Sequential ED-Design for BinaryDose–Response ExperimentsbyXiaoli YuM.Sc., The University of British Columbia, 2011B.Sc., The University of Alberta, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Statistics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2017c© Xiaoli Yu 2017AbstractDose–response experiments and subsequent data analyses are often carriedout according to optimal designs for the purpose of accurately determininga specific effective dose (ed) level. If the interest is the dose–response re-lationship over a range of ed levels, many existing optimal designs are notaccurate. In this dissertation, we propose a new design procedure, calledtwo-stage sequential ED-design which directly and simultaneously targetsseveral ed levels. We use a small number of trials to provide a tentativeestimation of the model parameters. The doses of the subsequent trials arethen selected sequentially, based on the latest model information, to maxi-mize the efficiency of the ed estimation over several ed levels.Although the commonly used logistic and probit models are convenientsummaries of the dose–response relationship, they can be too restrictive.We introduce and study a more flexible albeit slightly more complex three-parameter logistic dose-response model. We explore the effectiveness of thesequential ED-design and the D-optimal design under this model, and de-velop an effective model fitting strategy. We develop a two-step iterativealgorithm to compute the maximum likelihood estimate of the model pa-rameters. We prove that the algorithm iteration increases the likelihoodvalue, and therefore will lead to at least a local maximum of the likelihoodiiAbstractfunction. We also study the numerical solution to the D-optimal design forthe three-parameter logistic model. Interestingly, all our numerical solutionsto the D-optimal design are three-point-support distributions.We also discuss the use of the ED-design when experimental subjectsbecome available in groups. We introduce the group sequential ED-design,and demonstrate how to construct this design. The ED-design has a naturalextension to more complex models and can satisfy a broad range of thedemands that may arise in applications.iiiLay SummaryDose–response experiments are routinely conducted in the early phase ofclinical trials. The most common goal of these experiments is to collectinformation about the relationship between the dosage of an investigationaldrug and the responses of patients. This goal is often accomplished by ac-curately determining a specific dose level. In medical research, it is alsoimportant to determine the effective and safe dose range so that it is highenough to induce desired beneficial effects, and low enough to avoid poten-tial adverse effects. Motivated by this observation, we propose a new designprocedure that simultaneously estimates several dose levels. We demon-strate how to carry out this design. We find that the new design comparesfavourably with many existing designs that we are aware of.ivPrefaceThis thesis is under the supervision of Drs. Jiahua Chen and Rollin Brant.A paper based on Chapter 3 of the dissertation has been published in Yuet al. (2016). Chapter 4 is based on a manuscript in preparation coauthoredwith Dr. Jiahua Chen. Dr. Brant raised the research problem and Dr. Chenhelped to formulate the idea and suggested specific approaches in thesemanuscripts. Both had given me valuable constructive criticisms, helpedme to organize my ideas. They supported, encouraged and inspired meduring the writing of the manuscripts.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Dose–response experiments and design issues . . . . . . . . . 11.2 Major contributions . . . . . . . . . . . . . . . . . . . . . . . 61.3 Outline of the dissertation . . . . . . . . . . . . . . . . . . . 82 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Dose–response curves and parametric models . . . . . . . . . 11viTable of Contents2.2 Optimal designs . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 D-optimal design under the logistic model . . . . . . . . . . . 212.4 Sequential D-optimal design . . . . . . . . . . . . . . . . . . 232.5 Three-phase sequential design . . . . . . . . . . . . . . . . . 252.5.1 First stage . . . . . . . . . . . . . . . . . . . . . . . . 262.5.2 Second stage . . . . . . . . . . . . . . . . . . . . . . . 262.5.3 Third stage . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Up-and-down design . . . . . . . . . . . . . . . . . . . . . . . 292.7 Biased-coin up-and-down design . . . . . . . . . . . . . . . . 302.8 Group up-and-down design . . . . . . . . . . . . . . . . . . . 322.9 Accelerated biased-coin design . . . . . . . . . . . . . . . . . 332.10 Generalized Po´lya Urn design . . . . . . . . . . . . . . . . . 343 Two-stage Sequential ED-Design . . . . . . . . . . . . . . . . 363.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 New criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 The pilot experiment . . . . . . . . . . . . . . . . . . 393.3 Sequential ED-design under the logistic model . . . . . . . . 393.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . 433.4.1 Detailed specifications . . . . . . . . . . . . . . . . . . 443.4.2 Performance comparison when the response model iscorrectly specified . . . . . . . . . . . . . . . . . . . . 453.4.3 Performance comparison when the response model ismis-specified . . . . . . . . . . . . . . . . . . . . . . . 503.5 Limiting design as n increases . . . . . . . . . . . . . . . . . 54viiTable of Contents3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 583.7 R-code for the ED-design and simulation . . . . . . . . . . . 594 ED-design under the Three-parameter Logistic Model . . 674.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . 674.2 Three-parameter logistic model . . . . . . . . . . . . . . . . . 694.3 Maximum likelihood estimation . . . . . . . . . . . . . . . . 704.4 Potential designs for the three-parameter logistic model . . . 754.4.1 Up-and-down design . . . . . . . . . . . . . . . . . . . 764.4.2 D-optimal design . . . . . . . . . . . . . . . . . . . . 764.4.3 Vertex Direction Method(VDM) . . . . . . . . . . . . 784.5 Two-stage sequential ED-design . . . . . . . . . . . . . . . . 824.6 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . 854.6.1 The three-parameter model is both the assumed andthe truth for the dose–response experiment . . . . . . 874.6.2 Effects of fitting a three-parameter model when a two–parameter logistic model suffices . . . . . . . . . . . . 914.6.3 Effects under model misspecification . . . . . . . . . . 954.7 Numerical example . . . . . . . . . . . . . . . . . . . . . . . 1064.8 Limiting design as n increases . . . . . . . . . . . . . . . . . 1094.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 1124.10 R-code for simulations . . . . . . . . . . . . . . . . . . . . . . 1135 Group Sequential ED-Design . . . . . . . . . . . . . . . . . . 1225.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 122viiiTable of Contents5.2 Two-stage group sequential ED-design under the logistic re-gression model . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.3.1 Detailed specifications . . . . . . . . . . . . . . . . . . 1275.3.2 Performance comparison when the response model iscorrectly specified . . . . . . . . . . . . . . . . . . . . 1275.3.3 Performance comparison when the response model ismis-specified . . . . . . . . . . . . . . . . . . . . . . . 1305.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 1336 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . . 1356.1 Data structure . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.1.1 The likelihood . . . . . . . . . . . . . . . . . . . . . . 1376.2 Maximum likelihood estimation . . . . . . . . . . . . . . . . 1396.2.1 Data structure and assumptions . . . . . . . . . . . . 1406.3 Asymptotic properties of the maximum likelihood estimate . 1426.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 1487 Contributions and Future Research . . . . . . . . . . . . . . 1497.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 152Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155ixList of Tables3.1 Simulated RMSEs under the logistic model targeting rangeed25–ed75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Simulated RMSEs under the logistic model targeting rangeed10–ed40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3 Simulated RMSEs under probit mis-specified as logistic . . . 513.4 Simulated RMSEs under probit mis-specified as logistic . . . 523.5 Simulated RMSEs under the mis-specified logistic model . . . 543.6 Simulated RMSEs under the mis-specified logistic model . . . 554.1 D-optimal design under three-parameter Logistic model. . . . 784.2 D-optimal design under the three-parameter probit model. . . 794.3 Simulated RMSEs under the three-parameter model (α =−6.265, β = 0.055, λ = 0.5) . . . . . . . . . . . . . . . . . . . 894.4 Simulated RMSEs under the three-parameter model (α =−14.148, β = 0.1, λ = 2) . . . . . . . . . . . . . . . . . . . . . 904.5 Simulated RMSEs when fitting a simple logistic model whena two–parameter model suffices. . . . . . . . . . . . . . . . . . 934.6 Simulated RMSEs when fitting a three-parameter model whena two–parameter model suffices. . . . . . . . . . . . . . . . . . 94xList of Tables4.7 Simulated RMSEs under the three parameter model (α =−6.265, β = 0.055, λ = 0.5). . . . . . . . . . . . . . . . . . . . 974.8 Simulated RMSEs under the three-parameter model (α =−6.265, β = 0.055, λ = 0.5). . . . . . . . . . . . . . . . . . . . 984.9 Simulated RMSEs under the three-parameter model (α =−14.148, β = 0.1, λ = 2). . . . . . . . . . . . . . . . . . . . . . 994.10 Simulated RMSEs under the three-parameter model (α =−14.148, β = 0.1, λ = 2). . . . . . . . . . . . . . . . . . . . . . 1004.11 Simulated RMSEs under probit mis-specified as logistic (α =−6.265, β = 0.055, λ = 0.5). . . . . . . . . . . . . . . . . . . . 1024.12 Simulated RMSEs under probit mis-specified as logistic (α =−6.265, β = 0.055, λ = 0.5). . . . . . . . . . . . . . . . . . . . 1034.13 Simulated RMSEs under probit mis-specified as logistic ( α =−14.148, β = 0.1, λ = 2). . . . . . . . . . . . . . . . . . . . . . 1044.14 Simulated RMSEs under probit mis-specified as logistic ( α =−14.148, β = 0.1, λ = 2). . . . . . . . . . . . . . . . . . . . . . 1054.15 Number of subjects examined and showing the wheezing symp-tom for British coal miners. . . . . . . . . . . . . . . . . . . . 1075.1 Simulated RMSEs under the logistic model targeting rangeed25–ed75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2 Simulated RMSEs under the logistic model targeting rangeED10–ED40. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3 Simulated RMSEs under probit mis-specified as logistic . . . 1325.4 Simulated RMSEs under probit mis-specified as logistic . . . 133xiList of Figures2.1 A sample dose–response curve. . . . . . . . . . . . . . . . . . 132.2 Two sample logistic regression curves. . . . . . . . . . . . . . 143.1 Histogram of the ED-design for (a) estimating ed25, ed50,and ed75 when the response curve is correctly specified aslogistic; (b) estimating ed25, ed50, and ed75 when the re-sponse is mis-specified; The x-axes correspond to the ed levels. 563.2 Histogram of the ED-design for (c) estimating ed10, ed25,and ed40 when the response curve is correctly specified as lo-gistic, and (d) estimating ed10, ed25, and ed40 when the re-sponse is mis-specified. The x-axes correspond to the ed levels. 574.1 Dose–response curves in the simulation . . . . . . . . . . . . . 884.2 Observed Data and the fitted curve for British Coal Miners . 1084.3 Histogram of the ED-design with respect to ed levels (α =−6.265, β = 0.055, λ = 0.5) for (a) estimating ed10, ed25,and ed40; (b) estimating ed25, ed50, and ed75, and (c) es-timating ed60, ed75, and ed90. . . . . . . . . . . . . . . . . . 110xiiList of Figures4.4 Histogram of the ED-design with respect to ed levels (α =−14.148, β = 0.1 and λ = 2) for (a) estimating ed10, ed25,and ed40; (b) estimating ed25, ed50, and ed75, and (c) es-timating ed60, ed75, and ed90. . . . . . . . . . . . . . . . . . 111xiiiAcknowledgementsFirst, I would like to express my sincere gratitude to my principal supervisorDr. Jiahua Chen, who has inspired and guided me throughout my Ph.D.study. Thank you, Dr. Chen, for your invaluable suggestions, guidance, andcontinuing support. I also would like to express my gratitude to my co–supervisor Dr. Rollin Brant for his suggestions and encouragement duringthe course of my Ph.D. study.Secondly, I would like to extend my appreciation to my supervisory com-mittee, Dr. Lang Wu for his suggestions and support throughout my Ph.D.study.Most importantly, I thank my parents and husband for their whole-hearted love and support. They have dedicated all their time and energyto help me take care of my daughter, whom was born when I started myPhD study. They also have encouraged me and brought me much joy andhappiness. Without the help from my supervisor Dr. Chen or the supportfrom my beloved family, the writing of this thesis would never have beenpossible.xivTo Yi, Hui, Zhennian, and MiaxvChapter 1Introduction1.1 Dose–response experiments and design issuesWhen a stimulus is administrated to a subject, some changes in the subjectmight be observed immediately or after a certain exposure time. Studyingdose–response relationship and developing dose–response models are of cen-tral importance in various applications. Dose–response experiments collectdata on the level or dosage of the stimulus applied and the response of thesubject. The information collected is used for model development and todetermine “safe”, “hazardous” and beneficial levels or dosages for investi-gational drugs, pollutants, foods, and other substances to which humans,other organisms, or non-living systems are exposed. These conclusions arethe basis for public policy or safety manuals.In drug developments, dose–response experiments are involved in bothPhases I and II clinical trials. The main goal of clinical trials is to uncoverthe relationship between the doses of an investigational drug and the proba-bility of toxicity or beneficial responses of patients in the target population.The dose given to a patient is ideally high enough to induce the desired re-sponse and low enough to avoid potential adverse effects (Dette et al., 2005;Dragalin et al., 2008a). An easily understandable example is the dosage11.1. Dose–response experiments and design issuesan anesthesiologist must decide: it must be low enough to not harm theirpatients, but high enough to induce the desired anesthetic effect. The ac-curate dose–response relationship is clearly vital in guiding the selection ofthe dose levels in such practices (Pace et al., 2007).Poor understanding of the underlying dose–response relationship in PhaseI/II clinical trials may result in selecting the wrong target doses to be usedin Phase III large scale confirmatory clinical trials. This may cause seriousethical and financial consequences. Selecting too high a dose may cause alarge number of toxic responses in experimental subjects, and choosing toolow a dose may fail to establish adequate efficacy. Both can lead to unwar-ranted failure to obtain the regulatory approval of an investigational drug(Dette et al., 2008; Bretz et al., 2010). We refer to Ting (2006) and Bretzet al. (2008) for additional general discussion on issues and challenges indose–response experiments in the context of the drug development process.The dose–response relationship also plays a vital role in other applica-tions. In pyrotechnics applications, we must have a thorough understandingof the sensitivity of a new explosive to the stress of a shock to avoid catas-trophic consequences. See Dror and Steinberg (2008) and Wu and Tian(2013) for more vivid descriptions. In pyrotechnics experiments, the stresslevel may be the drop height of an explosive, or the pressure on a pile ofammunition. The response is either explosion or nonexplosion (Wu andTian, 2013). For example, in testing the sensitivity of new pyrotechnics toignition, each sample is assumed to have a threshold stress level. Ignitionpulses that are larger than this level will ignite the sample. Ignition pulsesthat are smaller than this level will not ignite the sample. Repeated testing21.1. Dose–response experiments and design issueson any sample is not possible, because the pulse that is not large enough tocause ignition will damage the sample. To estimate the parameters of theunderlying response model, samples are tested at various stress levels andtheir responses are observed. Researchers then analyze the data to obtainan estimate of the model parameters (Neyer, 1994). See Neyer (1994), Drorand Steinberg (2008) and Wu and Tian (2013) for more vivid descriptions.The task of accurately characterizing the dose–response relationship maynot appear to be challenging. One may simply administer various dose levelsto a large number of subjects. The data on the responses of these subjectswould likely give a clear picture of the dose–response relationship. Thispractice is clearly not feasible. In drug developments it will hurt a largenumber of patients or volunteers. In other applications, the cost can beunacceptably high. We may not be able to find sufficient resources to runthe experiment, and it may take a long time to complete.Fortunately, experience and our intuition indicate that the relationshipbetween the dose level and the probability of response is smooth and mono-tone. Based on this belief, statistical design theory can be used to maximizethe information content of each experimental run/trial. A well-designed ex-periment reduces the cost and saves time in the drug development processor other applications.In the general context, an optimal design maximizes the expected infor-mation content in the anticipated data given a fixed number of experimentalruns. When a parametric model is selected for the relationship between theresponse variable and design variables (also called explanatory variables, co-variates, or dosage in drug development examples), an optimal design often31.1. Dose–response experiments and design issuesaims to maximize the Fisher information by running experiments at specificlevel combinations of the design variables. If the parametric model is linear,the Fisher information does not depend on the true parameter values of themodel. Hence, the optimal design is possible without the knowledge of thetrue parameter values (Montgomery, 2008).The dose–response relationship is apparently nonlinear. No matter howlow or how high a dose level is, the probability of responding to a stimulustakes a value between 0 and 1. For this reason, a nonlinear parametricmodel is often selected. Suppose a specific dose–response relationship suchas logistic is assumed. Under this model, the Fisher information is a functionof unknown parameters, as well as the specific design. Without knowledgeof the specific parameter values, it is not possible to determine whether adesign maximizes the Fisher information. When the parameter values areknown, it is at least possible in principle to find the design that gives themost efficient estimation of the model parameters (Wu, 1985b,a; Ford et al.,1985; Sitter and Fainaru, 1997; Sitter and Forbes, 1997).Clearly, if we knew the parameter values, there would be no point torun experiments to estimate them. To overcome this dilemma, one mayfirst run a pilot study in which the dose levels are selected based on priorknowledge. The resulting data will provide an improved estimate of themodel parameters over the prior guess. The optimal design based on thefitted model is then used for selecting the dose levels of further trials. Thisis called a two-stage design.Because the precision of the parameter estimation based on a pilot studyis necessarily low, the resulting second stage design may markedly differ from41.1. Dose–response experiments and design issuesan optimal one. To overcome this shortcoming, a full sequential approachcan be used; the parameter estimates are updated after each trial of theexperiment, and used to determine appropriate dose levels for the subsequenttrials.Suppose the only model assumption we wish to make is the monotonicitybetween the response probability and the dose level. The popular up-and-down design works well under such a nonparametric model assumption. Itaddresses both ethical and safety concerns as well (Anderson et al., 1946;Be´ke´sy, 1947; Dixon and Mood, 1948).The up-and-down design is also of sequential nature: depending on theoutcome of the current run/trial, the dose of the next trial will be madeone level higher or lower. By the appropriate choice of several criteria, themajority runs will concentrate on the target dose level. An appropriateestimate of the target dose level wii thereby be obtained.Despite its long history, design theory for binary experiments remainsan active research area. For recent developments, see Li and Wiens (2011),Wang et al. (2013), Wang et al. (2015), and Wu and Tian (2013).Wang et al. (2015) considered a two-stage sequential D-optimal design.They proposed first obtaining a tentative estimate of the model parameters.The D-optimality criterion