Examples of optimal design 4.5.3 The optimal sampling package 4.5.4 Sensitivity