is then used to select the dose level of every ad-ditional subject. Wu and Tian (2013) presented a three-phase sequentialdesign. The first phase aims to ensure a viable fitted model, and the sec-ond phase chooses the dose levels to satisfy D-optimality. The third phaseclusters the dose levels around the target ed level.In this dissertation, we wish to contribute to the literature of optimal51.2. Major contributionsdesigns concerning the dose–response experiment and the estimation of thedose–response relationship. The motivation of the research problem is fromthe following observations. While many designs are optimal if estimating themedian effective dose level is the sole goal of the experiment, they are not thebest when a range of ed levels are targeted. In many applications, it is de-sirable to accurately determine several ed levels. For example, Rosenbergerand Grill (1997) studied the dose–response experiment problem where ed50is the primary target, but ed25 and ed75 or other ed levels are also of inter-est. This prompted them to propose a new sequential design, and apply thisdesign to a psychophysical experiment where the objective was to observehow patients respond to a range of stimulus levels. From this consideration,we propose a new criterion and the corresponding sequential solution to itsimplementation. We research the usefulness of the new method in achievinghigher precision for estimating the dose–response relationship over a specificrange of interest. More specific details are given in the next section.1.2 Major contributionsThe most important contribution of this dissertation is the introduction ofa new optimality criterion. Traditionally, when a parametric dose–responsemodel is assumed, we often search for designs which enable us to most ac-curately estimate the model parameters. In applications, we consider, theultimate goal of the investigation is to accurately determine the various ef-fective dose levels. These two goals are closely related but not equivalent.Based on this consideration, we propose a new design criterion which we im-61.2. Major contributionsplement it sequentially. Because we directly aim at the accurate estimationof the effective dose levels, we call it sequential ED-design.We conduct extensive computer simulation to demonstrate that the pro-posed sequential ED-design indeed improves the efficiency of the experimentby changing the optimality target from model parameters to effective doselevels of interest compared with many existing designs. A paper based onthis part of the dissertation has been published in Yu et al. (2016).The logistic regression model is the most popularly assumed dose–responsemodel in applications. Either the proposed sequential ED-design or otherexisting sequential designs often require some preliminary estimate of the pa-rameter values based on a pilot experiment with a small number of runs/trials.The maximum likelihood approach is often the choice to give a preliminaryestimate of the model parameters based on the pilot data. A technical hur-dle is that the maximum likelihood estimate under the logistic regressionmodel may fail to exist when the data have a specific configuration, i.e., thelikelihood does not attain its maximum at a infinite parametric value. Thisis even more likely for pilot data. In this dissertation, we employ the ideaof adding pseudo observations. This approach takes advantage of our priorknowledge of the dose–response curve and enables a maximum likelihoodlike estimate for any data configurations.Based on these results, we further explore the application of the proposedsequential ED-design to a more flexible three-parameter logistic regressionmodel. We develop a two-step iterative algorithm to compute the maximumlikelihood estimate of the model parameters. We prove that the algorithmiteration increases the likelihood value and therefore will lead to at least71.3. Outline of the dissertationa local maximum of the likelihood function. We also study the numericalsolution to the D-optimal design for the three-parameter logistic regres-sion model. It is of interest to find that all our numerical solutions to theD-optimal design are three-point-support distributions. Simulation resultsindicate that the more flexible three–parameter logistic regression model canbe easily implemented, and the sequential ED-design remains effective.In addition to these achievements, we discuss the use of ED-design whenexperimental subjects become available in groups. For instance, two patientsmay become available for the next experimental trial at the same time. Wemay be required to decide their appropriate dose levels simultaneously.1.3 Outline of the dissertationThe thesis is organized as follows. In the next chapter we introduce somenotation and some parametric dose–response models. We give a general re-view of the corresponding optimal design theory. We derive the analyticaland numerical results for locally D-optimal designs under the standard lo-gistic model. We give a comprehensive review of some existing sequentialdesign procedures of dose–response experiments such as the up-and-downand related designs, and discuss their advantages and shortcomings.In Chapter 3, we introduce our two-stage sequential ED-design. We use asmall number of trials to provide a tentative estimation of the model param-eters. The dose levels of the subsequent trials are then selected sequentially,based on the latest model information, to maximize the efficiency of theed estimation over several ed levels. Some details of the ED-design under a81.3. Outline of the dissertationlogistic regression model are given. Simulations indicate that the ED-designcompares favorably with several existing designs under various scenarios. Inaddition, we provide some simulation evidence for the limiting ED-designwhen the sample size n goes to infinity. It appears that as a distributionover the dose range, the design has a limit with two support points.In Chapter 4, we introduce the three-parameter logistic model. Some de-tails of the ED-design under the three-parameter logistic regression modelare given. We investigate the effectiveness of the sequential ED-design,the D-optimal design, and the up-and-down design under this model, anddevelop an effective model fitting strategy. Simulations show that the com-bination of the proposed model and the data analysis strategy performswell. When the logistic model is correct, using the more complex modelsuffers hardly any efficiency loss. When the three-parameter model holdsbut the logistic model is violated, the new approach can be more efficient.In addition, we apply the new approach to a real dataset.In Chapter 5, we introduce a group sequential ED-design, and show howto construct it. Simulation studies indicate that our group ED-design com-pares favorably with several existing group design procedures under variousscenarios.In Chapter 6, the asymptotic properties of the two-stage sequential ED-design are investigated. The method of maximum likelihood is one of theclassical methods of estimation. We present some general results on theasymptotic properties of the maximum likelihood estimators following a two-stage sequential design. We provide evidence that the maximum likelihoodestimators from the two-stage sequential design exist and have the usual91.3. Outline of the dissertationasymptotic properties (i.e., consistency, asymptotically normality).Chapter 7 summarizes the dissertation, adds some conclusions and dis-cusses areas of future research.10Chapter 2PreliminariesTo explain our idea clearly, let us first introduce some specific notation andconcepts.2.1 Dose–response curves and parametric modelsWe use X or x to denote the dose level of a drug or a stimulus in a dose–response experiment. We use Y for the random outcome of the response.In toxicity studies, we put Y = 1 if the subject has toxicity reaction, andY = 0 otherwise. In the study of drug efficacy, we put Y = 1 if the desiredmedical effect is achieved, and Y = 0 otherwise. At this moment, we onlyconsider the situation where the response is not a vector. Namely, we donot consider multiple responses.Suppose a stimulus at dosage X = x is applied to a subject/recipient,and the outcome is Y . The dose–response relationship is defined to be thefunctionpi(x) = P (Y = 1|X = x).When the dose–response relationship pi(x) for a stimulus is fully and pre-cisely determined, the user may decide on a suitable level of the stimulus in112.1. Dose–response curves and parametric modelsapplications. She will have full knowledge of the risk of a catastrophic eventwhen a specific dosage is applied. She can also be nearly certain of when adesired effect will occur by applying a high enough level of the stimulus.We assume that pi(x) is a monotone increasing function. Conceptually,pi(0) = 0 and pi(∞) = 1. It is very unlikely to be true in most applications,especially in medical fields. In applications, some dose levels are of particularinterest to scientists. For instance, ed50 is a dose level at which 50% ofsubjects/recipients respond (Y = 1). In other words, it is the dosage suchthatpi(ed50) = 0.5.In general, the effective dose level edγ for some γ ∈ (0, 100) is the x valuesuch thatpi(edγ) = γ/100.We use ed25, ed50, ed75 and so on for dosages at which 25%, 50% and75% of the subjects respond. See Figure 2.1 for an illustration of the dose–response relationship.It is feasible to have the dose–response relationship pi(x) estimated basedon binary dose–response experiments nonparametrically. This practice issafe against potential model misspecification. However, nonparametric in-ference generally has lower efficiency compared with parametric inference.Hence, a parametric model assumption is often imposed if it can provide agood description of the dose–response relationship. There are many com-monly used models for this purpose. We limit our review on a few specificones.122.1. Dose–response curves and parametric models80 100 120 1400.00.20.40.60.81.0Doses(mg/kg)Probability of ReponseED50 = 114 mg/kgFigure 2.1: A sample dose–response curve.Logistic regression model. Under the logistic regression model assump-tion, we postulate the dose–response curve satisfyinglogit{pi(x)} = log [ pi(x)1− pi(x)]= α+ βx (2.1)for some parameter α and β. It is seen that under this model,pi(x) =exp(α+ βx)1 + exp(α+ βx). (2.2)132.1. Dose–response curves and parametric modelsFor any γ ∈ (0, 100), we haveedγ =logit(γ/100)− αβ(2.3)In particular, we have ed50 = −α/β.When x =∞, we have pi(∞) = 1 with β > 0. We do not generally havepi(0) = 0. This does not seem to be a problem in most applications. We willdiscuss other properties of the logistic regression model later.0 50 100 150 2000.00.20.40.60.81.0Dose Levels (mg/kg)Probability of Response α = − 6.265, β = 0.055α = − 14.148, β = 0.1Figure 2.2: Two sample logistic regression curves.142.1. Dose–response curves and parametric modelsProbit regression model. Under the probit regression model assumption,we postulate the dose–response curve satisfyingpi(x) = Φ(α+ βx) (2.4)where Φ(·) is the cumulative distribution function of the standard normaldistribution, and α and β are two model parameters.The probit regression model can be motivated by the existence of a latentvariable. Suppose there exists an auxiliary random variableZ = α+ βX + such that has the standard normal distribution. Suppose a positive re-sponse to the stimulus occurs only when Z > 0. In this case, we havepi(x) = P (Z > 0|X = x) = P (α+ βX + > 0) = Φ(α+ βx).In applications, the latent variable may be regarded as some unobservedstress index. The subject will respond to a stimulus only if its level exceedssome threshold value.Three-parameter logistic regression model. The two-parameter logis-tic regression model in (2.1) can be easily generalized to allow additionalflexibility:logit{piλ(x)} = log [ piλ(x)1− piλ(x)]= α+ βx (2.5)152.1. Dose–response curves and parametric modelsfor a parameter λ > 0. El-Saidi (1993) proposed the use of this modelfor the doseresponse relationship. Note that when λ < 0, we will find1−piλ(x) < 0. Then the logit function is not defined. Hence, the restrictionon λ is a mathematical necessity.A two-parameter logistic regression model has some build-in symmetry.For instance, it satisfiesedγ + ed(100− γ) = −2αβassuming β 6= 0. Such a restriction is hard to justify in applications. In-troduction of the parameter λ helps to soften this restriction without over-complicating the system. Under the proposed model, the effective dose levelat γ is given byedγ =logit((γ/100)λ)− αβ. (2.6)An explicit expression of the dose–response relationship ispi(x) = P{Y = 1|X = x} ={ exp(α+ βx)1 + exp(α+ βx)}1/λ. (2.7)Clearly, we may also introduce the three–parameter probit model in asimilar fashion. For the purpose of this dissertation, we focus on the three–parameter logistic regression model. It will be seen that our idea is generallyapplicable.162.2. Optimal designs2.2 Optimal designsAn experimental design is a plan for the set of level combinations of explana-tory/design variables. Namely, it specifies the number of experimental trials,running at specific level combinations. For example, in a dose–response ex-periment, a simple plan is to run 20 trials at dose level x1, and another 10trials at dose level x2, with the total number of trials n = 30. The doselevel is the explanatory/design variable in this example. The explanatoryvariable can be vector valued, for instance, when a trial is running withtwo drugs being administered together. The combination of dose levels ofdrugs A and B is a “level combination”. To develop optimal design theoryfor dose–response experiments, we first review concepts of generic optimaldesigns.Consider the situation where the conditional distribution of the responsevariable Y given the value of the explanatory variable X = x has a paramet-ric form f(y;x, θ) in the experiment to be carried out. Suppose observationsat independently selected X values x1, x2, . . . , xn are obtained and denotedas y1, . . . , yn. In this case, the likelihood function is given by`n(θ) =n∑i=1log f(yi;xi, θ). (2.8)One may then estimate θ by its maximum likelihood estimator θˆ. Undergeneral regularity conditions on the conditional density function and somerestrictions on the design, namely the configuration of {x1, x2, . . . , xn}, the172.2. Optimal designsmaximum likelihood estimator is asymptotically normal. That is, as n→∞,I−1/2n (θˆ − θ)→ N(0, I)where In is the Fisher information matrix defined asIn(θ) =n∑i=1E{∂ log f(yi;xi, θ)∂θ}{∂ log f(yi;xi, θ)∂θ}τ.We have adopted the convention that θ is regarded as the true parametervalue.The asymptotic result implies that the variance matrix of θˆ is approx-imately given by I−1n . The precision of the estimator θˆ is higher when Inis larger. Because In is a matrix, its magnitude is not well-defined. At thesame time, a matrix with a large determinant is deemed large in commonsense. Hence, a popular optimality criterion is to search for a design suchthat{det[In(θ)] : x1, . . . , xn ∈ X}is maximized. Here X is the space of possible x values. We usually callX the design space. The outcome of the design is a set of specific levelcombinations {x1, . . . , xn}. The resulting design is called D-optimal: D fordeterminant.The determinant of a symmetric matrix is the product of its eigenvalues.A positive definite matrix is seen as large if the sum of its eigenvalues is large.Recall that the sum of eigenvalues of a matrix is called its trace. Hence, onemay also choose a design so that the trace of the Fisher information is182.2. Optimal designsmaximized:max{tr[In(θ)] : x1, . . . , xn ∈ X}.The corresponding solution is called A-optimal.Example: Consider the situation where the design variable X is one di-mensional, and a linear model is appropriate:y = θ0 + θ1x+ where has the standard normal distribution. Based on n independentobservations {(xi, yi) : i = 1, 2, 3, . . . , n}, the Fisher information matrix isgiven byIn(θ) = n ∑xi∑xi∑x2i .One may notice that the Fisher information does not depend on the unknownparameter θ. If X = {−1,+1} and n = 2k, then the optimal design is givenby{x1, . . . , xn} = {−1,−1, · · · ,−1; +1,+1, . . . ,+1}.In other words, the D-optimal design for the experiment is to collect data byrunning k trials at x = −1 and other k trials at x = +1. It is also convenientto regard this design as a uniform distribution on {−1, 1}. We may hencedenote a design as ξn.When it comes to nonlinear models, the solution to optimal designs isno longer so simple.192.2. Optimal designsExample: Consider the logistic regression model given by (2.1). In thiscase, we havelog f(y;x, θ) = y log{pi(x)}+ (1− y) log{1− pi(x)}where we have used new notation θ = (α, β)τ . The Fisher informationmatrix based on a single observation at X = x is therefore given byI(x) = pi(x){1− pi(x)} xpi(x){1− pi(x)}xpi(x){1− pi(x)} x2pi(x){1− pi(x)} .When n observations are obtained at x1, . . . , xn, the Fisher informationbecomesIn(ξn) =n∑i=1 pi(xi){1− pi(xi)} xipi(xi){1− pi(xi)}xipi(xi){1− pi(xi)} x2ipi(xi){1− pi(xi)} .Here we have introduced ξn for the design which subscribes trials at doselevels x1, . . . , xn. We pointed out earlier that ξn can be regarded as a uniformdistribution on x1, . . . , xn.Clearly, the Fisher information is a function of θ because pi(x) dependson θ. Consequently, the D-optimal design for the dose–response experimentdepends on θ. According to Sitter and Wu (1993), given θ, the D-optimaldesign under the logistic dose–response model is a uniform distribution oned17.6 and ed82.4. Because ed values depend on the true value of θ, theoptimal design cannot be directly used to guide the experiment unless the θvalue is known. Yet if the θ value is known, there is no need to conduct the202.3. D-optimal design under the logistic modelexperiment. Nevertheless, the result of the D-optimal design can be used inother ways. For instance, it reveals the limit of how efficient a design canbe based on the D-optimal criterion.To avoid the dilemma that a good design is possible only if the dose–response relationship is known, sequential approaches are often used. Thegeneral idea is simple: run a small pilot experiment based on the priorinformation of the applicants to obtain a rough idea on the dose–responserelationship. Updating our knowledge of the dose–response relationship, andselect the “optimal design” for the next stage of experiment. In the mostextreme case, a completely sequential design is used.2.3 D-optimal design under the logistic modelAs mentioned in the last section, the D-optimal design under the logisticdose–response model has two optimal dose levels, ed17.6 and ed82.4. Thederivation of these two optimal doses has been extensively studied, andis already available in the literature (See Abdelbasit and Plackett (1983);Minkin (1987); Sitter and Wu (1993); Mathew and Sinha (2001), amongothers). In this section, we shall only give a brief derivation of the D-optimaldesign under the logistic model.In order to get the D-optimal design, we need to maximize the determi-nant of the Fisher information matrix I(α, β). Let ai = α + βxi. We write212.3. D-optimal design under the logistic modelthe Fisher information matrix asI(α, β) = ∑ni=1 exp(−ai)(1+exp(−ai))2 ∑ni=1 xi exp(−ai)(1+exp(−ai))2∑ni=1 xiexp(−ai)(1+exp(−ai))2∑ni=1 x2iexp(−ai)(1+exp(−ai))2Minkin (1987) studied the following representation for the determinant ofthe above Fisher information,β2|I(α, β)| =[ n∑i=1exp(−ai)(1 + exp(−ai))2 ][ n∑i=1a2iexp(−ai)(1 + exp(−ai))2 ]−[ n∑i=1aiexp(−ai)(1 + exp(−ai))2 ]2.Let wi = exp(ai)/(1 + exp(ai))2. The above equation is simplified toβ2|I(α, β)| = ( n∑i=1wi)(n∑i=1wia2i)− ( n∑i=1wiai)2. (2.9)Using similar arguments in Abdelbasit and Plackett (1983), Minkin (1987)showed that the first term in equation (2.9) is maximized when ai satisfiesthe following equation,ai =(exp(ai) + 1)/(exp(ai)− 1),which is solved as ai = ±1.5434.Minkin (1987) claimed that when n is even, it is possible to simulta-neously maximize the first term in (2.9) and minimize the second termbeing subtracted, by assigning n/2 subjects to the dose corresponding toaj = 1.5434, and assigning the remaining n/2 subjects to the dose corre-222.4. Sequential D-optimal designsponding to aj = −1.5434. Therefore, the D-optimal design consists of twodoses x1 and x2, which satisfy α + βx1 = 1.5434, and α + βx2 = −1.5436.Hence,x1 = (1.5434− α)/β,andx2 = (−1.5434− α)/β.Note that the corresponding probabilities of responses for x1 and x2 arepi(−1.5434) = 0.176 and pi(1.5434) = 0.824. Thus, x1 and x2 correspond toed17.6 and ed82.4 doses, i.e., the corresponding optimal doses are ed17.6and ed82.4. For the probit dose–response model, the corresponding optimaldose levels are ed12.8 and ed87.2.The direct use of the above D-optimal design is not always plausible.First, the above derivation for the D-optimal design clearly shows that opti-mal designs rely on the complete knowledge of the model parameters whichare always unknown or no experiments are needed. Thus, optimal designsare often considered as benchmarks or reference points for comparing withalternative designs.2.4 Sequential D-optimal designThe D-optimal design requires complete knowledge of the dose–responserelationship. We can overcome this difficulty by implementing the designsequentially. The experiment runs in a trial-by-trial way with the next dosedetermined by the D-optimal criterion updated based on the most recent232.4. Sequential D-optimal designtrial results.Sequential D-optimal designs usually use a pilot study and maximumlikelihood to give an initial estimate of the model parameters. However, thisprocedure can be applied only if the MLE of the model parameters exist.The MLE may not exist under the logistic regression model. According toSilvapulle (1981), the MLE exists if there is an overlapping pattern in thedata. Hence, the sequential D-optimal design generally needs to start withan initial stage which ends when the trial results meet the condition for theexistence of the MLE (Silvapulle, 1981; Albert and Anderson, 1984; Santnerand Duffy, 1986).Wang et al. (2013) and Wang et al. (2015) are recent examples. Fol-lowing Neyer (1991, 1994) and Langlie (1963), they proposed a sequentialtwo–stage D–optimality design. Their designs consist of two stages: an ini-tial stage and a D–optimality stage. The first stage is designed to find anoverlap between stimuli that generate responses and those that generatenonresponses, and tentatively estimate the model parameters. The over-lap guarantees the existence of the MLE of the unknown model parameters(Silvapulle, 1981). The estimates will then be used in the next stage of theexperiment. In the second stage, the parameter estimation is updated aftereach additional trial. The subsequent design points are then selected se-quentially to maximize the determinant of the Fisher information matrix ofthe parameters. The procedure continues until the number of trials reachesthe predetermined sample size.Other relevant literatures on the sequential D-optimal design includeWu (1985a), Wu (1985b), Neyer (1991), Neyer (1994), Dror and Steinberg242.5. Three-phase sequential design(2008), Wu and Tian (2013), among others.2.5 Three-phase sequential designThe design proposed by Wu and Tian (2013) is another interesting sequentialapproach. They developed a three-phase sequential procedure to quickly andefficiently estimate a single ed level.Their design consists of three stages. The first and second stages provideinformation for an initial estimate of the dose–response model. The goal ofthe first stage is to quickly identify a reasonable experimental range bygenerating some responses and nonresponses, and to allocate the designpoints to find an overlap in the data. In the second stage, subsequent designpoints are then chosen to optimize the parameter estimation based on theD-optimal criterion. In the third stage, Robbins-Monro-Joseph procedure(Robbins and Monro, 1951; Joseph, 2004) is applied to cluster the designpoints around the unknown target ed level.Wu and Tian (2013) descried their procedures as follows. Let X or xdenote the dose level of a stimulus. Let Y be the random outcome. Inpyrotechnics study, Y = 1 if the experimental subject has exploded, andY = 0 otherwise. Wu and Tian considered the location-scale modelpi(x) = P (Y = 1|X = x) = f((x− µ)/σ). (2.10)where µ and σ are unknown parameters, and f is a known distributionfunction. Under this model, the effective dose level edγ for some γ ∈ (0, 100)252.5. Three-phase sequential designis the x value such thatpi(edγ) = γ/100.Hence,edγ = µ+ σf−1(γ/100).2.5.1 First stageTo implement the three-phase sequential design, a key ingredient is the up-date of the parameter estimate after each trial. The MLE is a popularchoice, however, it may not exist under the logistic regression model (Silva-pulle, 1981). To ensure the existence of MLE, the first stage aims to finda reasonable range of the design points by generating some responses andnonresponses, and to find an overlap in the data using a searching scheme.See Wu and Tian (2013) for more vivid descriptions of the first stage design.2.5.2 Second stageIn the second stage, the subsequent design points are selected based on theD-optimal criterion. The MLEs of the model parameters are updated aftereach additional trial.• Compute the MLE (µˆs, σˆs) of (µ, σ) based on the observed data(x1, y1), . . . , (xs, ys).• Denote xm = min(x1, . . . , xs) and xM = Max(x1, . . . , xs). To ensurethe estimates are in the design region, Wu and Tian suggested to262.5. Three-phase sequential designtruncate the estimates µˆs and σˆs asµ˜s = Max{xm,min(µˆs, xM )}andσ˜s = Min{σˆs, xM − xm}.Hence, µ˜s is in [xm, xM ], and σ˜s dose not exceed xM − xm.• Then, select the next design point xs+1 such that the determinantof the Fisher information matrix evaluated at (µ˜s, σ˜s) based on theexisting s trials and an extra trial at dose level xs+1 is maximized.• Suppose n1 runs are assigned for the first and second stage of thedesign. The above process will then be repeated until the number oftrials reaches the predetermined size n1.2.5.3 Third stageThe goal of the thirst stage is to cluster the design points around the un-known target ed level. Wu and Tian applied the Robbins-Monro-Josephprocedure. See Robbins and Monro (1951), Lai and Robbins (1979), Joseph(2004), and Wu and Tian (2013) for detailed descriptions of the Robbins-Monro-Joseph procedure. The third stage consists of two main steps.• The first step is to choose an initial value. Wu and Tian appliedxn1+1 = µ˜n1 + f−1(γ/100)σ˜n1272.5. Three-phase sequential designwhere µ˜n1 and σ˜n1 are MLEs of (µ, σ) based on the first n1 observations(x1, y1), . . . , (xn1 , yn1) obtained from the first and second stage of theexperiment. Denote the Fisher information matrix as In1(µ˜n1 , σ˜n1).Compute the inverse of Fisher information matrix.V = In1(µ˜n1 , σ˜n1)−1 = v00 v01v10 v11Here v00 = var(µ˜n1), v11 = var(σ˜n1), and v01 = v10 = cov(µ˜n1 , σ˜n1)are elements of the variance-covariance matrix of (µ˜n1 , σ˜n1). Then letτ21 = v00 + {f−1(γ/100)}2v11.• The second step is to cluster the design points around the targeted level. Denote yn1+1 be the observed response at xn1+1. Wu andTian proposed to select the subsequent design points based on theRobbins-Monro-Joseph iterative scheme (Joseph, 2004),xn1+i+1 = xn1+i − ai(yn1+i − bi), i ≥ 1.Here xn1+i and yn1+i are the ith design point and its correspondingresponse, and ai and bi are some positive constants. See Joseph (2004)and Wu and Tian (2013) for a general discussion on the choice of aiand bi.• Suppose n2 runs are assigned to the third stage of the experiment. Theabove procedure is then repeated until the number of trials reaches thepredetermined size n2.282.6. Up-and-down design2.6 Up-and-down designBoth the D-optimal design and the sequential D-optimal design are possibleonly if a parametric model is assumed for the dose–response relationship.Without a parametric model, the up-and-down design proposed by Dixonand Mood (1948) is an effective way of determining the median effectivedose level ed50. See Pace et al. (2007), among others.The following is a quick description of the up-and-down design. Prior tothe trial, K ordered discrete dose levels,Ω = {x1 < x2 < . . . < xK}are specified based on prior information. The experiment starts with thefirst subject at dose X(1) = x1, or at a level thought to be close to the truetarget dose such as ed50, or at a level selected randomly from Ω. Supposethat the first trial is completed at X(1) = xk. If the observed response ofthe first subject is Y1 = 1, the second subject is assigned to a lower doseX(2) = xk−1. If the observed response of the first subject is Y1 = 0, thesecond subject is assigned to a higher dose X(2) = xk+1. If X(1) = x1 orX(1) = xK , appropriate adjustments are made.Research shows that the up-and-down design tends to assign doses X(i)in the long run clustered unimodally around ed50. Because of this, after ntrials, one may estimate ed50 using the empirical mean µˆ = n−1∑ni=1X(i).Other estimators may also be used. For example, Brownlee et al. (1953)proposed to not include the initial dose X(1) in the calculation of the es-292.7. Biased-coin up-and-down designtimate, but include xn+1, the dose that would have been assigned to the(n+ 1)th subject:µˆ =1nn+1∑i=2X(i)Another commonly used nonparametric estimator for ed50 is the turningpoint estimator (See Wetherill (1963), Choi (1971), Choi (1990), amongothers). If x1, . . . , xn is a sequence of dose levels, we say that there is aturning point at time j, 1 < j < n, if xj−1 < xj and xj > xj+1 (peak)or if xj−1 > xj and xj < xj+1(trough), that is, the sequence of dose levelsturns from increasing to decreasing, or from decreasing to increasing. Let σdenote the difference between successive dose levels. Let t1, t2, . . . , denotethe doses at the turning points (peaks and troughs). Definewi = ti + σ/2 if ti is a troughti − σ/2 if ti is a peak.The turning point estimator for ed50 based on k turning points isw¯ =k∑i=1wi/kThe turning point estimator is widely used in anesthesia up-and-down stud-ies.2.7 Biased-coin up-and-down designThe standard up-and-down design is developed specifically for estimatinged50. If a different ed level is of interest, one may use the biased-coin302.7. Biased-coin up-and-down designup-and-down design of Durham and Flournoy (1994). Let the ed level ofinterest be edγ for some γ ∈ (0, 100). The generalization aims to clusterthe assigned doses around edγ to enable efficient estimation.The biased-coin up-and-down design is as follows. Let the dose level ofthe first subject be X(1).• Suppose the nth subject is assigned at dose X(n) = xk, and respondswith Yn = 0. For γ ≤ 0.5, the (n+ 1)th subject will be assigned at thelower dose X(n+ 1) = xk−1. Otherwise, the (n+ 1)th subject will berandomized with probability b = γ/(100− γ) to the next higher doselevel, and 1− b to the same dose.• Suppose the nth subject is assigned at dose X(n) = xk, and respondswith Yn = 1. For γ > 50, the (n+ 1)th subject will be assigned at thelower dose X(n+ 1) = xk−1. Otherwise, the (n+ 1)th subject will berandomized with probability b = (100− γ)/γ to the next higher doselevel, and 1− b to the same dose.Appropriate adjustments are made, if X(n) = x1 or X(n) = xK , where x1and xK are the lowest and highest dose levels specified in Ω.Durham and Flournoy (1994) showed that the assigned doses X(i) clus-ter unimodally around the target ed level in a biased-coin up-and-downexperiment. They suggested to use the mode of the assigned doses as anonparametric estimator of the target ed level.312.8. Group up-and-down design2.8 Group up-and-down designSometimes, a group of experimental subjects become available at the sametime. Hence, it may be desirable to assign the same dose level to all subjectsin this group. A group up-and-down design has been developed, and it isanother widely used sequential design in clinical trials. The design was firstdescribed by Anderson et al. (1946), followed by Wetherill (1963).Tsutakawa (1967a,b) analyzed the group up-and-down design with thegoal to estimate ed50. Gezmu and Flournoy (2006) generalized their method,and constructed the group up-and-down design to target any ed levels.The group up-and-down design proceeds with groups of s experimentalsubjects for some s. Let clow and cupper be the integers between 0 and s, suchthat 0 ≤ clow < cupper ≤ s. Constants clow and cupper are usually referred toas the cutoff points. Prior to the trial, K ordered discrete dose levels,Ω = {x1 < x2 < . . . < xK}are specified, just like the case of the standard up-and-down design.The experiment starts with the first group at dose X(1), where X(1)may be chosen as the lowest dose level in Ω. The responses of the subjectsfrom the first group are used to determine the dose level assigned to thenext group.• If there are at most clow subjects in the first group with response Y = 1at dose X(1) = xk, the second group are assigned to the next higherdose X(2) = xk+1.322.9. Accelerated biased-coin design• If there are at least cupper subjects in the first group with responseY = 1 at dose X(1) = xk, the second group are assigned to the nextlower dose X(2) = xk−1.• Otherwise, the second group are assigned to the same dose level.If X(n) is at the lowest dose x1 or highest dose xK , appropriate adjustmentsare made.Given the target effective dose level edγ, Gezmu and Flournoy (2006)studied the choice of the cutoff points clow and cupper, and the group size sso that the assigned doses cluster around edγ. Similar to the biased-coinup-and-down designs, the mode of the assigned doses are suggested as anonparametric estimator of the target edγ under the group up-and-downdesign.2.9 Accelerated biased-coin designThe sequential designs discussed above are easy to implement and widelyused in dose–response experiments. In these designs, the dose assigned to asubject depends on the response of the preceding subject. Therefore, a newsubject cannot enter the trial until the preceding subject has responded. Toutilize these sequential designs, a subject’s response needs to be observedquickly, otherwise, the above designs may cause long trial duration, whichis obvious not desirable in clinical trial practice.Motivated by this, Stylianou and Follmann (2004) proposed to modifythe biased-coin up-and-down design to deal with the situation that a sub-332.10. Generalized Po´lya Urn designject’s response to a stimulus is not observed quickly. Their idea is to assigndoses to subjects as they enter the trial based on the response of the lastsubject who has completed the trial. This modification allows researchersto evaluate several subjects simultaneously.Their design follows the dose assigning paradigm of the biased-coin up-and-down design, and is referred to as the accelerated biased-coin up-and-down design. If a subject enters the trial before the response of the precedingsubject has been observed, the new subject will be assigned to a dose basedon the last observed response.Stylianou and Follmann (2004) compared the accelerated design withthe biased-coin up-and-down design, and found that the accelerated designgreatly reduces the duration of the trial, and does not affect the estimationprecision of the target ed level, when estimated by an isotonic regressionestimator (Stylianou and Flournoy (2002)).2.10 Generalized Po´lya Urn designRosenberger and Grill (1997) proposed a sequential design procedure basedon the generalized Po´lya urn (GPU) model from Athreya and Ney (2012)to efficiently estimate ed50, while potentially estimate other ed levels suchas ed25 and ed50.The GPU model can be used to design dose–response experiments (Rosen-berger, 1996; Rosenberger and Grill, 1997). A number of does levels areprespecified. The procedure starts with a urn containing a population ofparticles. Each particle is labeled with a dose level. A particle is drawn at342.10. Generalized Po´lya Urn designrandom and its dose level is assigned to the experimental subject. If theexperimental subject responds (or does not respond), one particle for eachof the next k lower (or higher) doses are added to the urn. The procedureis then repeated until the end of the experiment.Rosenberger and Grill (1997) suggested k = 5 and referred to this schemeas the 5-up/5-down rule. They found the design points are unimodally dis-tributed around ed50. To better estimate other ed levels such as ed25 anded50, they proposed to alter the design so that the dose levels would spreadout further. Based on the simulation result of Rosenberger and Grill (1997),the GPU design is efficient for estimating ed50. However, its performancefor estimating other ed levels such as ed25 and ed75 are variable when thenumber of trials is small, but its performance improves when the numberincreases.The sequential designs discussed above are simple to implement in prac-tice. However, they all aim to estimate a single ed level of dose–responsecurves. If one is interested in knowing the dose–response relationship overa dose range, these designs are not most appropriate.35Chapter 3Two-stage SequentialED-Design3.1 IntroductionIn the last two chapters, we have introduced and reviewed many dose–response models and some related design issues. We pointed out that manyclassical designs such as the up-and-down design and its generalizations inthe literature aim to most accurately estimate a specific ed level. The D-optimal and other optimal designs aim to most accurately estimate modelparameters. While a more accurate parameter estimation should generallylead to more accurate characterization of the dose–response relationship,these two goals are not equivalent. For this reason, we propose a new ap-proach in designing a binary experiment, and investigate whether it leadsto a more efficient model estimation from this angle of interest.More specifically, we study a situation where the dose–response relation-ship over a range of ed levels is of interest. We believe such a relationshipcan be well characterized after several carefully chosen ed levels are accu-rately estimated simultaneously. Based on these considerations, we propose363.2. New criteriona two-stage sequential ED-design.In the next section, we will give a detailed description of the proposeddesign.3.2 New criterionAssume that the dose–response relationship is given by the conditional prob-ability mass function f(y;x, θ) where θ is the model parameter. We assumethat the total number of trials n will be used to obtain data for the modelfitting.Under the parametric model assumption, each ed level ξ is a smoothfunction of the unknown parameter θ: ξ = g(θ). Suppose that followingsome scheme, i trials have been carried out. Let θˆi be an estimate of θbased on the data obtained from the first i trials. When i is large, thevariance-covariance matrix of θˆi is well approximated by the inverse of theFisher information Ii(θ). The variance of g(θˆi) is therefore approximately{5gτ (θ)}{I−1i (θ)}{5g(θ)} (3.1)where 5g(θ) is the gradient of g(·).When i is small, the approximate variance (3.1) is not accurate. Nev-ertheless, it remains a good metric of the relative informativeness of thedata collected so far. Because of this, we propose to select dose (i + 1) tominimize the total observed variance of several ed levels chosen by the user.Let Ii(θ; +x) be the Fisher information based on the first i trials and the373.2. New criterionpotential (i+1)th trial to be run at dose level x. The proposed ED-criterionis to select the next dose level x which minimizesK0∑j=1{5gτj (θˆi)}{I−1i (θˆi; +x)}{5gj(θˆi)} (3.2)where gj(θ) = ξj are K0 selected target ed levels. In this thesis, we useK0 = 3, and our criterion will not be affected if other K0 values are selected.Starting from some K initial pilot trials, the sequential ED-criterionis used to select dose levels for the (K + 1)th, (K + 2)th trials until theexperiment terminates.Let {(Xi, Yi), i = 1, 2, . . . , n} be the doses and responses. Under asequential design such as our proposed one, they are not independent ofeach other. However, the dependence of Yi on (Y1, X1), . . . , (Yi−1, Xi−1)is only through Xi. As a consequence, the likelihood constructed from(Y1, X1), . . . , (Yi, Xi) retains a product form∏ir=1 f(Yr;Xr, θ) despite thedependence structure of the data arising from the sequential design (Chaud-huri and Mykland, 1993, 1995). Hence, the likelihood based on the proposedsequential design is identical to that arising from the independent observa-tions. A general discussion on the validity of this likelihood function will begiven in Chapter 6.The ED-design can be implemented with θ estimated by its MLE se-quentially. It is natural to have all target edγ and the entire dose–responsecurve estimated based on the likelihood method. We now discuss detailsunder the logistic model.383.3. Sequential ED-design under the logistic model3.2.1 The pilot experimentTo apply the proposed ED-design, we need a pilot experiment to give us aninitial parameter estimation. We propose to identify a dose range coveringthe anticipated dose levels ed10 to ed90. We then create set Ω with k doselevels from ed10 to ed90. The data collected in the pilot experiment willbe used to provide an initial parameter estimation for the implementationof the proposed ED-design sequentially.3.3 Sequential ED-design under the logisticmodelRecall that the logistic dose–response model assumes thatlogit{pi(x)} = log{pi(x)/(1− pi(x))} = α+ βx. (3.3)Under this model, θ = (α, β)τ and the probability mass functionP (Y = 1;X = x, θ) = f(1;x, θ) = pi(x)andP (Y = 0;X = x, θ) = f(0;x, θ) = 1− pi(x)withpi(x) = [1 + exp(−(α+ βx))]−1.To implement our proposed ED-design and other sequential designs, a393.3. Sequential ED-design under the logistic modelkey ingredient is the update of the parameter estimate after each trial. TheMLE is a popular choice in the literature. However, the MLE may not existunder the logistic regression model. Suppose A1 is the set of dose levels atwhich the subjects responded to the stimulus and A0 is the set of dose levelsat which the subjects did not respond. The MLE exists only if the convexhulls of A1 and A0 overlap (Silvapulle, 1981; Albert and Anderson, 1984).Particularly in simulation studies or the early stages of a sequential ex-periment, the above condition may not be satisfied. Various suggestionshave been made in the literature. For instance, one may extend the pilotexperiment until the data collected permit a valid MLE. In this dissertationwe investigate a novel approach.In most applications, the user has some idea on a sufficiently low doselevel at which the subject will not respond (Y = 0), and a high enough doselevel at which the subject will respond (Y = 1). Our idea is to make useof such prior information in a non-Bayesian way. For this purpose, let s1and s2 be the anticipated ed01 and ed99 values. Based on this, we createfour pseudo-outcomes: at dose level s1, we create two weighted responses ofY = 0 and Y = 1 with weights 0.99 and 0.01; at dose level s2, we create twoweighted responses of Y = 0 and Y = 1 with weights 0.01 and 0.99. After Kobservations obtained from some pilot experiment, they are expanded withthese four pseudo-outcomes. It can be seen that the resulting A1 and A0 haveoverlapping convex hulls. Hence, the MLE based on the expanded data setunder the logistic regression model always exists. The pseudo-outcomes areclearly based on our prior knowledge, which gives this approach a Bayesianflavour. However, the prior information is not accommodated as a prior403.3. Sequential ED-design under the logistic modeldistribution on the parameter value.Given a dose level x, the Fisher information based on a single trial underthe logistic model is given by pi(x){1− pi(x)} xpi(x){1− pi(x)}xpi(x){1− pi(x)} x2pi(x){1− pi(x)} .The Fisher information after n trials with doses x1, . . . , xn is given byIn(α, β) =n∑i=1 pi(xi){1− pi(xi)} xipi(xi){1− pi(xi)}xipi(xi){1− pi(xi)} x2ipi(xi){1− pi(xi)} .If an additional trial were carried out at dose level x, the Fisher informationwould beIn(α, β; +x) = In(α, β) + pi(x){1− pi(x)} xpi(x){1− pi(x)}xpi(x){1− pi(x)} x2pi(x){1− pi(x)} .Note that the ed level is related to the model parameter byξγ = EDγ = gγ(α, β) =logit(γ/100)− αβ.Under the current model, for a generic ed level ξ, (3.2) becomesv00 + 2ξv01 + ξ2v11β2(3.4)where vij are the elements of I−1(α, β).Recall that we use Ii(α, β; +x) for the Fisher information after i trials,413.3. Sequential ED-design under the logistic modeland another trial at the proposed dose level x. We usevˆ00i (x), vˆ01i (x), vˆ10i (x), vˆ11i (x)for the elements of Ii(αˆ, βˆ; +x) where αˆ, βˆ are estimated parameter valuesbased on the first i trials and the pseudo-observations.In the second stage of our sequential ED-design, we choose the (i+ 1)thdose level asxi+1 = arg minx3∑j=1{vˆ00i (x) + 2ξˆj vˆ01i (x) + ξˆ2j vˆ11i (x)}.Clearly, the sequential ED-design can be used for any number of ed levels.The numerical computation of xi+1 is also easy: a simple linear search suf-fices. Hence, the new design can satisfy a broad range of the demands thatmay arise in applications.More specially, let us suppose ed25, ed50 and ed75 are the target ed lev-els. We now demonstrate how to select the next dose level in the ED-design.Denote ED25, ED50 and ED75 as ξj , j = 1, 2, 3. Note thatξj =logit(pi)− αβwith pi being one of 0.25, 0.50 and 0.75.The ξj value can be written as a function of α and β, i.e., ξj = g(α, β).Let ξˆj = g(αˆk, βˆk), via the delta-method, one can easily obtained the asymp-423.4. Simulation studiestotic variance of g(αˆk, βˆk) as follows.var(g(αˆk, βˆk)) = 5g(αˆk, βˆk)T I−1k (x; αˆk, βˆk)5 g(αˆk, βˆk) (3.5)where5g(αˆk, βˆk) is the gradient of g(αˆk, βˆk), and I−1k (x; αˆk, βˆk) is the inverseof the Fisher information matrix. Then the asymptotic variance of ξˆj , forj = 1, 2, 3, isvar(ξˆj) =v00 + 2ξˆjv01 + (ξˆj)2v11βˆ2k(3.6)Here v00 = var(αˆk), v11 = var(βˆk), and v01 = v10 = cov(αˆk, βˆk) are elementsof the variance-covariance matrix of (αˆk, βˆk),V = I−1k (x; αˆk, βˆk) = v00 v01v10 v11Then the (k+1)th dose level is determined by minimizing the total varianceof ξˆj , for j = 1, 2, 3, i.e.,xk+1 = arg minx3∑j=1var(gj(αˆk, βˆk)).3.4 Simulation studiesWe conduct simulations to investigate the performance of the ED-design.We compare the new design with existing designs including the standardup-and-down design, the D-optimal design, and the two-stage D-optimaldesign.433.4. Simulation studies3.4.1 Detailed specificationsWe investigate the performance of the designs through a hypothetical dose–response experiment with a binary outcome. For any given design, theresponse values of the experiment are generated at the dose levels prescribedby the design according to the assumed dose–response relationship. The goalof the experiment is to estimate the dose–response curve, namely f(y; θ, x).The detailed specifications are as follows.1. Up-and-down design: As discussed in Section 2.6, this design placesthe doses on a grid of prespecified dose levels:Ω = {x1, . . . , xK}for some K. In this simulation, we choose K = 7 with x1 and xK beingthe anticipated ed01 and ed90. The choice of the first dose level willbe decided case by case in the simulation.2. D-optimal design: According to Sitter and Wu (1993), a D-optimaldesign for the logistic response curve is a uniform distribution on twodose levels: ed17.6 and ed82.4. Hence, we assign half of the subjectsto ed17.6 and half to ed82.4, since the data generating dose–responsecurves are known in the simulation.3. Two-stage D-optimal design: We first form a grid of nine doses fromthe anticipated ed10 to ed90. The first stage is carried out at themiddle k = 7 dose levels. The subsequent dose x is chosen to maximizedet{Ii(θˆi; +x)}.443.4. Simulation studies4. Sequential ED-design: The first seven trials are at the dose levels forthe two-stage D-optimal design. The subsequent dose x is chosen tominimize (3.2).The two-stage D-optimal design of Wang et al. (2015) has a complexscheme for its first stage. We have replaced this with our own more practicalfirst-stage design. In all cases, we obtain the MLE for θ at the conclusion ofthe n trials. We repeat the simulation N times for all designs. The RMSEsare computed as follows:RMSE(ξˆj) =√√√√N−1 N∑r=1(ξˆrj − ξj)2,where ξˆrj is the estimate of ξj in the rth repetition. The overall RMSE iscomputed asRMSE =√√√√ 3∑j=1RMSE2(ξˆj).In this study, we choose N = 1000 and the sample sizes n = 30, 60, and 120.3.4.2 Performance comparison when the response model iscorrectly specifiedIn applications, we do not know the true form of the dose–response curve orthe corresponding parameter values. Yet all designs must start with a guessof the true response curve. In this section, we consider the situation wherethe observed response curve agrees well with the true curve. In particular,453.4. Simulation studieswe generate data according to the logistic regression modellogit[pi(x)]= −6.265 + 0.055x. (3.7)These parameter values are taken from Gezmu and Flournoy (2006). Theyillustrated their group up-and-down design using the same example fromFlournoy (1993). The drug studied in the example is cyclophosphamide,measured in mg/kg. This response model was constructed from expert opin-ion as described in Flournoy (1993). Under this model, ed25= 94, ed50 =114, and ed75 = 134.The details of the four designs to be simulated under this model are asfollows:• For the up-and-down design, the specific dose levels are x1 = 34 andx7 = 154. The dose range is given byΩ = {34, 54, 74, 94, 114, 134, 154}.The initial dose level is set to x4 = 94.• For the D-optimal design, the optimal dose levels are ed17.6 = 86 anded82.4 = 143.• For the two-stage D-optimal design and the ED-design, we use thefollowing grid of K = 7 doses in the first stage:Ω = (84, 94, 104, 114, 124, 134, 144).463.4. Simulation studiesTable 3.1: Simulated RMSEs under the logistic model targeting range ed25–ed75n ED-design Two-stage D Up-and-down D-optimal30Total 17.14 18.05 18.12 20.18ED25 10.55 10.88 10.48 12.48ED50 8.12 9.03 8.06 9.89ED75 10.79 11.21 12.39 12.4060Total 12.54 12.83 12.79 12.63ED25 7.80 7.67 7.90 7.70ED50 5.89 6.42 5.53 6.15ED75 7.85 8.03 8.40 7.90120Total 8.74 9.01 9.16 8.87ED25 5.46 5.51 5.89 5.49ED50 4.09 4.50 3.88 4.41ED75 5.46 5.53 5.85 5.39In the first simulation, we choose ed25, ed50, and ed75 as the targets.The results are given in Table 3.1.The results show that our ED-design has the lowest total RMSE whenn = 30. Its RMSE is generally lower when n = 60 and n = 120, but thedifferences are smaller. Our design is noticeably superior to the two-stageD-optimal design. If ed50 is the target, then the up-and-down approach iscompetitive or superior. However, even in this case, our ED-design is amongthe best.It may be surprising that the D-optimal design based on a known dose–473.4. Simulation studiesresponse relationship is not the best. This may be explained by the fact thatD-optimal designs aim to maximize the determinant of the Fisher informa-tion matrix. However, the performance measure in this simulation is theRMSE. Indeed, the averages of the determinants of the Fisher informationof the ED-design divided by n2 for n = 30, 60, and 120 are 15.4, 14.4, and14.6. The corresponding value for the D-optimal design is 17.08.In the second simulation, we consider the situation where a lower rangeof ed levels is of interest. We use ed10, ed25, and ed40 as the targets. Thesimulation settings remain the same except that the initial dose level forthe biased-coin up-and-down design is set to x3 = 74. This approach canselect only one target dose level in each simulation. We simulate all threepossibilities and Table 3.2 gives the results for each case.Adjusting the target ed levels of our ED-design has the desired effect.The resulting data enable much more efficient estimation over the targetrange in terms of the total RMSE. In comparison with both D-optimal de-signs, it has the lowest RMSE at both ed10 and ed25. The difference issmaller at ed40, but our design has the lowest RMSE levels in two of thethree sample sizes simulated. Tuning the biased-coin up-and-down designto specific ed levels improves its results. Particularly when n = 120, theup-and-down design achieved the lowest RMSE at the targeted ed level.However, this is at the cost of lower precision at the other ed levels. If thegoal is to determine a single ed level, the up-and-down design is the bestapproach.We repeat the first and second simulation studies 20 times under thesame simulation setting. We note that the resulting total RMSEs and indi-483.4. Simulation studiesTable 3.2: Simulated RMSEs under the logistic model targeting range ed10–ed40.n ED-design Two-stage DUp-and-downD-optimalTargetED10TargetED25TargetED4030Total 17.93 20.81 21.82 18.81 20.24 23.58ED10 12.68 15.13 11.61 13.18 15.59 17.53ED25 8.93 10.84 11.15 8.92 9.71 12.15ED40 9.01 9.29 14.74 10.02 8.51 10.0760Total 13.25 15.23 17.38 13.25 15.02 15.79ED10 9.48 11.13 8.57 9.69 11.90 11.96ED25 6.52 7.89 8.98 6.09 7.11 8.02ED40 6.56 6.75 12.17 6.69 5.78 6.46120Total 9.35 10.76 13.38 9.36 10.57 10.87ED10 6.73 7.90 6.14 6.93 8.52 8.02ED25 4.55 5.58 6.96 4.24 4.92 5.63ED40 4.63 4.70 9.64 4.66 3.88 4.69vidual RMSEs are quite similar. For example, when ed25, ed50, and ed75are the targets, the standard error between the resulting total RMSEs is0.12 for n = 30; when ed10, ed25, and ed40 are the targets, the standarderror between the resulting total RMSEs is 0.26 for n = 30.493.4. Simulation studies3.4.3 Performance comparison when the response model ismis-specifiedIn applications, the dose–response relationship is unknown. In this section,we consider the case where the observed response curve is mis-specified.Specifically, we consider the case where the observed dose–response rela-tionship is logistic but the true model is probit. Thus, we generate dataaccording to the probit model,probit(pi) = Φ−1(pi) = −6.265 + 0.055x (3.8)where Φ(·) is the cumulative distribution function of the standard normaldistributionΦ(z) =1√2pi∫ z−∞exp{−12x2}dx.Note that under model (3.8), we haveedγ =Φ−1(γ/100)− αβ.Under this model, ed25 = 102, ed50 = 114, and ed75 = 127. The dose–response curve is assumed to be model (3.7), and the simulation is otherwiseidentical to that in the last section. The results are presented in Tables 5.3and 5.4. For ed25–ed75, we set ed50 as the target for the up-and-downdesign. For ed10–ed40, we target each level separately, as before.With the same regression coefficients, the probit model has a steeperslope in the range ed25–ed75 compared with the logistic model. The se-quential designs seem to have some ability to recover from the mis-specified503.4. Simulation studiesTable 3.3: Simulated RMSEs under probit mis-specified as logistictargeting ED range 25–75.n ED-design Two-stage D Up-and-down D-optimal30Total 10.08 10.59 11.29 18.93ED25 6.25 6.33 6.70 11.45ED50 4.86 5.36 5.03 10.20ED75 6.23 6.59 7.57 11.0960Total 7.16 7.46 7.63 13.98ED25 4.53 4.56 4.71 8.23ED50 3.38 3.76 3.46 7.50ED75 4.39 4.55 4.90 8.46120Total 5.00 5.05 5.39 9.13ED25 3.09 3.09 3.38 5.47ED50 2.38 2.54 2.43 4.88ED75 3.13 3.09 3.42 5.44model, and their RMSEs are hence lower than those in the last section. OurED-design clearly has the best overall performance in both ranges. Theup-and-down design again has good performance at the target ed level butpoorer performance overall. Targeting ed25 achieves the best trade-off.Both the logistic and probit models are symmetric in the ed levels: edγ+ ed(100− γ) = 2 × ed50 for any γ ∈ (0, 100).In the following example, we generate data according to the modellogit[pi(x)]= −11.95 + 1.12√x, (3.9)513.4. Simulation studiesTable 3.4: Simulated RMSEs under probit mis-specified as logistictargeting ED range 10–40.n ED-design Two-stage DUp-and-downD-optimalTargetED10TargetED25TargetED4030Total 11.16 12.53 14.54 11.62 11.81 20.76ED10 7.80 9.19 6.96 7.55 8.82 13.81ED25 5.59 6.46 7.84 5.92 5.87 11.46ED40 5.69 5.57 10.07 6.56 5.24 10.4460Total 7.62 8.65 10.73 8.03 8.45 14.93ED10 5.34 6.41 4.84 5.57 6.38 9.88ED25 3.78 4.44 5.71 3.99 4.21 8.23ED40 3.90 3.74 7.68 4.20 3.59 7.58120Total 5.30 5.84 7.70 5.56 5.94 9.85ED10 3.76 4.32 3.53 3.83 4.41 6.51ED25 2.59 3.01 4.14 2.79 3.02 5.47ED40 2.69 2.55 5.45 2.91 2.59 4.98but we again analyze the data under model (3.7). We choose the aboveparameter values so that the corresponding ed levels (i.e., ed25, ed50, anded75) roughly match the ed levels derived from model (3.7). Note thatunder model (3.9),ξj =[ logit(pi)− αβ]2.Under this model, ed25 = 94, ed50 = 114, and ed75= 136. As discussed523.4. Simulation studiesin Section 2.6, for the first stage, we use a grid of K = 7 doses:Ω = (84, 94, 104, 114, 124, 134, 144).The subsequent dose x is chosen to minimize (3.2).In this simulation, we first consider the situation where the lower ed lev-els are of interest, choosing ed10, ed25, and ed40 as the targets. For thebiased-coin up-and-down design, we select ed25 as the target with the ini-tial dose level at x2 = 94. This choice gives the best performance. Thesimulation is repeated for ed25, ed50, and ed75. The results are given inTables 3.5 and 3.6.The results show that our ED-design has a lower total RMSE when thelower ed levels are of interest. It helps to improve efficiency over the wholerange of interest and is particularly efficient at ed10. When the middleed levels are of interest, the ED-design remains competitive, and none ofthe three designs is a clear winner.533.5. Limiting design as n increasesTable 3.5: Simulated RMSEs under the mis-specified logistic modeln ED-design Two-stage D Up-down D-optimal30Total 16.47 19.22 17.68 24.40ED10 11.61 13.94 11.76 18.14ED25 8.13 10.00 8.64 12.55ED40 8.39 8.68 9.99 10.4360Total 12.33 14.57 12.76 16.53ED10 8.82 10.75 8.89 12.52ED25 5.88 7.49 6.06 8.34ED40 6.28 6.39 6.88 6.87120Total 8.87 10.29 9.29 11.57ED10 6.17 7.39 6.44 8.65ED25 4.20 5.27 4.46 5.84ED40 4.80 4.86 4.99 4.993.5 Limiting design as n increasesWe have so far focused on the performance of the ED-design. Recall that adesign for a binary dose–response experiment is equivalently a probabilitydistribution on the design space or dose range. The D-optimal design isknown to be a two-support-point design at ed17.6 and ed82.4 under thelogistic regression model. Because of its sequential nature, the ED-designhas many more support points. When n goes to infinity, it is possible thatthe limiting distribution has two support points.We simulated the ED-design for two scenarios with n = 5000, first tar-543.5. Limiting design as n increasesTable 3.6: Simulated RMSEs under the mis-specified logistic modeln ED-design Two-stage D Up-down D-optimal30Total 17.53 17.56 19.12 20.29ED25 10.23 10.00 10.70 12.55ED50 8.06 8.63 8.54 10.02ED75 11.72 11.56 13.35 12.4160Total 13.18 12.83 13.58 13.55ED25 7.82 7.49 7.53 8.34ED50 5.99 6.28 5.81 6.61ED75 8.76 8.32 9.69 8.40120Total 9.26 9.61 9.41 9.49ED25 5.36 5.27 5.55 5.84ED50 4.36 4.93 4.01 4.83ED75 6.16 6.35 6.45 5.70geting ed25, ed50, ed75 and then ed10, ed25, ed40. In the first scenariothe observed and true response curves agree and are logistic. In the secondscenario the observed dose–response model is logistic and the true model isprobit. The resulting histograms are given in Figures 3.1 and 3.2. There areclear indications that in both scenarios, the limiting distribution is binomialaround ed20 and ed80 when the target ed levels are ed25, ed50, and ed75.We hope to prove this in the future.553.5. Limiting design as n increases(a)0500150025001 6 11 21 36 56 76 91 96(b)0500150025001 6 11 22 36 51 66 80 91 96Figure 3.1: Histogram of the ED-design for (a) estimating ed25, ed50, anded75 when the response curve is correctly specified as logistic; (b) estimat-ing ed25, ed50, and ed75 when the response is mis-specified; The x-axescorrespond to the ed levels.563.5. Limiting design as n increases(c)0500150025001 6 11 21 36 56 76 91 96(d)0500150025001 6 11 22 36 51 66 80 91 96Figure 3.2: Histogram of the ED-design for (c) estimating ed10, ed25, anded40 when the response curve is correctly specified as logistic, and (d) es-timating ed10, ed25, and ed40 when the response is mis-specified. Thex-axes correspond to the ed levels.573.6. Concluding remarks3.6 Concluding remarksIn dose–response experiments, there may be insufficient or inaccurate knowl-edge of the dose–response curve for the dose levels to be chosen properly.Dose–response information gathered from such an experiment is often unre-liable.We have therefore proposed a two-stage sequential ED-design for suchexperiments that unitize a second stage sequential experiment to compensatefor the scarcity or inaccuracy of the dose–response information in the firststage experiment.Our design simultaneously targets several ed levels of the underlyingdose–response curve. We propose that the dose–response relationship canbe well described by accurately estimating several ed levels simultaneously.Simulations are conducted to investigate the performance of the proposeddesign under various scenarios. They are designed to mimic a real dose–response experiment with the goal to estimate the unknown dose–responsecurve over a wide dose range. Simulations show that in general our designis more robust and compares favourably with existing designs.Although the commonly used logistic and probit models are convenientsummaries of the dose–response relationship, they can be too restrictive.Our ED-design has a natural extension to more complex models. This willbe seen in the next chapter.583.7. R-code for the ED-design and simulation3.7 R-code for the ED-design and simulation# define functionq.logit <- function (x) {p = (logit(x) - alpha ) / betareturn (p)}q.probit <- function (x) {p = (qnorm(x) - alpha ) / betareturn (p)}logit <-function(p) log(p/(1-p))g <- function(x, alpha, beta) alpha + beta*xl <- function(g) exp(g)/(1+exp(g))p <- function(x, alpha, beta) l(g(x, alpha, beta))# Function f.fisher() computes the fisher information matrixf.fisher <- function(x) {a <- alpha_hat + beta_hat * xfisher <- matrix(c(sum(exp(a)/(1+exp(a))^2),sum(x*exp(a)/(1+exp(a))^2),sum(x*exp(a)/(1+exp(a))^2),sum(x^2*exp(a)/(1+exp(a))^2)), 2, 2)return(fisher)593.7. R-code for the ED-design and simulation}# function f.dose() selects the next dose at the second stagef.dose <- function (x){a.sel <- alpha_hat + beta_hat * xfisher.sel <- fisher + matrix(c(exp(a.sel)/(1+exp(a.sel))^2,x*exp(a.sel)/(1+exp(a.sel))^2,x*exp(a.sel)/(1+exp(a.sel))^2,x^2*exp(a.sel)/(1+exp(a.sel))^2), 2, 2)cov.sel <- solve(fisher.sel)v00.sel <- cov.sel[1,1]v01.sel <- cov.sel[1,2]v11.sel <- cov.sel[2,2]# varianevar1.sel <- (v00.sel + 2*v01.sel*mu_hat1 + v11.sel*mu_hat1^2) / beta_hat ^ 2var2.sel <- (v00.sel + 2*v01.sel*mu_hat2 + v11.sel*mu_hat2^2) / beta_hat ^ 2var3.sel <- (v00.sel + 2*v01.sel*mu_hat3 + v11.sel*mu_hat3^2) / beta_hat ^ 2sum.se <- var1.sel + var2.sel + var3.selreturn(sum.se)}# initial designn1 <- 7603.7. R-code for the ED-design and simulationdelta <- (max-min)/(n1+1)c <- seq (from = 1, to = n1, by = 1)x1 <- min + delta * c[1]x2 <- min + delta * c[2]x3 <- min + delta * c[3]x4 <- min + delta * c[4]x5 <- min + delta * c[5]x6 <- min + delta * c[6]x7 <- min + delta * c[7]dose.initial <- c(x1, x2, x3, x4, x5, x6, x7)# proposed.R Function# add two pseudo points on each boundary point# ensure the existence of MLEy_pes1 = 1y_pes2 = 0x_pes1 = (logit (0.01) - alpha) / betax_pes2 = (logit (0.99) - alpha) / betay_pes3 = 0y_pes4 = 1x_pes3 = (logit (0.01) - alpha) / betax_pes4 = (logit (0.99) - alpha) / betafor (k in 1:m)613.7. R-code for the ED-design and simulation{# change with different response modelsyhat <- rbinom(length(dose.initial), 1, p(dose.initial, alpha, beta))data <- data.frame(rbind(cbind (y_pes1, x_pes1), cbind (y_pes2, x_pes2),cbind (y_pes3, x_pes3), cbind (y_pes4, x_pes4),cbind (yhat, dose.initial)))names(data)[1] <- paste("y")names(data)[2] <- paste("x")# weight for the initial experimentweight <- c(0.01, 0.01, 0.99, 0.99, rep (1, 7))fit <- glm(y ~ x, weights = weight, data = data, family = binomial)alpha_hat <- fit$coef[[1]]beta_hat <- fit$coef[[2]]r <- dose.p (fit, p = ed)mu_hat1 <- r[[1]]mu_hat2 <- r[[2]]mu_hat3 <- r[[3]]# Second stage# Select the next dose levelfor (j in 1:(n-7)){w <- c(0.01, 0.01, 0.99, 0.99, rep (1, 7+j-1)) # weight# calculate fisher information623.7. R-code for the ED-design and simulationx <- data $ xa <- alpha_hat + beta_hat * xfisher <- matrix(c(sum(w*exp(a) / (1+exp(a))^2),sum(w*x*exp(a) / (1+exp(a))^2),sum(w*x*exp(a) / (1+exp(a))^2),sum(w*x^2*exp(a)/ (1+exp(a))^2)), 2, 2)# choose the next dosagefor (i in 1:length(t)){sum[i] <- f.dose(t[i])}opt <- t[which(sum == min (sum))]# change with different response modelsprob <- p(opt, alpha, beta)y <- rbinom (1, 1, prob)data <- rbind (data, data.frame (y, x = opt))weight_opt <- c(0.01, 0.01, 0.99, 0.99, rep (1, 7 + j))fit <- glm(y ~ x, data, weights = weight_opt, family = binomial)alpha_hat <- fit$coef[[1]]beta_hat <- fit$coef[[2]]mu_hat1 <- r[[1]]mu_hat2 <- r[[2]]633.7. R-code for the ED-design and simulationmu_hat3 <- r[[3]]}weight_final <- c(0.01, 0.01, 0.99, 0.99, rep (1, n))fit <- glm(as.factor(y) ~ x, data = data,weights = weight_final, family = binomial)r <- dose.p (fit, p = ed)mu_temp1[k] <- r[[1]]mu_temp2[k] <- r[[2]]mu_temp3[k] <- r[[3]]mu_temp [k, ] <- c(mu_temp1[k], mu_temp2[k], mu_temp3[k])diff1[k] <- (mu_temp1[k] - mu1) ^ 2diff2[k] <- (mu_temp2[k] - mu2) ^ 2diff3[k] <- (mu_temp3[k] - mu3) ^ 2diff[k] <- diff1[k] + diff2[k] + diff3[k]datatotal[, 2*k -1] <- c(data[, 1])datatotal[, 2*k] <- c(data[, 2])print(k)}# General settingsrm(list=ls())graphics.off()library(stats)library(MASS)643.7. R-code for the ED-design and simulationlibrary(logistf)n <- 30m <- 1000alpha <- -6.2647min <- 74max <- 154ed <- c(0.25, 0.50, 0.75)a1 <- (logit(0.01)-alpha)/betaa2 <- (logit(0.99)-alpha)/betat <- seq(from = a1, to = a2, by = 5)mu1 <- (logit (ed[1]) - alpha) / betamu2 <- (logit (ed[2]) - alpha) / betamu3 <- (logit (ed[3]) - alpha) / betamu <- c(mu1, mu2, mu3)# define objectsdiff1 <- rep (0, m)diff2 <- rep (0, m)diff3 <- rep (0, m)diff <- rep (0, m)mu_temp1 <- rep (0, m)mu_temp2 <- rep (0, m)mu_temp3 <- rep (0, m)mu_hat1 <- rep (0, m)mu_hat2 <- rep (0, m)653.7. R-code for the ED-design and simulationmu_hat3 <- rep (0, m)sum <- rep(0, length(t))# main part# proposed two-stagesource("initialDesign.R")source("proposed.R")# outputsmse1 <- mean(diff1)mse2 <- mean(diff2)mse3 <- mean(diff3)mse <- mean(diff) #average msesqrt(c(mse1, mse2, mse3, mse))66Chapter 4ED-design under theThree-parameter LogisticModelIn the last chapter, we have explored the two-stage sequential ED-designunder the logistic and probit models. Simulation studies show the ED-designis more robust and compares favourably with existing designs. Although thecommonly used logistic and probit models are convenient summaries of thedose–response relationship, they can be too restrictive in applications. OurED-design has a natural extension to more complex models, and we willexplore this in this chapter.4.1 Problem descriptionNaturally, if the model is mis-specified in an application, the optimal designis then misguided, and the resulting data analyses may cause an unreliableestimation of the ed levels. One way to lower this risk is to design the ex-periment that targets an accurate estimation of a range of ed levels, instead674.1. Problem descriptionof a single median dose level or model parameters as we discussed in the lastchapter.Another apparent approach to lower the risk of model misspecificationis to apply a more flexible and hence more complex dose–response model.The choice of such a model invariably reflects a trade-off between the modelflexibility and inference efficiency. A nonparametric model has ultimate flex-ibility, and therefore is free from the risk of model misspecification. How-ever, it likely needs more trials to achieve the same estimation precisioncompared with the analyses under approximately valid parametric modelassumptions. Commonly used logistic or probit models are simple and havegood mathematical and statistical properties. They are satisfactory in manyapplications. Nevertheless, their model assumptions do impose some severerestrictions on the dose–response relationship. Hence, a mildly more com-plex model can be useful to lower the risk of model misspecification if it doesnot complicate the issues related to optimal designs and data analyses, aswell as maintaining good efficiency in estimating the ed levels.In this chapter, we show that the three-parameter logistic regressionmodel goes some distance in this direction. We investigate the effectivenessof the sequential ED-design, the D-optimal design, and the up-and-downdesign under this model, and develop an effective model fitting strategy. Wedevelop an easy way to implement an iterative numerical algorithm withguaranteed convergence for computing the maximum likelihood estimationof the model parameters. The sequential ED-design can be implementedafter some laborious but simple mathematical derivations. Although we haveyet to generate any theory on its D-optimal designs, a numerical procedure684.2. Three-parameter logistic modelvia the well-developed vertex direction method (VDM) works well.Simulation studies show that the combination of the proposed model andthe data analysis strategy performs well. When the logistic model is correct,applying the more complex model suffers hardly any efficiency loss. Whenthe three-parameter model holds but the logistic model is violated, the newapproach is more efficient. Our research is a useful addition to the toolboxof the dose–response experiment.4.2 Three-parameter logistic modelStatisticians and scientists are keenly aware that both the logistic and probitmodels can be poor approximations of the true dose–response relationshipin an application. A more flexible model can be advantageous if it does notcause complex issues. The three-parameter logistic dose–response modelintroduced in Chapter 2 ideally meets this demand. El-Saidi (1993) havealready proposed the use of this model for the dose–response relationship.The three-parameter logistic regression model assumes thatlogit(piλ(x)) = ln{ piλ(x)1− piλ(x)}= α+ βx. (4.1)We require λ > 0 to ensure piλ(x) is between 0 and 1, and do not placerestrictions on α and β.We note that when λ = 1, the three-parameter model becomes the com-monly used logistic model. In this case, for any γ ∈ (0, 100), the model694.3. Maximum likelihood estimationsatisfiesedγ + ed(100− γ) = −2αβassuming β 6= 0. Such a restriction is hard to justify in applications.Introduction of parameter λ helps to soften this restriction without over-complicating the system. Under this model, the effective dose level at γ isgiven byedγ =logit((γ/100)λ)− αβ. (4.2)An explicit expression of dose–response relationship ispi(x) = P{Y = 1|X = x} ={ exp(α+ βx)1 + exp(α+ βx)}1/λ. (4.3)As discussed in previous chapters, many sequential designs, including theED-design, contain a step to update the estimation of the model parameters.The maximum likelihood estimate is a common choice. For this reason, weinvestigate the problem of parameter estimation via maximum likelihood inthe next Section.4.3 Maximum likelihood estimationLet (xi, yi) : i = 1, . . . , n be observations from a dose–response experiment,and assume model (4.1). Under commonly used designs, the log-likelihoodbased on this data set is given by`n(θ) =n∑i=1{yi ln(pi(xi))+ (1− yi) ln(1− pi(xi))}704.3. Maximum likelihood estimationwhere θ = (α, β, λ)τ .When λ = 1 is fixed, the model becomes the usual logistic model, and`n(θ) is known to be concave in α and β. The concavity permits a simplenumerical solution to the maximum likelihood estimate of α and β. Weremark that when xi’s corresponding to y = 1 is completely separated fromthose corresponding to y = 0, the maximum point βˆ = ±∞. The problemcan be easily addressed by adding some informative pseudo observations assuggested in Chapter 3. This technique will also be used for the procedurebeing developed.After some investigation, we find that given any value of λ, the loglikelihood remains concave in α and β. Given any α and β, the log-likelihoodis concave in λ. Because of these properties, the following two-loop iterativenumerical algorithm works nicely. We propose to start the algorithm withthe initial value λ(0) = 1, and set k = 0. Let be a small positive value suchas 10−5.1. Let`(k)n (α, β) = `n(α, β, λ(k)).Use an iterative algorithm to solve(α(k+1), β(k+1)) = arg maxα,β`(k)n (α, β).2. Defineai =exp(α(k+1) + β(k+1)xi)1 + exp(α(k+1) + β(k+1)xi)714.3. Maximum likelihood estimationand`(k)n (λ) =n∑i=1{(1− yi) ln(1− aλi ) + λyi ln(ai)}.Use an iterative algorithm to solveλ(k+1) = arg maxλ`(k)n (λ).If `n(θ(k+1)) − `n(θ(k)) ≤ , stop and report θ(k+1) and `n(θ(k+1)).Otherwise, set k = k + 1 and go back to Step 1.In the above presentation, we have used `(k)n (α, β) and `(k)n (λ) as twodifferent functions. We pointed out that the objective functions in bothloops are concave that guarantee the convergence of any sensible iterativeprocedures that we may use in these two steps, and hence of the entirealgorithm. We state the concave conclusions in two lemmas, and start withthe simpler one.Lemma: Function `(k)n (λ) in Step 2 is concave in λ given any data set (xi, yi)for i = 1, 2, . . . , n with n ≥ 1.Proof: To prove the concavity, it suffices to show that the second derivativeof this function is always non-negative. Some straightforward algebra showsthat∂`(k)n (λ)∂λ=n∑i=1(yi − aλi )(ln ai)1− aλiand subsequently,∂2`(k)n (λ)∂λ2=n∑i=1aλi (yi − 1)(ln ai)2(1− aλi )2≤ 0724.3. Maximum likelihood estimationsince yi ≤ 1 for all i. Therefore, the function is concave as claimed.Lemma: Given any data set (xi, yi) for i = 1, 2, . . . , n with n ≥ 1, the ob-jective function `(k)n (α, β) in Step 1 is concave in α, β, under the assumptionλ > 0.Proof: For notational simplicity, we will drop the superscript (k) and sub-script n from `(k)n (α, β), and denote it simply as `(α, β) in this proof. Westart working on the case where n = 1 so that we further drop summationand subindex i.To prove this result, it suffices to show that the Hessian matrixH = −∂2`∂α2∂2`∂α∂β∂2`∂α∂β∂2`∂β2is positive definite. For this purpose, we note that∂`∂α={ ypi(x)− (1− y)1− pi(x)}pi(x)∂α=(y − pi(x))(1− piλ)λ(1− pi(x)) .and∂2`∂α2={(y − 1)(1− piλ(x))λ(1− pi(x))2 −λpiλ−1(y − pi(x))1− pi(x)}pi(x)∂α=1λ2pi(x)(1− piλ(x)){(y − 1)(1− piλ(x))(1− pi(x))2 −λpiλ−1(y − pi(x))1− pi(x)}.We first show that the above second derivative is less than or equal to0. Note that the first factor in ∂2`/∂α2 is nonnegative. So we only need to734.3. Maximum likelihood estimationdetermine the sign of the second factor. We consider the cases of y = 1 andy = 0 separately.(a) When y = 1, the first term in the second factor vanishes, and thesecond term is clearly less than or equal to 0.(b) When y = 0, the second factor becomesλpiλ(1− pi(x))− (1− piλ(x))(1− piλ(x))2 .Denote its numerator as f(λ) whose derivative is given byf ′(λ) = piλ(1− pi(x)) + λpiλ(1− pi(x)) lnpi(x) + piλ(x) lnpi(x)= piλ(x){1− pi(x) + (1 + λ− λpi(x)) lnpi(x)}≤ piλ(x){lnpi(x) + (1 + λ− λpi(x)) lnpi(x)}= λpiλ(x)(1− pi(x)) lnpi(x) ≤ 0where we have made use of the inequality 1 − pi(x) ≤ − lnpi(x). Combinedwith the fact that f(0) = 0, we find f(λ) ≤ 0 for all λ ≥ 0. This furtherimplies ∂2`/∂α2 ≤ 0 when y = 0.Combining (a) and (b), noticing that y is either 0 or 1, we formally statethat for all λ > 0,∂2`∂α2≤ 0.To finish the proof, we note that∂2`∂α∂β= x∂2`∂α2;∂2`∂β2= x2∂2`∂α2.744.4. Potential designs for the three-parameter logistic modelTherefore, in the sense of being nonnegative definiteness, we findH = −∂2`∂α2∂2`∂α∂β∂2`∂α∂β∂2`∂β2 = − ∂2`∂α21 xx x2 ≥ 0.When the design contains n dose levels, the Hessian matrix is the sum of nnonnegative definite matrices. Hence it remains nonnegative definite. Thiscompletes the proof.By these two lemmas, `n(θ(k)) is an increasing sequence in k with anupper bound 0. Hence, `n(θ(k)) has a finite limit as k → ∞. The corre-sponding θ(k) is almost guaranteed to converge to at least a local maximumpoint. Rigorous discussion on global maximum can be tedious and distrac-tive. We do not pursue the issue in this dissertation.4.4 Potential designs for the three-parameterlogistic modelThe choice of a new model does not lead to new design issues, but some ad-ditional technical work. All optimality criteria introduced previously remaineffective under the three-parameter logistic model (4.1). We merely work onexisting procedures under the new model. In the following, we selectivelydiscuss some particulars.754.4. Potential designs for the three-parameter logistic model4.4.1 Up-and-down designThe up-and-down design and its variations do not require a parametricmodel on the dose–response relationship pi(x). The design is used for thepurpose of accurately estimating a specific effective dose level edγ, and com-monly the target is γ = 50. The design requires a user to choose before handa grid of dose levelsΩ = {x1, . . . , xK} (4.4)for some K based on prior information on pi(x) so that x1 < edγ < xK .The experiment starts with assigning a stimulus at level xj in Ω to thesubject. If the subject responds, the level is moved down to xj−1, andotherwise up to xj+1. Special rules are needed if xj is on the boundaryof Ω. Variations are needed such as staying at xj with a specific positiveprobability related to the target edγ. A nonparametric estimate of edγ maybe used. Our experience shows that such estimators are not efficient. Formore informative comparison, we obtain the maximum likelihood estimate(MLE) under the assumed model, and estimate edγ in the simulation, even ifthe data are obtained under the up-and-down design in this chapter. Clearly,introducing the three-parameter model leads to no new issues.4.4.2 D-optimal designAs pointed out in previous chapters, the variance-covariance matrix of theMLE of the parameter θ is well approximated by I−1n (θ) when the number ofruns n is large, where In(θ) is the Fisher information. A D-optimal designis a design which maximizes the determinant of In(θ). As far as we are764.4. Potential designs for the three-parameter logistic modelaware, there have been no direct results on the D-optimal design for thethree-parameter logistic model. In this section, we do not aim to give atheoretical solution to the D-optimal design for the three-parameter logisticmodel. Rather, we provide a numerical approach to get approximate D-optimal designs. Solutions to the D-optimal designs will be used in oursimulation studies.Sitter and Wu (1993) showed that under the (two-parameter) logisticresponse model, Ψ∗ is a uniform distribution on ed17.6 and ed82.4; andunder the probit model, Ψ∗ is a uniform distribution on ed12.8 and ed87.2.We do not have a comparable theory for the D-optimal design under thenew model but point out that a vertex direction method (VDM) remainseffective for numerical solutions. This method will be illustrated in the nextsection.We implemented VDM as an R function for the three-parameter logisticmodel (4.1). The resulting D-optimal design for α = −6.265, β = 0.055 andλ = 0.5 is a uniform distribution on ed2, ed35 and ed91. The resulting D-optimal design for α = −14.148, β = 0.1 and λ = 2 is a uniform distributionon ed6, ed55, and ed95. These two designs are used in simulations for thepurpose of comparison. We also implemented VDM for the three-parameterprobit model. Under the three-parameter logistic and probit models, designpoints and design weights change with different λ values. A number of λvalues are given in Tables 4.1 and 4.2. For more discussions of VDM, werefer to Fedorov (1972), Wynn (1972), and Wu (1978).774.4. Potential designs for the three-parameter logistic modelTable 4.1: D-optimal design under three-parameter Logistic model.Model Three-parameter Logistic Modelα = −6.265, β = 0.055 α = −14.148, β = 0.1λ = 0.5ED2 ED35 ED91 ED2 ED35 ED910.33 0.33 0.33 0.33 0.33 0.33λ = 0.75ED3 ED39 ED92 ED3 ED39 ED920.33 0.33 0.33 0.33 0.33 0.33λ = 1ED3 ED43 ED93 ED3 ED43 ED930.33 0.33 0.33 0.33 0.33 0.33λ = 1.25ED4 ED46 ED94 ED4 ED46 ED940.33 0.33 0.33 0.33 0.33 0.33λ = 1.5ED5 ED49 ED94 ED5 ED49 ED940.33 0.33 0.33 0.33 0.33 0.33λ = 1.75ED5 ED52 ED95 ED5 ED52 ED950.33 0.33 0.33 0.33 0.33 0.33λ = 2ED6 ED55 ED95 ED6 ED55 ED950.33 0.33 0.33 0.33 0.33 0.334.4.3 Vertex Direction Method(VDM)A number of algorithms have been proposed for numerical computation ofthe D-optimal design. In this dissertation, we apply a well-known iterativestrategy, vertex direction method (see Fedorov (1972), Wynn (1972), andWu (1978)) to numerically compute the D-optimal design under the three-parameter logistic regression model. Let’s consider a finite design spaceX of permissible dose levels, x1, . . . , xk. A design is a set of dose levelsx1, . . . , xk together with how often they are applied: m1, . . . ,mk, ignoring784.4. Potential designs for the three-parameter logistic modelTable 4.2: D-optimal design under the three-parameter probit model.Model three-parameter probit Modelα = −6.265, β = 0.055 α = −14.148, β = 0.1λ = 0.5ED4 ED53 ED97 ED4 ED53 ED970.33 0.33 0.33 0.33 0.33 0.33λ = 0.75ED4 ED55 ED98 ED4 ED55 ED980.33 0.33 0.33 0.33 0.33 0.33λ = 1ED4 ED57 ED98 ED4 ED57 ED980.33 0.33 0.33 0.33 0.33 0.33λ = 1.25ED5 ED59 ED98 ED5 ED59 ED980.33 0.33 0.33 0.33 0.33 0.33λ = 1.5ED5 ED60 ED98 ED5 ED60 ED980.33 0.33 0.33 0.33 0.33 0.33λ = 1.75ED5 ED61 ED98 ED5 ED61 ED980.33 0.33 0.33 0.33 0.33 0.33λ = 2ED5 ED62 ED98 ED5 ED62 ED980.33 0.33 0.33 0.33 0.33 0.33the order of the runs. This design is characterize by a distribution Ψ onX whose probability mass function is given by ψ(xj) = mj/n. To reflectthe dependence on Ψ, we denote the Fisher information as I(Ψ) here. A Ψdegenerates at x is denoted as δx, and we use I(x) for I(δx). It is seen thatI(Ψ) =∫XI(x)dΨ(x).794.4. Potential designs for the three-parameter logistic modelThe popular D-optimal design is defined to beΨ∗ = arg max{ln[det(I(Ψ))]}.Define the directional derivativeD(Ψ;x) = lim→0+ln[det(I((1− )Ψ) + δx)]− ln[det(I(Ψ))].It is known that Ψ∗ is the D-optimal design if and only if D(Ψ∗;x) ≤ 0for any x. Starting from an initial design Ψ = Ψ(0), VDM searches forx∗ = arg maxD(Ψ;x) and∗ = arg max ln[det(I((1− )Ψ + δx∗)}.Iterate to obtain Ψ(k), k = 1, 2, . . . until the determinant stops increasing.In particular, for the three-parameter logistic regression model (4.1),D(Ψ;x) = tr(I−1(Ψ)I(x))− 3where the constant 3 is the dimension of I. Additional details for the explicitexpression of I will be given at the end of this section.We implemented VDM as an R function for the three-parameter logisticmodel (4.1). The steps of VDM applied in our simulations are as follows. Inthe experiment, suppose mi patients are assigned to doses xi, i = 1, . . . , k,and∑ki=1mi = n. Here k is the number of distinct dose levels. The corre-804.4. Potential designs for the three-parameter logistic modelsponding experimental design isξ =x1 x2 · · · xkm1 m2 · · · mkDenote wj = mj/n as the design weight for xj , j = 1, . . . , k. Letw = (w1, . . . , wk) ∈ Ω = {w :k∑i=1wi = 1, wi ≥ 1}be the vector of design weights. Give a parametric model, the goal of theD-optimal design is to find an allocation of weights, w1, . . . , wk, to the de-sign points, x1, . . . , xk, such that the determinant of the Fisher informationmatrix of the model parameters is maximized.1. Choose an initial designξ =x1 x2 · · · xkw(1)1 w(1)2 · · · w(1)kIn the simulation, we choose to have equal weights at all n designpoints. Let n = 200. The dose level xi is chosen based on the assumeddose–response model used in the simulation. Then the initial designwe choose isξ =1 2 · · · 2001/200 1/200 · · · 1/200814.5. Two-stage sequential ED-design2. For the current design, w(t)1 , w(t)2 , . . . , w(t)k , t = 1, 2, . . . , find the indeximax with the maximum directional derivative, that is,D(imax, w(t)) = max1≤i≤nD(i, w(t)).Then set the new design weight at the (t+ 1)th iteration asw(t+1)i =(1− δ(t))wi i 6= imax(1− δ(t))wi + δ(t) i = imaxwhereδ(t) =D(imax, w(t))/m− 1D(imax, w(t))− 1.Here m = 3 for the three-parameter logistic regression model.3. Repeat step 2 untilD(i, w(t))−m ≤ .Here is a small positive constant.Each iteration of the vertex directional derivative method moves the weightw in the direction of a design point at which the directional derivative is thelargest.4.5 Two-stage sequential ED-designThe D-optimal design as well as many other optimal designs focus on theprecision of the parameter estimation under the assumed model. The form of824.5. Two-stage sequential ED-designthe parameter in consideration is generally chosen as the one permitting themost convenient analytical presentation of the dose–response model. Underthe three-parameter logistic regression model, for instance, one naturallytakes θ = (α, β, λ) as the target parameter. In some applications, we aremore interested in the precise estimation of ed levels. Hence, the proposedED-design can be easily applied here.Let ξ = g(θ), for some smooth function g with gradient function 5g(θ).The variance of its MLE is approximately{5gτ (θ)}{I−1(θ)}{5g(θ)} (4.5)where 5g(θ) is the gradient of g(·). The sequential ED-design aims tominimizem∑j=1{5gτj (θ)}{I−1(θ)}{5gj(θ)} (4.6)with ξj = gj(θ) being m selected ed levels. Clearly, this solution dependson the value of the unknown parameter θ. Hence, a sequential version isneeded.Suppose the experiment has been carried out at dose levels x1, . . . , xiwith response values y1, . . . , yi. Let θˆi be the intermediate MLE of θ basedon the data obtained from the first i trials. Let Ii(θˆi; +x) be the Fisherinformation based on the first i trials and the potential (i+ 1)th trial to berun at dose level x. We choose the next dose level x that minimizesv(i;x) =m∑j=1{5gτj (θˆi)}{I−1i (θˆi; +x)}{5gj(θˆi)}. (4.7)834.5. Two-stage sequential ED-designRepeat this rule until i = n.The sequential ED-design needs an initial set of trials and the corre-sponding θˆ. We recommend and use a uniform initial design on a set ofdosage Ω = {x1, . . . , xK} for K = 7 with x1 and xK being equally spacedgrids between the perceived ed01 and ed99.Some algebra for implementation of the ED-design. The numericalvalue of v(i;x) can be computed based on the following simple mathematicalresults. Denote the single observation log-likelihood as`(θ) = y lnpi(x) + (1− y) ln (1− pi(x)).We have∂`(x)∂θ=(y − pi(x))pi(x)(1− pi(x))∂pi(x)∂θ.The contribution of a single trial at dose level x to the Fisher informationmatrix is therefore given byI(θ;x) = E[ (y − pi(x))pi(x)(1− pi(x))]2{∂pi(x)∂θ}{∂pi(x)∂θ}τ=1pi(x)(1− pi(x)){∂pi(x)∂θ}{∂pi(x)∂θ}τand ∂pi(x)/∂θ have three entries given by∂pi(x)∂α= λ−1pi(x)(1− piλ(x))∂pi(x)∂β= λ−1xpi(x)(1− piλ(x)) = x∂pi(x)∂α∂pi(x)∂λ= −λ−1pi(x) lnpi(x).844.6. Simulation studiesFinally, for ξγ = edγ, we haveg(θ; γ) =logit((γ/100)λ)− αβOg(θ; γ) = − 1β(1, ξ,− ln(γ/100)1− (γ/100)λ)τ.These calculations lead to a simple way to evaluate v(i;x) and the numericalsolution to its minimization.4.6 Simulation studiesWe conduct simulation studies to demonstrate several issues related to theuse of the extended three-parameter logistic regression model (4.1) for thedose–response experiment. We repeat the simulation N = 1000 times for allmodel/design combinations. The sample sizes are chosen to be n = 30, 60,and 120. We choose three effective dose levels each time as the targets forestimation and obtain their MLEs. Under each model/design setting, wecompute the RMSE of a single ed level asRMSE(ξˆj) =√√√√N−1 N∑r=1(ξˆrj − ξj)2,where ξˆrj is the estimate of ξj in the rth repetition. The total RMSE iscomputed asRMSE =√√√√ 3∑j=1RMSE2(ξˆj).854.6. Simulation studiesThree designs are included in the simulation. One is the up-and-down designwhose implementation does not depend on the model, but a specific targeted level will be indicated in the summary of the simulation result. Wechoose a set of doses Ω = {x1, . . . , x7} with x1 and x7 being equally spacedgrids between the anticipated ed01 and ed99. We simulate on the D-optimaldesign and the sequential ED-design, assuming the relevant knowledge of thedose–response model as discussed in the last section. For the sequential ED-design, we use a uniform initial design on Ω as specified for the up-and-downdesign.We wish to use simulations to answer several questions related to thecombination of the ED-design and the three-parameter model under thedose–response experiment.The first question is how the ED-design fairs. Does it have any advantagecompared with other potential designs under a three-parameter model? Asyou will see, the ED-design works very well in this respect.The second question is whether the use of the three-parameter modelnecessary? Based on our simulation, if one focuses on the precision of theestimation of ed levels over a local region as we did in the last chapter, wecan observe the advantage of employing the correct three-parameter logisticmodel.Finally, if the true dose–response relationship is a two-parameter logisticmodel, how much efficiency do we lose by using a more complex three-parameter logistic? Our simulation shows the loss is very limited or canhardly be noticed.We now present our simulation results in three subsequent sections.864.6. Simulation studies4.6.1 The three-parameter model is both the assumed andthe truth for the dose–response experimentWe generate data according to the three-parameter logistic regression model(4.1) with α = −6.265, β = 0.055, λ = 0.5, and α = −14.148, β = 0.1,λ = 2 in two separate simulations. The corresponding dose–response curvesare depicted in Figure 4.1.The simulations are done for each of the three sets of the targeting doselevels: (a) ed25, ed50, ed75; (b) ed10, ed25, ed40; and (c) ed60, ed75,ed90.The use of the up-and-down design requires us to identify a targeted level. In our simulation, we always take the middle one as its target.The ed01 and ed99 values under λ = 0.5 is (74, 211); The ed01 and ed99values under λ = 2 is (49, 180). These values are used in determiningthe first stage design Ω. The simulation results are reported in Tables 4.3and 4.4.We take note that the RMSEs under each design decrease when n in-creases. Their sizes are not dramatically different, but those of the D-optimaldesign are higher. The sequential ED-design has the best overall perfor-mance. The up-and-down design gives the lowest RMSEs for some singleed level. These are expected as the D-optimality aims at the precise esti-mation of θ, not ed levels, and the up-and-down design is never intendedfor the current purpose: estimating ED levels under a parametric model.Nevertheless, it is nice to find that the sequential ED-design works well.874.6. Simulation studies0 50 100 150 2000.00.20.40.60.81.0Dose LevelsProbability of Responseα = − 6.265, β = 0.055, λ = 0.5α = − 14.148, β = 0.1, λ = 20 50 100 150 2000.00.20.40.60.81.0Dose LevelsProbability of Response α = − 6.265, β = 0.055λ = 0.5λ = 1λ = 1.25λ = 1.5λ = 2Figure 4.1: Dose–response curves in the simulation884.6. Simulation studiesTable 4.3: Simulated RMSEs under the three-parameter model (α = −6.265, β = 0.055,λ = 0.5)Size n = 30 n = 60 n = 120Design ED Up-down D-opt ED UP-down D-opt ED Up-down D-optTotal 13.32 16.20 16.82 9.87 10.33 11.96 6.93 7.42 8.44ED10 8.62 9.39 11.33 6.66 6.79 8.16 4.71 4.97 5.82ED25 6.78 8.09 9.03 4.78 5.05 6.34 3.23 3.61 4.45ED40 7.55 10.43 8.54 5.50 5.92 6.02 3.93 4.15 4.19Total 14.02 15.68 16.97 10.49 10.85 11.95 7.47 7.69 8.46ED25 8.20 8.00 9.03 6.15 5.83 6.33 4.46 4.38 4.45ED50 6.72 7.68 8.72 4.83 5.06 6.16 3.48 3.51 4.29ED75 9.17 11.09 11.42 7.00 7.62 8.05 4.88 5.25 5.77Total 17.36 21.59 22.27 12.63 13.49 16.01 8.89 9.26 11.61ED60 9.00 9.08 8.94 6.29 6.01 6.45 4.39 4.24 4.60ED75 8.69 10.64 11.23 5.92 6.45 8.09 4.07 4.34 5.82ED90 12.04 16.44 17.03 9.22 10.21 12.21 6.57 7.00 8.93894.6. Simulation studiesTable 4.4: Simulated RMSEs under the three-parameter model (α = −14.148, β = 0.1,λ = 2)Size n = 30 n = 60 n = 120Design ED Up-down D-optimal ED Up-down D-optimal ED Up-down D-optimalTotal 16.76 18.61 21.35 12.26 12.98 15.84 8.92 9.09 11.05ED10 12.01 12.16 15.58 9.03 9.00 11.63 6.61 6.66 8.22ED25 8.34 9.44 11.23 5.79 6.38 8.24 4.12 4.45 5.66ED40 8.21 10.46 9.32 5.93 6.85 6.91 4.35 4.30 4.75Total 13.05 14.65 16.23 9.36 10.01 11.78 6.76 6.92 8.06ED25 8.76 8.67 11.23 6.37 6.52 8.24 4.69 4.65 5.66ED50 6.37 6.32 8.48 4.42 4.42 6.28 3.12 3.25 4.28ED75 7.28 9.97 8.10 5.25 6.18 5.61 3.74 3.96 3.82Total 12.32 15.42 16.30 8.67 9.98 11.54 6.23 6.66 8.30ED60 6.84 7.22 7.93 4.81 4.99 5.77 3.46 3.52 3.89ED75 6.24 7.63 8.10 4.14 4.96 5.61 2.83 3.27 3.82ED90 8.12 11.28 11.72 5.90 7.08 8.28 4.35 4.61 6.26904.6. Simulation studies4.6.2 Effects of fitting a three-parameter model when atwo–parameter logistic model sufficesWhen the logistic model is appropriate, but a three-parameter model isassumed in the design and analysis, the results are likely suboptimal. Inthis section, we use simulations to examine the degree of the efficiency loss.We simulate dose–response data from the logistic regression model with twosets of designs, and analyze the data: one set is under the two-parameterlogistic regression model, and the other set is under the three-parameterlogistic regression model. We only include the D-optimal design and thesequential ED-design in this simulation. The up-and-down design is notincluded, because it does not depend on the model assumption, althoughthe data analysis could be performed under a model assumption.In this simulation, we generate data from the two-parameter model:logit(pi(x)) = −6.265 + 0.055x.The results are presented in Tables 4.5 and 4.6. Table 4.5 is obtained underthe correct two-parameter logistic regression model assumption. The D-optimal design in this case is a uniform distribution on ed17.6 and ed82.4which are known to us though not known in applications. Table 4.6 isobtained under the also correct three-parameter logistic regression modelassumption, though it is more complex than needed.According to these results, we notice that the use of the ED-design hasadvantages compared with the D-optimal design. The simulated RMSEs un-der the ED-design are always lower than those under the D-optimal design.914.6. Simulation studiesThe efficiency gain can be as much as 40%.In addition, the use of the more complex and necessary three-parametermodel does not hurt the efficiency significantly. In the case of targetinged10, ed25, and ed40, the total RMSE increases from 17.93 to 18.36 whenthe sample size n = 30. This loss is below 2.4%. The worst case is whenn = 120, the total RMSE increases from 9.35 to 9.88. The loss is below5.7%.In comparison, the efficiency of the D-optimal design can be affectedmore markedly. In the case when ed60, ed75 and ed90 are targeted with n =120, the loss is as high as 18%. When n = 30, the use of the more complexthree-parameter logistic regression model makes the D-optimal design moreefficient. This might be due to the fact that the initial design takes up alarge proportion of the number of trials.Overall speaking, if the ED-design is used, the use of a more than nec-essary three-parameter logistic regression model does not hurt much of theefficiency in estimating ed levels.924.6. Simulation studiesTable 4.5: Simulated RMSEs when fitting a simple logistic model when atwo–parameter model suffices.Fit a Simple Logistic Regression ModelDesign ED-design D-optimal designSize 30 60 120 30 60 120Total 17.93 13.25 9.35 23.58 15.79 10.87ED10 12.68 9.48 6.73 17.53 11.96 8.02ED25 8.93 6.52 4.55 12.15 8.02 5.63ED40 9.01 6.56 4.63 10.07 6.46 4.69Total 17.14 12.54 8.74 20.18 12.63 8.87ED25 10.55 7.80 5.46 12.48 7.70 5.49ED50 8.12 5.89 4.09 9.89 6.15 4.41ED75 10.79 7.85 5.46 12.40 7.90 5.39Total 18.91 13.42 9.40 26.03 15.76 10.58ED60 9.49 6.64 4.86 9.09 6.45 4.62ED75 9.39 6.51 4.57 12.87 8.01 5.47ED90 13.40 9.67 6.63 20.72 11.94 7.79934.6. Simulation studiesTable 4.6: Simulated RMSEs when fitting a three-parameter model when atwo–parameter model suffices.Fit the three-parameter Logistic ModelDesign ED-design D-optimalSize 30 60 120 30 60 120Total 18.36 13.63 9.88 23.56 17.50 12.41ED10 12.30 9.60 7.03 17.29 13.04 9.32ED25 9.55 6.57 4.68 12.03 8.76 6.18ED40 9.74 7.10 5.13 10.56 7.72 5.37Total 17.52 12.33 8.86 19.99 14.52 10.22ED25 10.64 7.80 5.66 12.03 8.76 6.18ED50 8.45 5.48 4.04 10.33 7.56 5.23ED75 11.06 7.81 5.48 12.17 8.77 6.23Total 19.00 13.48 9.92 23.77 17.25 12.49ED60 9.77 6.97 5.08 10.57 7.70 5.33ED75 9.43 6.36 4.51 12.17 8.77 6.23ED90 13.30 9.64 7.22 17.47 12.70 9.42944.6. Simulation studies4.6.3 Effects under model misspecificationIn this section, we investigate the effect of two kinds of model misspec-ification. One is when the dose–response relationship satisfies the three-parameter logistic regression model with λ 6= 1. The other is when thedose–response relationship is not even a three-parameter logistic model.In both situations, we compute two sets of RMSEs of ed estimates. Oneset is when the design and analysis are done under the three-parameter lo-gistic regression model assumption; the other set is done under the usualtwo-parameter logistic regression model assumption. We wish to demon-strate that the design and analysis based on the three-parameter logisticregression model leads to a more accurate estimation of the target ed levels.For the first kind of model misspecification, we generate data from twothree-parameter models:logit(piλ(x)) = −6.265 + 0.055x (4.8)with λ = 0.5, andlogit(piλ(x)) = −14.148 + 0.1x (4.9)with λ = 2. Under model (4.8), ed25 = 114, ed50 = 130, and ed75 =148 when λ = 0.5; and under model (4.9), ed25 = 114, ed50 = 130, anded75 = 144 when λ = 2. The results are presented in Tables 4.7 and 4.8,and Tables 4.9 and 4.10 . Tables 4.7 and 4.9 are obtained under the two-parameter logistic regression model assumption. The D-optimal design inthis case is a uniform distribution on ed17.6 and ed82.4. Tables 4.8 and 4.10954.6. Simulation studiesare obtained under the correct three-parameter logistic regression modelassumption.According to these results, we notice that the use of the ED-design isnoticeably superior to the D-optimal design in both situations. The simu-lated RMSEs under the ED-design are always lower than those under theD-optimal design. The efficiency gain can be as much as 30%.In addition, the use of the more complex three-parameter model whendata are generated under the three-parameter logistic model gains the effi-ciency significantly. In the case of λ = 2, targeting ed25, ed50, and ed75,the total RMSE decreases from 15.57 to 14.02 when the sample size n = 30.This gain is as much as 11%. Overall speaking, the use of the more com-plex three-parameter logistic regression model makes the ED-design moreefficient in estimating ed levels.For the second kind of model misspecification, we generate data fromthe three-parameter probit model:pi(x) = Φ1/λ(−6.265 + 0.055x)with λ = 0.5, andpi(x) = Φ1/λ(−14.148 + 0.1x)λ = 2. Under this model, ed25 = 114, ed50 = 124, and ed75 = 134 whenλ = 0.5; and ed25 = 86, ed50 = 102, and ed75 = 117 when λ = 2. Thesimulation results are presented in Tables 4.11 and 4.12, and Tables 4.13and 4.14. Tables 4.11 and 4.13 are obtained under the two-parameter lo-gistic regression model assumption. The D-optimal design in this case is964.6. Simulation studiesTable 4.7: Simulated RMSEs under the three parameter model (α =−6.265, β = 0.055, λ = 0.5).Fit a Simple Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 13.87 9.96 7.16 16.62 13.38 11.04ED10 10.14 7.08 4.99 11.75 9.50 7.77ED25 6.53 4.74 3.43 8.63 7.15 6.09ED40 6.86 5.17 3.82 7.97 6.14 4.94Total 15.57 11.14 7.65 15.72 11.46 8.69ED25 9.45 6.63 4.30 8.63 7.15 6.09ED50 6.64 5.06 3.57 8.12 5.82 4.30ED75 10.44 7.39 5.22 10.34 6.81 4.47Total 18.98 12.89 8.98 19.80 13.86 10.23ED60 8.16 5.78 4.23 8.65 5.82 3.90ED75 8.74 5.90 4.21 10.34 6.81 4.47ED90 14.73 9.89 6.71 14.50 10.57 8.34974.6. Simulation studiesTable 4.8: Simulated RMSEs under the three-parameter model (α =−6.265, β = 0.055, λ = 0.5).Fit the three-parameter Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 13.32 9.87 6.93 16.82 11.96 8.44ED10 8.62 6.66 4.71 11.33 8.16 5.82ED25 6.78 4.78 3.23 9.03 6.34 4.45ED40 7.55 5.50 3.93 8.54 6.02 4.19Total 14.02 10.49 7.47 16.97 11.95 8.46ED25 8.20 6.15 4.46 9.03 6.34 4.45ED50 6.72 4.83 3.48 8.72 6.16 4.29ED75 9.17 7.00 4.88 11.42 8.05 5.77Total 17.36 12.63 8.89 22.40 15.83 11.48ED60 9.00 6.29 4.39 9.33 6.58 4.63ED75 8.69 5.92 4.07 11.42 8.05 5.77ED90 12.04 9.22 6.57 16.86 11.94 8.78984.6. Simulation studiesTable 4.9: Simulated RMSEs under the three-parameter model (α =−14.148, β = 0.1, λ = 2).Fit a Simple Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 17.43 12.39 8.99 24.82 15.78 12.04ED10 14.00 9.78 6.98 19.42 12.80 10.33ED25 7.50 5.40 3.95 12.51 7.47 5.11ED40 7.19 5.35 4.06 9.07 5.42 3.50Total 13.80 9.57 6.95 16.25 10.09 6.93ED25 9.43 6.64 4.76 12.51 7.47 5.11ED50 5.95 4.25 3.47 7.52 4.69 3.13ED75 8.14 5.43 3.70 7.13 4.90 3.49Total 13.06 8.75 6.35 15.63 10.30 7.10ED60 6.27 4.57 3.49 6.59 4.37 3.06ED75 6.22 4.08 2.96 7.13 4.90 3.49ED90 9.63 6.25 4.40 12.25 7.93 5.37994.6. Simulation studiesTable 4.10: Simulated RMSEs under the three-parameter model (α =−14.148, β = 0.1, λ = 2).Fit the three-parameter Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 16.76 12.26 8.92 21.35 15.84 11.05ED10 12.01 9.03 6.61 15.58 11.63 8.22ED25 8.34 5.79 4.12 11.23 8.24 5.66ED40 8.21 5.93 4.35 9.32 6.91 4.75Total 13.05 9.36 6.76 16.23 11.78 8.06ED25 8.76 6.37 4.69 11.23 8.24 5.66ED50 6.37 4.42 3.12 8.48 6.28 4.28ED75 7.28 5.25 3.74 8.10 5.61 3.82Total 12.32 8.67 6.23 16.30 11.54 8.30ED60 6.84 4.81 3.46 7.93 5.77 3.89ED75 6.24 4.14 2.83 8.10 5.61 3.82ED90 8.12 5.90 4.35 11.72 8.28 6.261004.6. Simulation studiesa uniform distribution on ed12.8 and ed87.2. Tables 4.12 and 4.14 areobtained under the three-parameter logistic regression model assumption.Both model assumptions are incorrect. However, we wish to demonstratethe use of the three-parameter logistic model makes the ED-design moreefficient in estimating ed levels.According to the simulation result, we notice that the use of the ED-design is noticeably superior to the D-optimal design in both situations. Thesimulated RMSEs under the ED-design are always lower than those underthe D-optimal design. In the case of λ = 2, targeting ed10, ed25, and ed40,the total RMSE increases from 8.39 to 12.89 when the sample size n = 30.The efficiency gain is as much as 54%.In addition, the use of the more complex model when data are generatedunder the three-parameter probit model gains the efficiency noticeably. Inthe case of λ = 2, targeting ed60, ed75, and ed90, the total RMSE decreasesfrom 8.29 to 7.12 when the sample size n = 30. This efficiency gain is asmuch as 16%. Overall speaking, the use of the more complex model makesthe ED-design more efficient in estimating ed levels, when the model ismis-specified.1014.6. Simulation studiesTable 4.11: Simulated RMSEs under probit mis-specified as logistic (α =−6.265, β = 0.055, λ = 0.5).Fit a Simple Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 9.27 6.29 4.31 9.02 6.81 7.48ED10 6.68 4.49 3.01 5.44 3.98 4.69ED25 4.36 3.08 2.11 4.85 3.86 4.37ED40 4.72 3.16 2.24 5.31 3.95 3.85Total 9.73 6.22 4.38 10.29 7.28 6.40ED25 6.22 3.91 2.72 4.85 3.86 4.37ED50 4.18 2.71 2.00 5.72 4.06 3.52ED75 6.21 4.01 2.80 7.05 4.65 3.07Total 10.17 6.71 4.45 12.75 8.38 5.64ED60 5.05 3.29 2.21 6.19 4.23 3.25ED75 4.78 3.17 2.14 7.05 4.65 3.07ED90 7.41 4.90 3.21 8.63 5.55 3.441024.6. Simulation studiesTable 4.12: Simulated RMSEs under probit mis-specified as logistic (α =−6.265, β = 0.055, λ = 0.5).Fit the three-parameter Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 8.58 5.75 4.25 12.36 7.65 4.95ED10 5.76 3.94 2.94 7.24 4.40 3.14ED25 4.42 2.85 2.06 7.11 4.41 2.77ED40 4.57 3.08 2.27 7.06 4.43 2.64Total 8.57 6.00 4.25 12.09 7.56 4.83ED25 5.17 3.70 2.68 7.11 4.41 2.77ED50 4.27 2.85 2.00 6.99 4.41 2.65ED75 5.34 3.77 2.61 6.84 4.28 2.94Total 9.14 6.52 4.58 12.55 7.71 5.26ED60 4.77 3.45 2.47 6.89 4.36 2.72ED75 4.49 3.07 2.12 6.84 4.28 2.94ED90 6.37 4.60 3.22 7.94 4.70 3.421034.6. Simulation studiesTable 4.13: Simulated RMSEs under probit mis-specified as logistic ( α =−14.148, β = 0.1, λ = 2).Fit a Simple Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 8.52 5.65 3.70 8.75 5.95 4.31ED10 6.33 4.13 2.67 6.24 4.19 3.05ED25 3.97 2.69 1.79 4.63 3.10 2.16ED40 4.09 2.75 1.83 4.04 2.87 2.14Total 8.25 5.39 3.52 7.51 5.80 4.77ED25 5.42 3.50 2.28 4.63 3.10 2.16ED50 3.40 2.36 1.57 3.93 3.00 2.41ED75 5.20 3.36 2.17 4.43 3.89 3.50Total 8.29 5.30 3.55 8.29 7.05 6.24ED60 3.97 2.66 1.79 3.99 3.27 2.80ED75 3.87 2.51 1.71 4.43 3.89 3.50ED90 6.17 3.84 2.55 5.76 4.88 4.331044.6. Simulation studiesTable 4.14: Simulated RMSEs under probit mis-specified as logistic ( α =−14.148, β = 0.1, λ = 2).Fit the three-parameter Logistic Regression ModelDesign ED-design D-optimal DesignSize 30 60 120 30 60 120Total 8.39 5.43 3.13 12.89 7.62 4.33ED10 5.81 3.80 2.13 9.13 5.16 2.80ED25 4.24 2.67 1.57 6.67 4.02 2.39ED40 4.32 2.82 1.68 6.19 3.91 2.28Total 7.33 5.21 3.66 10.47 6.59 4.21ED25 4.59 3.21 2.31 6.67 4.02 2.39ED50 3.67 2.57 1.78 5.98 3.82 2.24ED75 4.37 3.19 2.21 5.42 3.57 2.66Total 7.12 5.06 3.65 9.63 6.39 4.99ED60 3.97 2.77 2.05 5.76 3.70 2.29ED75 3.76 2.51 1.71 5.42 3.57 2.66ED90 4.56 3.41 2.48 5.48 3.79 3.541054.7. Numerical example4.7 Numerical exampleTo illustrate the ideas described in this chapter, we consider a real clinicalstudy. Brown (1982) assumed a logistic response curve to model the rela-tionship between the wheezing symptom and the age of British coal miners.The number of subjects being examined and the number of subjects showingthe symptom are copied in Table 4.15.We first fit the observed data using model (4.1). The MLE of the modelparameters are αˆ = −1.798, βˆ = 0.044, and λˆ = 0.400. We then fit theobserved data using a simple logistic model. The MLE of the model param-eters are αˆ = −4.225 and βˆ = 0.065. For comparison, we draw the observeddata and the two fitted curves in Figure 4.2. The result of Figure 4.2 showsthat the three-parameter logistic model (4.1) is a better fit than the simplelogistic model.1064.7. Numerical exampleTable 4.15: Number of subjects examined and showing the wheezing symp-tom for British coal miners.Group AgeNumberExaminedNumbershowingsymptomproportionshowingsymptom1 22 1952 104 0.0532 27 1791 128 0.0723 32 2113 231 0.1094 37 2783 378 0.1365 42 2274 442 0.1946 47 2393 593 0.2487 52 2090 649 0.3118 57 1750 631 0.3619 62 1136 504 0.4471074.7. Numerical example30 40 50 600.10.20.30.4 AgeProportion respondingα = − 4.2247, β = 0.0652, λ = 1α = − 1.7976, β = 0.0442, λ = 0.4Figure 4.2: Observed Data and the fitted curve for British Coal Miners1084.8. Limiting design as n increases4.8 Limiting design as n increasesWe have so far focused on implementing the ED-design under the three-parameter logistic regression model. Recall that a design for a binary dose–response experiment is a probability distribution on the design space or doserange. From Section 4.4.2, the D-optimal design is roughly a three-support-point design under the three-parameter logistic regression model. Because ofits sequential nature, the ED-design likely has many more support points.When n goes to infinity, it is possible that the limiting distribution hasthree support points. We simulated the ED-design for two scenarios withn = 1000, first targeting ed25, ed50, ed75, and then targeting ed10, ed25,ed40, and targeting ed60, ed75, ed90. In the first scenario, we consider thecase where the data are generated under the three-parameter logistic modelwith α = −6.265, β = 0.055 and λ = 0.5. In the second scenario, we setα = −14.148, β = 0.1 and λ = 2. The resulting histograms are given inFigures 4.3 and 4.4. There are clear indications that in both scenarios, thelimiting distribution has three support points around ed6, ed32 and ed86when the target ed levels are ed25, ed50, and ed75, or ed10, ed25, anded40. We hope to prove this in the future.1094.8. Limiting design as n increases(a)0501502501 6 16 31 52 67 81 91 96(b)0501001502002501 6 16 31 52 67 81 91 96(c)0501001502001 6 16 31 52 67 81 91 96Figure 4.3: Histogram of the ED-design with respect to ed levels (α =−6.265, β = 0.055, λ = 0.5) for (a) estimating ed10, ed25, and ed40; (b)estimating ed25, ed50, and ed75, and (c) estimating ed60, ed75, and ed90.1104.8. Limiting design as n increases(d)0501502501 6 11 21 35 57 77 91(e)0501001502002501 6 11 21 35 57 77 91(f)0501001502002501 6 11 21 35 57 77 91Figure 4.4: Histogram of the ED-design with respect to ed levels (α =−14.148, β = 0.1 and λ = 2) for (a) estimating ed10, ed25, and ed40; (b)estimating ed25, ed50, and ed75, and (c) estimating ed60, ed75, and ed90.1114.9. Concluding remarks4.9 Concluding remarksWe explore the use of the three-parameter logistic regression model for dose–response experiments. We show that the sequential ED-design can be easilycarried out under this model assumption, and the resulting data analysis isvery effective.Simulation results show that the three-parameter logistic regression modelis an effective extension of the commonly used logistic regression model with-out leading to more complex data analysis issues.Simulation studies also show that the combination of the proposed modeland data the analysis strategy works well. When the logistic model is correct,applying the more complex model suffers hardly any efficiency loss. Whenthe three-parameter model holds but the logistic model is violated, the newapproach gains substantial ground. Our finding leads to a useful additionto the toolbox of the dose–response experiment.1124.10. R-code for simulations4.10 R-code for simulationsrm(list=ls())library(MASS)# define objectsm = 1000n = 30diff1 <- rep (0, m)diff2 <- rep (0, m)diff3 <- rep (0, m)diff <- rep (0, m)mu_temp1 <- rep (0, m)mu_temp2 <- rep (0, m)mu_temp3 <- rep (0, m)mu_temp <- matrix(0, m, 3)t <- seq(from = 40, to = 200, by = 1)sum <- rep(0, length(t))alpha <- -6.2647beta <- 0.05478lambda <- 0.5maxit <- 50ed <- c(0.25, 0.50, 0.75)dose.initial <- seq(from=(logit(0.01^lambda)-alpha)/beta,to=(logit(0.99^lambda)-alpha)/beta, by=20)mu1 <- (logit (ed[1]^lambda) - alpha) / beta1134.10. R-code for simulationsmu2 <- (logit (ed[2]^lambda) - alpha) / betamu3 <- (logit (ed[3]^lambda) - alpha) / beta# Define functions# Function f.lr.p() computes the probability vector# under the logistic modelf.lr.p <- function (x, alpha, beta, lambda) {prob = (exp(alpha + beta * x)/(1+exp(alpha + beta *x)))^(1/lambda)return (prob)}f.lr.l <- function(x, y, theta, lambda) {alpha <- theta[1]beta <- theta[2]ai <- (exp(x * beta + alpha) / (1+ exp(x * beta + alpha)))w <- c(0.1, 0.1, 0.9, 0.9, rep (1, length(x)-4))l <- sum(w * (-y * log(ai^(1/lambda)) - (1 - y) * log(1 - ai^(1/lambda))))return(l)}# function for the parameter estimationf.est <- function(x, y, theta){lambda.start = 11144.10. R-code for simulationstheta.start <- thetamle = optim(theta.start, f.lr.l, x = x, y = y, lambda = lambda.start)alpha_hat <- mle$par[[1]]beta_hat <- mle$par[[2]]theta.hat = c(alpha_hat, beta_hat)mle = optimize(f.lr.l, c(0.01,100), x = x, y = y, theta = theta.hat)lambda_hat <- mle$minimumlike.1 <- mle$objectivei <- 0 # initial iteration indexdiff.like <- 1while (i <= maxit & diff.like > 1e-6){i <- i + 1mle = optim(theta.hat, f.lr.l, x = x, y = y, lambda = lambda_hat)alpha_hat <- mle$par[[1]]beta_hat <- mle$par[[2]]theta.hat = c(alpha_hat, beta_hat)mle = optimize(f.lr.l, c(0.01,100), x = x, y = y, theta = theta.hat)lambda_hat <- mle$minimumlike.2 <- mle$objectivediff.like <- like.1 - like.2like.1 <- like.2}para=c(alpha_hat, beta_hat, lambda_hat, like.2)}1154.10. R-code for simulationsfisher <- function (x,theta,lambda){w <- c(0.01, 0.01, 0.99, 0.99, rep (1, length(x)-4))alpha <- theta[1]beta <- theta[2]pi <- f.lr.p(x, alpha, beta, lambda)v00 <- sum(w*lambda^(-2) * pi * (pi-1)^(-1)* (1-pi^lambda)^2)v11 <- sum(w*x^2 * lambda^(-2) * pi * (pi-1)^(-1) * (1-pi^lambda)^2)v22 <- sum(w*log(pi)^2 * lambda^(-2) * pi * (pi-1)^(-1))v01 <- sum(w*x * lambda^(-2) * pi * (pi-1)^(-1) * (1-pi^lambda)^2)v02 <- sum(w*log(pi) * lambda^(-2) * pi * (1-pi)^(-1) * (1-pi^lambda))v12 <- sum(w*x * log(pi) * lambda^(-2) * pi * (1-pi)^(-1) * (1-pi^lambda))fisher.infor <- - matrix(c(v00, v01, v02, v01, v11,v12, v02, v12, v22), 3,3)return(fisher.infor)}# dose selection functionf.dose.box <- function (x, theta, lambda, fisher.infor, ed){alpha <- theta[1]beta <- theta[2]pi <- f.lr.p(x, alpha, beta, lambda)v00 <- sum(lambda^(-2) * pi * (pi-1)^(-1)* (1-pi^lambda)^2)1164.10. R-code for simulationsv11 <- sum(x^2 * lambda^(-2) * pi * (pi-1)^(-1) * (1-pi^lambda)^2)v22 <- sum(log(pi)^2 * lambda^(-2) * pi * (pi-1)^(-1))v01 <- sum(x * lambda^(-2) * pi * (pi-1)^(-1) * (1-pi^lambda)^2)v02 <- sum(log(pi) * lambda^(-2) * pi * (1-pi)^(-1) * (1-pi^lambda))v12 <- sum(x * log(pi) * lambda^(-2) * pi * (1-pi)^(-1) * (1-pi^lambda))fisher.sel <- fisher.infor - matrix(c(v00, v01, v02, v01,v11, v12, v02, v12, v22), 3,3)cov.sel <- solve(fisher.sel)v00.sel <- cov.sel[1,1]v11.sel <- cov.sel[2,2]v22.sel <- cov.sel[3,3]v01.sel <- cov.sel[1,2]v02.sel <- cov.sel[1,3]v12.sel <- cov.sel[2,3]# varianemu_hat1 <- (logit (ed[1]^lambda) - alpha) / betamu_hat2 <- (logit (ed[2]^lambda) - alpha) / betamu_hat3 <- (logit (ed[3]^lambda) - alpha) / betavar1.sel <- (v00.sel + v11.sel*mu_hat1^2+ (log(ed[1])/(1-ed[1]^lambda))^2*v22.sel+ 2*v01.sel*mu_hat1 - 2*v02.sel*log(ed[1])/(1-ed[1]^lambda)- 2*v12.sel*mu_hat1*log(ed[1])/(1-ed[1]^lambda)) / beta ^ 21174.10. R-code for simulationsvar2.sel <- (v00.sel + v11.sel*mu_hat2^2+ (log(ed[2])/(1-ed[2]^lambda))^2*v22.sel+ 2*v01.sel*mu_hat2-2*v02.sel*log(ed[2])/(1-ed[2]^lambda)- 2*v12.sel*mu_hat2*log(ed[2])/(1-ed[2]^lambda)) / beta ^ 2var3.sel <- (v00.sel + v11.sel*mu_hat3^2+ (log(ed[3])/(1-ed[3]^lambda))^2*v22.sel+ 2*v01.sel*mu_hat3-2*v02.sel*log(ed[3])/(1-ed[3]^lambda)- 2*v12.sel*mu_hat3*log(ed[3])/(1-ed[3]^lambda)) / beta ^ 2sum.se <- var1.sel + var2.sel + var3.selreturn(sum.se)}# main party_pes1 = 1y_pes2 = 0x_pes1 = (logit (0.01^lambda) - alpha) / betax_pes2 = (logit (0.99^lambda) - alpha) / betay_pes3 = 0y_pes4 = 1x_pes3 = (logit (0.01^lambda) - alpha) / betax_pes4 = (logit (0.99^lambda) - alpha) / beta1184.10. R-code for simulationsfor (k in 1:m){# initial design# generate datayhat <- rbinom(length(dose.initial), 1,f.lr.p(dose.initial, alpha, beta, lambda))data <- data.frame(rbind(cbind (y_pes1, x_pes1), cbind (y_pes2, x_pes2),cbind (y_pes3, x_pes3), cbind (y_pes4, x_pes4),cbind (yhat, dose.initial)))names(data)[1] <- paste("y")names(data)[2] <- paste("x")theta <- c(alpha, beta)out <- f.est (data$x, data$y, theta)alpha_hat <- out[1]beta_hat <- out[2]lambda_hat <- out[3]# Second stage# Select the next dose levelfor (j in 1:(n-length(dose.initial))){# calculate fisher informationtheta <- c(alpha_hat, beta_hat)fisher.info <- fisher(data$x, theta, lambda_hat)1194.10. R-code for simulationsfor (i in 1:length(t)){sum[i] <- f.dose.box(t[i], theta, lambda_hat,fisher.info, ed)}opt <- t[which(sum == min (sum))]prob <- f.lr.p(opt, alpha, beta, lambda)y <- rbinom (1, 1, prob)data <- rbind (data, data.frame (y, x = opt))# new functiontheta <- c(alpha_hat, beta_hat)out <- f.est (data$x, data$y, theta)alpha_hat <- out[1]beta_hat <- out[2]lambda_hat <- out[3]}theta <- c(alpha_hat, beta_hat)out <- f.est (data$x, data$y, theta)alpha_hat <- out[1]beta_hat <- out[2]lambda_hat <- out[3]mu_temp1[k] <- (logit (ed[1]^lambda_hat) - alpha_hat) / beta_hat1204.10. R-code for simulationsmu_temp2[k] <- (logit (ed[2]^lambda_hat) - alpha_hat) / beta_hatmu_temp3[k] <- (logit (ed[3]^lambda_hat) - alpha_hat) / beta_hatmu_temp [k, ] <- c(mu_temp1[k], mu_temp2[k], mu_temp3[k])diff1[k] <- (mu_temp1[k] - mu1) ^ 2diff2[k] <- (mu_temp2[k] - mu2) ^ 2diff3[k] <- (mu_temp3[k] - mu3) ^ 2diff[k] <- diff1[k] + diff2[k] + diff3[k]print(k)}sqrt(mean(diff1))sqrt(mean(diff2))sqrt(mean (diff3))sqrt(mean(diff))121Chapter 5Group Sequential ED-Design5.1 IntroductionThe standard up-and-down design of Dixon and Mood (1948) has been in-troduced in Chapter 2. Under this design, the dose of the next trial is onelevel higher or lower than that of the current trial depending on the outcomeof the current subject. Much of the discussion of the up-and-down design(Dixon and Mood, 1948; Derman, 1957; Durham and Flournoy, 1994, 1995;Durham et al., 1997; Stylianou and Flournoy, 2002) focused on the situationwhere there is only one subject at each trial: only one subject is admitted,and its response is used to choose the next dose level.The idea of the up-and-down design can also be used so that multi-ple subjects are admitted to the experiment at each trial. Anderson et al.(1946); Wetherill (1963); Tsutakawa (1967b); Gezmu and Flournoy (2006)among others, proposed a group version of the up-and-down design. Theyconsidered the problem that a group of m subjects are admitted to a singledose level at each trial. They proposed to determine the dose level for thenext trial based on the number of positive responses at the current trial.If the number of positive responses in the group is less than or equal to athreshold s, the dose level of the next trial will be moved one level higher.1225.1. IntroductionIf the number of positive responses is greater than or equal to a thresholdt, the dose of the next trial will be moved one level lower. Otherwise, thedose level for the next trial stays the same. With appropriate values of thegroup size m, and thresholds s and t, Gezmu and Flournoy (2006) showedthat the group up-and-down design allocates a large number of dose levelsaround the prespecified target dose.Clinical trials involving developing new drugs include all drug devel-opment experiments that are conducted on human beings. They can lastfor many years with high financial and human costs. “Phase II/III StudyTrends and Market Outlook (2016-2020)” reported that large drug develop-ment Sponsors (R&D $500M+) spent $465,725,000 in Phase II/III trials in2015 on average. Despite the high financial costs of clinical trials, there aregreater human costs. New drugs studied in clinical trials can be unsafe andinferior. It is unethical to make any decision lightly. Thus, it is desirable todesign the experiments to speed up the clinical trial, and prevent waste ofvaluable resources. In practice, clinical trials are carried out over a certainperiod of time, and individuals often enter the trial sequentially in groups.Thus, for ethical, scientific and economic reasons, it is often more practicaland natural to treat individuals sequentially in groups.The sequential ED-design proposed in Chapter 3 has been developed tohave one subject admitted at each trial. It appears that the same idea workswhen a group of subjects admits at each trial. In this chapter, we proposea group scheme and call it two-stage group sequential ED-design. The firststage will be a fix-point design. A group of subjects will be assigned tosome pre-selected dose levels. The responses of these subjects will be used1235.2. Two-stage group sequential ED-design under the logistic regression modelto obtain a rough parameter estimation. At the second stage, we use a groupsequential procedure. We look for the most informative dose combinationfor the next group of subjects with respect to some criterion. This procedureis then repeated until the experimental subjects are exhausted.A simulation study is conducted to investigate the performance of theproposed two-stage group sequential ED-design under various scenarios. Wemimic real dose–response experiments with the goal of accurately estimatingthe unknown dose–response curve over a wide dose range.5.2 Two-stage group sequential ED-design underthe logistic regression modelWe only consider the case where each group is made of two subjects andtwo different dose levels are permitted. The idea can be used to groups withdifferent sizes and more dose levels. We investigate three possible ways toselect two doses at each trial.Suppose ξj , j = 1, 2, 3 are the target ed levels. Each ed level is a functionof the model parameters α and β under the logistic regression model: ξj =gj(α, β) for some smooth function gj . Let αˆk, βˆk be the maximum likelihoodestimators of the model parameters based on the outcomes of the first ktrials. As it has been pointed out before, the variance of ξˆ = g(αˆk, βˆk) isconceptually approximated byvar(ξˆ) = 5g(αˆk, βˆk)T I−1k (x; αˆk, βˆk)5 g(αˆk, βˆk)1245.2. Two-stage group sequential ED-design under the logistic regression modelwhere 5g(αˆk, βˆk) is the gradient of g(αˆk, βˆk) and Ik(x; αˆk, βˆk) is the Fisherinformation matrix based on the existing k trials and an extra trial at doselevel x, at the model parameter value αˆk, βˆk. Each of the next three proposeddose selection methods aims to minimize some criterion.1. Let ξˆ2 be the maximum likelihood estimate of ξ2 based on the outcomesof the first k trials. We computex1 = arg minx≤ξˆ2{ 3∑j=15g(αˆk, βˆk)T I−1k (x; αˆk, βˆk)5 g(αˆk, βˆk)}x2 = arg minx≥ξˆ2{ 3∑j=15g(αˆk, βˆk)T I−1k (x; αˆk, βˆk)5 g(αˆk, βˆk)}.(5.1)A simple linear search can be used to find solutions easily.2. We select two doses to minimize the total anticipated asymptotic vari-ance of the target ed levels in the second stage of our group sequentialED-design:(x1, x2) = arg minx1,x2{ 3∑j=15g(αˆk, βˆk)T I−1k (x1, x2; αˆk, βˆk)5 g(αˆk, βˆk)}.(5.2)where Ik(x1, x2; αˆk, βˆk) is the Fisher information matrix based on theexisting k trials and two extra trials at dose levels x1 and x2, at themodel parameter value αˆk, βˆk.1255.3. Simulation3. Remember that3∑j=15g(αˆk, βˆk)T I−1k (x; αˆk, βˆk)5 g(αˆk, βˆk)represents the potential total asymptotic variance of ξˆj , j = 1, 2, 3 afterthe first k trials and one single additional trial at x. Our experienceindicates that, as a function of x, it can have two local minima. Wepropose to choose two dose levels for the next group of two subjects atthese two local minima. We again use a simple linear search for thispurpose.5.3 SimulationWe conduct simulations to investigate the performance of the group sequen-tial ED-design. We compare the new design with several existing designs,and repeat the simulation N = 1000 times for all model/design combina-tions. The simulation sample sizes are chosen to be n = 30, 60, and 120. Wechoose three effective dose levels each time as the targets and obtain theirMLEs. Under each model/design setting, we compute the RMSE of a singleed level asRMSE(ξˆj) =√√√√N−1 N∑r=1(ξˆrj − ξj)2,where ξˆrj is the estimate of ξj in the rth repetition. The total RMSE iscomputed asRMSE =√√√√ 3∑j=1RMSE2(ξˆj).1265.3. Simulation5.3.1 Detailed specificationsThe detailed simulation specifications are as follows. Four designs are in-cluded in the simulation: the up-and-down design, the group up-and-downdesign, the sequential ED-design, and the group ED-design. Similar to thesetting of Chapter 3.4, for the up-and-down design, we choose K = 7 equalspaced grids for Ω between the anticipated ed01 and ed90. For the groupED-design and the sequential ED-design, the initial design is uniform on Ω.The group up-and-down design uses the same Ω as its grids. The choice ofthe first dose level, thresholds s and t, and the group size s will be specifiedlater in the simulation.5.3.2 Performance comparison when the response model iscorrectly specifiedIn this section, we consider the situation where the assumed response curveagrees with the true curve. We generate data according to the followinglogistic regression modellogit[pi(x)]= −6.265 + 0.055x.Under this model, ed25 = 94, ed50 = 114, and ed75 = 134.The details of the four designs under this model are as follows:• For the up-and-down design, the specific dose levels are x1 = 34 andx7 = 154. The dose range is given byΩ = {34, 54, 74, 94, 114, 134, 154}.1275.3. SimulationThe initial dose level is set to x4 = 94.• For the ED-design, we use the following grid of K = 7 doses in thefirst stage:Ω = (84, 94, 104, 114, 124, 134, 144).• For the group up-and-down design, the dose range is given byΩ = {34, 54, 74, 94, 114, 134, 154}.The initial dose level is set at x4 = 94. We choose thresholds s = 0,t = 2, and group size s = 2.• For the group ED-design, we use the following grid of K = 8 doses inthe first stage:Ω = (74, 84, 94, 104, 114, 124, 134, 144).The subsequent doses are chosen according to each of the three pro-posed dose selection methods.In the first simulation, we set ed25, ed50, and ed75 as the target doselevels. The simulation results are given in Table 5.1. The results show thatthe group sequential ED-design is noticeably superior to the group up-and-down design with lower total RMSEs when n = 30, n = 60 and n = 120.The differences are getting smaller as n increases.Its individual RMSEs is generally lower except targeting ed50 whenn = 60 and n = 120. The group ED-design based on the third selection1285.3. Simulationmethod has lower RMSEs than the group ED-design with the first and sec-ond selection methods. If ed25 is the target, the group ED-design (method3) is superior than the ED-design with lower RMSE. Comparing with theup-and-design, group ED-design (method 3) is generally superior with lowertotal RMSEs.Table 5.1: Simulated RMSEs under the logistic model targeting range ed25–ed75nEDGroup ED-designGroup UD Up-and-downM 1 M 2 M 330Total 17.14 18.60 18.04 17.80 22.81 18.12ED25 10.55 10.79 11.36 10.20 11.75 10.48ED50 8.12 7.86 8.94 8.66 9.24 8.06ED75 10.79 12.95 10.80 11.75 17.23 12.3960Total 12.54 13.16 12.93 12.73 14.67 12.79ED25 7.80 7.79 8.41 7.42 8.67 7.90ED50 5.89 5.89 6.31 6.05 5.69 5.53ED75 7.85 8.81 7.52 8.40 10.37 8.40120Total 8.74 9.02 9.00 8.84 9.92 9.16ED25 5.46 5.49 5.73 5.22 6.25 5.89ED50 4.09 4.30 4.35 4.23 3.80 3.88ED75 5.46 5.72 5.40 5.74 6.69 5.85In the second simulation, we consider the situation where a lower rangeof ed levels is of interest. We take ed10, ed25, and ed40 as the target1295.3. Simulationdose levels. The simulation settings remain the same except that the initialdose level for the up-and-down design is set at x3 = 74. This approach canselect only one target dose level in each simulation. We simulated all threepossibilities, and the simulation results are given in Table 5.2.The results show that the group ED-design is noticeably superior to thegroup up-and-down design with lower total RMSEs. In comparison with theup-and-down design, the group ED-design is generally superior with lowerRMSEs. Tuning the biased-coin up-and-down design to specific ed levelsimproves its results. Particularly when targeting ed25, the up-and-downdesign achieved lower RMSEs. If ed25 is the target, the group ED-design(method 1) is superior to the ED-design with lower RMSEs in all threesample sizes simulated.5.3.3 Performance comparison when the response model ismis-specifiedIn applications, the dose–response relationship is unknown. In this section,we consider the case where the observed response curve is mis-specified.We investigate the performance of the group ED-design. Specifically, weconsider the case where the observed dose–response relationship is logisticbut the true model is probit. Thus, we generate data according to the probitmodel (3.8). Recall that under model (3.8), we haveedγ =Φ−1(γ/100)− αβ,1305.3. SimulationTable 5.2: Simulated RMSEs under the logistic model targeting range ED10–ED40.n EDGroup-ED design Up-and-downGroup UDM 1 M 2 M 3TargetED10TargetED25TargetED4030Total 17.93 19.58 20.16 19.07 21.82 18.81 20.24 24.11ED10 12.68 15.13 15.19 13.62 11.61 13.18 15.59 19.00ED25 8.93 8.92 10.02 9.50 11.15 8.92 9.71 11.24ED40 9.01 8.66 8.67 9.37 14.74 10.02 8.51 9.7060Total 13.25 14.39 14.44 13.38 17.38 13.25 15.02 17.69ED10 9.48 10.94 10.87 9.72 8.57 9.69 11.90 14.38ED25 6.52 6.33 7.15 6.49 8.98 6.09 7.11 8.34ED40 6.56 6.88 6.28 6.51 12.17 6.69 5.78 6.05120Total 9.35 9.36 10.06 9.13 13.38 9.36 10.57 12.60ED10 6.73 6.65 7.68 6.57 6.14 6.93 8.52 10.28ED25 4.55 4.42 4.92 4.36 6.96 4.24 4.92 5.99ED40 4.63 4.88 4.25 4.61 9.64 4.66 3.88 4.15and ed25 = 102, ed50 = 114, and ed75 = 127. The dose–response curveis given by Model (3.7), and the simulation is otherwise identical to thatin the last section. The results are presented in Tables 5.3 and 5.4. Fored25–ED75, we set ed50 as the target for the up-and-down design. Fored10–ED40, we target each level separately, as before.In the next simulation, we repeat the simulation with the target doselevels changed to ed10, ed25, and ed40. In comparison with the groupup-and-down design, the group ED-design clearly has the best overall per-formance in both ranges. The up-and-down design again has good perfor-1315.3. SimulationTable 5.3: Simulated RMSEs under probit mis-specified as logistictargeting ED range 25–75.nEDGroup ED-designGroup UD Up-and-downM 1 M 2 M 330Total 10.08 11.29 11.03 11.29 13.45 11.29ED25 6.25 6.77 6.82 6.68 7.14 6.70ED50 4.86 5.15 5.64 5.66 5.74 5.03ED75 6.23 7.42 6.58 7.12 9.84 7.5760Total 7.16 7.58 7.85 7.78 8.53 7.63ED25 4.53 4.69 5.05 4.67 4.94 4.71ED50 3.38 3.48 3.97 3.83 3.53 3.46ED75 4.39 4.83 4.01 4.91 6.00 4.90120Total 5.00 5.32 5.57 5.46 5.73 5.39ED25 3.09 3.22 3.53 3.18 3.55 3.38ED50 2.38 2.53 2.78 2.69 2.34 2.43ED75 3.13 3.40 3.29 3.53 3.84 3.42mance at the target ed level, but poorer performance overall. Targetinged25 achieves the best trade-off.The group ED-design based on the third selection method has lower totalRMSEs than the other two methods. The sequential ED-design has the bestperformance, except when targeting ed40 with n = 120.1325.4. Concluding remarksTable 5.4: Simulated RMSEs under probit mis-specified as logistictargeting ED range 10–40.n EDGroup-ED design Up-and-downGroup UDM 1 M 2 M 3TargetED10TargetED25TargetED4030Total 11.16 11.74 12.13 11.49 14.54 11.62 11.81 14.65ED10 7.80 8.39 8.84 7.99 6.96 7.55 8.82 11.08ED25 5.59 5.78 6.11 5.62 7.84 5.92 5.87 7.24ED40 5.69 5.83 5.63 6.05 10.07 6.56 5.24 6.2760Total 7.62 8.20 8.53 7.75 10.73 8.03 8.45 10.08ED10 5.34 5.92 6.25 5.40 4.84 5.57 6.38 7.98ED25 3.78 3.84 4.25 3.80 5.71 3.99 4.21 4.89ED40 3.90 4.18 3.94 4.05 7.68 4.20 3.59 3.74120Total 5.30 5.32 5.86 5.46 7.70 5.56 5.94 7.21ED10 3.76 3.74 4.40 3.78 3.53 3.83 4.41 5.75ED25 2.59 2.67 2.89 2.69 4.14 2.79 3.02 3.53ED40 2.69 2.68 2.57 2.88 5.45 2.91 2.59 2.555.4 Concluding remarksClinical trials are planned experiments on human beings with high financialand human costs. It is desirable to design the experiments to speed up thetrial and prevent waste of valuable resources. In practice, individuals oftenenter the trial sequentially in groups. Hence, it is often more practical totreat individuals by groups. Motivated by this observation, in this chapter,we propose a group sequential ED-design.Our group ED-design has a natural extension to more complex models.1335.4. Concluding remarksMoreover, the group ED-design can also be used for any ed levels, and meeta broad range of the demands that may arise in applications. Simulationsshow that in general our design is more robust, and compares favourablywith existing designs.134Chapter 6Asymptotic PropertiesIn dose–response experiments with sequential designs, doses administratedto experimental subjects are selected depending on previous doses and re-sponses. Data generated from such experiments are not independent. De-spite the dependence structure arising from such designs, the likelihood isidentical to the one derived from independent observations. For discussionsin this respect, see Chaudhuri and Mykland (1993, 1995); Stylianou andFlournoy (2002); Hu and Rosenberger (2006); Fedorov and Leonov (2013);Rosenberger and Lachin (2015).In this chapter, we first investigate the likelihood function derived fromthe dependent observations generated from dose–response experiments withsequential designs. We present the derivation of the likelihood function forthe up-and-down experiment.Next we study the asymptotic properties of the maximum likelihoodestimators from designs with certain properties. When independent andidentically distributed observations are available, it is well known that undersome regularity conditions, maximum likelihood estimators are the solutionsto their score functions. They are consistent and asymptotically normal. Wepresent some general results on the asymptotic properties of the maximum1356.1. Data structurelikelihood estimators following a two-stage sequential design. We provideevidence that the maximum likelihood estimators from the two-stage se-quential design exist, and have the usual asymptotic properties.6.1 Data structureFollowing Hu and Rosenberger (2006), we begin with a useful data structurewhich facilitates the derivation of the likelihood. Consider an experimentwith n experimental subjects. Each subject is assigned to a stimulus atone of K dose levels. Suppose that subjects are assigned sequentially andrespond immediately. Let T = (T1, . . . , Tn)T be a matrix of randomizationsequence, where Ti = (Ti1, . . . , TiK) is a vector of zeroes with a 1 in the jthentry, if jth dose level is assigned to the ith subject.Let Y = (Y1, . . . , Yn)T be a matrix of responses, where Yi = (Yi1, . . . , YiK)is a sequence of responses which would be observed, if every dose level isassigned to the ith subject independently. Note that only one element of Yiis observable.Let ti = (ti1, . . . , tiK) and yi = (yi1, . . . , yiK) be the realized dose as-signments and responses from the ith subject, i = 1, 2, . . . , n. The observeddata for the ith subject is zi = {ti,∑Kj=1 tijyij}. Note that Yij is observedonly if Tij = 1.1366.1. Data structure6.1.1 The likelihoodBased on the above data structure, Y1j , Y2j , . . . , Ynj , are independent andidentically distributed. Denote their density function asY1j ∼ f(·; θj) (6.1)where θj is the unknown parameter of interest. For s = 1, . . . , n, Ys isindependent of Y1, Y2, . . . , Ys−1, T1, . . . , Ts. However, Ts is dependent onY1, Y2, . . . , Ys−1, T1, . . . , Ts−1.Now we consider a sequentially designed dose–response experiment, inwhich doses assigned to subjects are selected sequentially based on previousresponses and doses. Let us first consider the situation where K = 2 andn = 1. The observed data in this case are z1 = {t1, t11y11 + t12y12}. WehaveP (Z1 = z1) =P (Z1 = y11) = f(y11; θ1) if t12 = 0P (Z1 = y12) = f(y12; θ2) if t12 = 1The likelihood function isL(θ1, θ2) = f(y11; θ1)t11f(y12; θ2)t12 .Next we consider the situation where K = 2 and n = 2. The ob-served data for these two subjects are z1 = {t1, t11y11 + t12y12}, and z2 ={t2, t21y21 + t22y22}. Since the dose assignment of the second subject is com-pletely determined by the dose assignment and response of the first subject,1376.1. Data structureP (T2 = t2|Z=z1) = 1. Hence, the likelihood function is given byL(θ1, θ2) = P (Z1 = z1, Z2 = z2)= P (Z2 = z2|Z1 = z1)P (Z1 = z1)= P (Z2 = z2|Z1 = z1, T2 = t2)P (T2 = t2|Z=z1)P (Z1 = z1)= P (Z2 = z2|T2 = t2)P (T2 = t2|Z=z1)P (Z1 = z1)= P (Z2 = z2|T2 = t2)P (Z1 = z1)= f(y21; θ1)t21f(y22; θ2)t22f(y11; θ1)t11f(y12; θ2)t12 .Following the same principle, the likelihood derived from the sequentialexperiment with K doses and n observations is as follows:L(θ1, . . . , θK) = f(y11; θ1)t11 × f(y12; θ2)t12 × . . .× f(y1K ; θK)t1K× f(y21; θ1)t21 × f(y22; θ2)t22 × . . .× f(y2K ; θK)t2K× f(y31; θ1)t31 × f(y32, θ2)t32 × . . .× f(y3K ; θK)t3K...× f(yn1; θ1)tn1 × f(yn2; θ2)tn2 × . . .× f(ynK ; θK)tnK=n∏i=1K∏j=1{f(yij ; θj)}tij .Note that this likelihood is identical to the one based on independent obser-vations.In the up-and-down experiment, Yij is a Bernoulli random variable. Itequals to 1 if the ith subject responds at the jth dose level, and 0 otherwise,j = 1, . . . ,K. Thus, f(·; θj) is the probability function of the Bernoulli dis-1386.2. Maximum likelihood estimationtribution, with the probability of success pj being the probability of responseat jth dose level. The likelihood function is therefore given byL(·) =n∏i=1K∏j=1[pyijj (1− pj)1−yij ]tij=n∏i=1K∏j=1ptijyijj (1− pj)tij(1−yij)=K∏j=1p∑ni=1 tijyijj (1− pj)∑ni=1 tij(1−yij).Let Nj =∑ni=1 Tij be the number of subjects assigned to dose level j,and let Sj =∑ni=1 YijTij be the number of subjects respond at dose level j.Let nj and sj be the observed values of Nj and Sj . The likelihood functionis therefore written asL(·) =K∏j=1psjj (1− pj)nj−sj (6.2)The above likelihood function is generally applicable.6.2 Maximum likelihood estimationAs discussed above, data generated from the sequential design are depen-dent. Chaudhuri and Mykland (1993) showed that despite the dependencestructure in the data, the resulting likelihood function is the same to theone derived from the independent observations. Following Chaudhuri andMykland (1993), we investigate the likelihood function from the sequentialED-design.1396.2. Maximum likelihood estimation6.2.1 Data structure and assumptionsFollowing Chaudhuri and Mykland (1993), we consider a dose–response ex-periment with the response variable Y and explanatory variable X whosevalues are chosen from a finite experiment space Ω. Let the response space,the set of all possible outcomes of the experiment, be R. Denote the condi-tional distribution of Y given X = x as f(y|θ, x). Let Θ be the parameterspace. Here θ ∈ Θ is the unknown parameter, and f is a known distribu-tion function. In addition, we assume f(y|θ, x) is smooth and regular sothat log f(y|θ, x) is differentiable in θ, and the Fisher information matrix,denoted as I(θ;x), exists finitely. The Fisher information matrix I(θ;x) is∫R[∇ log{f(y|θ, x)}][∇ log{f(y|θ, x)}]T µ(dy),where ∇ is the gradient operator, and µ is the usual counting measure.Recall from Chapter 3, the sequential ED-design consists of two stages.Assume that the total number of trials n is predetermined prior to theexperiment. Suppose that n1 trials are carried out in the first stage. Theremaining (n− n1) trials are carried out in the second stage. Compute themaximum likelihood estimator θˆn1 of the model parameter θ based on theobserved data (y1, x1), . . . , (yn1 , xn1) from the first n1 trials. In the secondstage, suppose ξj , j = 1, 2, 3, are the target ed levels. Each ed level is afunction of the model parameter θ, i.e., ξj = g(θ). Let θˆi−1 be the maximumlikelihood estimator of θ based on (y1, x1), . . . , (yi−1, xi−1), n1 + 1 ≤ i ≤ n.Let ξˆj = g(θˆi−1), via delta-method, the asymptotic variance of ξˆj can be1406.2. Maximum likelihood estimationapproximated by5g(θˆi−1)T I−1i−1(θˆi−1;x)5 g(θˆi−1)where 5g(θˆi−1) is the gradient of g(θˆi−1) and I−1i−1(θˆi−1;x) is the inverse ofthe Fisher information matrix after (i− 1) trials and the potential ith trialto be run at dose level x. Hence, for each i, n1 + 1 ≤ i ≤ n, the ith designpoint is determined byxi = arg minx3∑j=1var(ξˆj), j = 1, 2, 3.From the sequential scheme discussed above, the ith design point Xi isdetermined based on the past observations (Y1, X1), . . . , (Yi−1, Xi−1). Thus,the observations generated from this sequential design are no longer in-dependent. The standard asymptotic properties of the maximum likeli-hood estimates may not be applicable. However, the dependence of Yi on(Y1, X1), . . . , (Yi−1, Xi−1) is only through Xi. As a consequent, the likeli-hood constructed from (Y1, X1), . . . , (Yi, Xi) remains in the product form∏ir=1 f(Yr|θ,Xr), despite the dependence structure of the data arising fromthe sequential design (See Chaudhuri and Mykland (1993)). Hence, thelikelihood conducted from the sequential ED-design is identical to the onearising from the independent and identically distributed observations. As aresult, the resulting MLE of θ can be computed as usual. We denote the1416.3. Asymptotic properties of the maximum likelihood estimateMLE of θ based on (Y1, X1), . . . , (Yn, Xn) asθˆn = arg maxθ∈Θn∏r=1f(Yr|θ,Xr)In the next section, we investigate the asymptotic properties (i.e., con-sistency and asymptotically normality) of θˆn derived from the sequentialED-design. Following Chaudhuri and Mykland (1995), we identify some reg-ularity conditions which will guarantee the desirable asymptotic behavioursof the maximum likelihood estimates.6.3 Asymptotic properties of the maximumlikelihood estimateIn this section, we investigate the asymptotic properties of θˆn from thesequential ED-design. Following Chaudhuri and Mykland (1993, 1995), wesummarize the following general conditions on the response model f(y; θ, x).• Condition 1: The response space R does not depend on θ and x. Forevery y ∈ R and x ∈ Ω, log f(y; θ, x) is thrice continuously differen-tiable in θ at any θ ∈ Θ.• Condition 2: Let ∇ log f(y; θ, x) = G(y; θ, x) be the gradient vector oflog f(y|θ, x) with respective to θ. Then G(y; θ, x) should satisfy∫RG(y; θ, x)f(y; θ, x)µ(dy) = 0,1426.3. Asymptotic properties of the maximum likelihood estimateandsupx∈Ω∫R|G(y; θ, x)|2+tf(y; θ, x)µ(dy) <∞for some t > 0. Here | · | is the usual Euclidean norm.• Condition 3: Let H(y; θ, x) be the Hessian matrix of log f(y; θ, x) asthe second order partial derivatives of log f(y; θ, x) with respective toθ. Then∫RH(y; θ, x)f(y; θ, x)µ(dy)= −∫R(G(y; θ, x))(G(y; θ, x))Tf(y; θ, x)µ(dy)= −I(θ;x),andsupx∈Ω∫R|H(y; θ, x))|2f(y; θ, x)µ(dy) <∞.• Condition 4: For every θ ∈ Θ, there is an open neighbourhood N(θ) ⊂Θ, and a nonnegative random variable K(y; θ, x) which satisfiessupx∈Ω∫RK(y; θ, x)f(y; θ, x)µ(dy) <∞.Each of the third order partial derivatives of log f(y; θ′, x) with respec-tive to θ′ is dominated by K(y; θ, x) for all θ′ ∈ N(θ).In addition, Chaudhuri and Mykland (1995), presented the followingasymptotic results for the maximum likelihood estimate in a general adaptivesequential design setting. We summarize their findings in the following.1436.3. Asymptotic properties of the maximum likelihood estimateTheorem 1: Assume that Condition 1 through Condition 4 hold, and(Y1, X1), . . . , (Yn, Xn) are observations generated from an adaptive sequen-tial design. Denote λn as the smallest eigenvalue of1nn∑r=1I(θ;Xr).Suppose the design is that for some positive constant α < 1/4, nαλn remainsbounded away from zero in probability as n→∞ for any θ ∈ Θ. Then themaximum likelihood estimator θˆn of θ exists and is weakly consistent for θ.Theorem 2: Assume that Condition 1 through Condition 4 hold, and(Y1, X1), . . . , (Yn, Xn) are observations generated from an adaptive sequen-tial design. Suppose1nn∑r=1I(θ;Xr)p→ A as n→∞,where A is a nonrandom positive definite matrix. Then there exists a max-imum likelihood estimator θˆn of θ such that the distribution of√n(θˆn − θ)converges weakly to a multivariate normal distribution with mean zero andvariance-covariance matrix A−1 as n→∞.The proofs of Theorems 1 and 2 were given in Chaudhuri and Myk-land (1993, 1995). Their proofs utilize some standard martingale techniquesintroduced in Lai and Wei (1982) for an adaptive sequential design. Wepresent and summarize their proofs as follows.1446.3. Asymptotic properties of the maximum likelihood estimateProof of Theorem 1. Let {Fni, 1 ≤ i ≤ n} be an increasing sequence ofσ-fields generated by Y1, . . . , Yi, that is, Fni = σ(Y1, . . . , Yi). It follows fromCondition 2 that{ i∑r=1G(Yr; θ,Xr);Fni, 1 ≤ i ≤ n}is a square integrable martingale. Then,E∣∣ n∑r=1G(Yr; θ,Xr)∣∣2 = E( n∑r=1∣∣G(Yr; θ,Xr)∣∣2) = Op(n)as n→∞. Hence,n∑r=1G(Yr; θ,Xr) = Op(n1/2). (6.3)Next, consider the sequence{ i∑r=1[H(Yr; θ,Xr) + I(θ;Xr)];Fni, 1 ≤ i ≤ n}.Condition 3 implies that it is also a square integrable martingale. By asimilar argument,n∑r=1[H(Yr; θ,Xr) + I(θ;Xr)]= Op(n1/2). (6.4)For any δ > 0, let Nδ(θ) be the neighbourhood centred at θ with radius δ.Let δn = n−β for some α < β < (1/2)− α. Then, the smallest eigenvalue of1456.3. Asymptotic properties of the maximum likelihood estimatethe Hessian matrix ofnα−1n∑r=1log f(Yr|θ′, Xr)is negative and bounded away from zero in probability as n → ∞ for allθ′ ∈ Nδ(θ). This implies thatlimn→∞P( n∑r=1log{f(Yr; θ′, Xr)} is concave for θ′ ∈ Nδ(θ))= 1. (6.5)Next consider the third order Taylor expansion of the log-likelihood aroundθ,n∑r=1log f(Yr; θ′, Xr) =n∑r=1log{f(Yr; θ,Xr)}+ (θ′ − θ)T[ n∑r=1G(Yr; θ,Xr)]+ (θ′ − θ)T[ n∑r=1H(Yr; θ,Xr)](θ′ − θ) +Rn(θ′, θ).Under Condition 4, the remainder term in the above equation satisfiessupθ′:|θ′−θ|≤δn|Rn(θ′, θ)| = Op(nδ3n). (6.6)Now (6.5) and (6.6) imply that the probability of the event that the like-lihood equation has a root within the neighbourhood Nδ(θ) will tend to 1as n → ∞. Because the size of the neighbourhood can be made arbitrar-ily small, this further implies the consistency of the maximum likelihoodestimator.1466.3. Asymptotic properties of the maximum likelihood estimateProof of Theorem 2. Using the result in Theorem 1, let θˆn be a weaklyconsistent estimator for θ. Consider a first order Taylor expansion aroundθ,n−1n∑r=1G(Yr; θˆn, Xr) = n−1n∑r=1G(Yr; θ,Xr)+[n−1n∑r=1H(Yr; θ,Xr) + ∆n(θ)]T(θˆ − θ)where ∆n(θ) is a random matrix of size op(1) under Condition 4 and weakconsistency of θˆn. Following Conditions 2 and 3, and the design condition inTheorem 2, by the martingale central limit theorem (Hall and Heyde, 2014),n−1/2n∑r=1G(Yr, θ,Xr)converges weakly to a multivariate normal distribution with mean zero andvariance-covariance matrix A.Note that Conditions 1 through 4 are standard Cramer-type conditionsthat hold for a large class of models. The response models, f(y; θ, x),(e.g. standard logistic regression model) considered in this thesis for dose–response experiments are smooth and regular. These Cramer-type condi-tions 1 through 4 are satisfied. In addition, under the sequential ED-design,the design points are sequentially selected in the same fashion as in Chaud-huri and Mykland (1995). Hence, the sequence of design points satisfy theconditions assumed in Theorem 1. Other than the eigenvalue condition,all other conditions are satisfied. As a result, we have nearly proved thatthe maximum likelihood estimator derived from our sequential ED-design is1476.4. Concluding remarksconsistent and asymptotically normal.6.4 Concluding remarksFollowing Chaudhuri and Mykland (1995), we went through the derivation ofthe likelihood function based on the dependent observations generated fromthe proposed design. We show that it is identical to the one with indepen-dent observations. We identify some regularity conditions under which theresulting maximum likelihood estimators are consistent and asymptoticallynormal.148Chapter 7Contributions and FutureResearchDose–response experiments are routinely conducted in Phases I and II clini-cal trials to study the relationship between the doses of a stimulus and the re-sponses of experimental subjects. Estimating the underlying dose–responserelationship is the primary goal of dose–response experiments (Dette et al.,2005; Dragalin et al., 2008a,b).Accurately charactering the dose–response relationship is a key step inthe clinical development process of pharmaceutical drugs. Poor understand-ing of the underlying dose–response relationship may lead to select wrongtarget doses to be used in large scale confirmatory clinical trials, which maycause serious ethical and financial consequences. Selecting too high a dosemay cause potential toxicity to experimental subjects, and choosing too lowa dose may fail to establish adequate efficacy, and fail to obtain the regula-tory approval of the drug. See Bretz et al. (2008), Bretz et al. (2010), andDette et al. (2008).Statistical design theory is therefore developed to most effectively collectthe needed information while minimizing potential side effects. Despite its1497.1. Contributionslong history, design theory for binary dose–response experiments remains anactive research area.7.1 ContributionsThe first contribution of this dissertation is the introduction of a new op-timality criterion. Traditionally, when a parametric dose–response modelis assumed, we often search for designs which enable us to most accuratelyestimate the model parameters. In applications as we observed, the ultimategoal of the investigation is to accurately determine various ed levels. In thisdissertation, we take a new approach in designing a binary dose–responseexperiment. We consider a situation where the dose–response relationshipover a range of ed levels is of interest.We proposed a new design criterion which directly and simultaneouslytargets several ed levels. We believe such a relationship can be well char-acterized after several tactically chosen ed levels are accurately estimated.Based on these considerations, we propose a two-stage sequential design.The proposed sequential design is easy to implement in general and leadsto more efficient estimation of ed levels. Because our design is sequentialand aims to efficiently estimate several chosen ed levels, we call it two-stagesequential ED-design or simply ED-design.We conducted extensive computer simulations to demonstrate that theproposed sequential ED-design indeed improves the efficiency of the experi-ment by changing the optimality target from estimating model parametersto ed levels of interest compared with many existing designs. First, we1507.1. Contributionsconfirm that our ED-design is indeed more efficient for estimating severaltargeted ed levels based on the total root mean square error (RMSE) orthe individual RMSEs. The D-optimal design and its sequential version donot have this flexibility, and the up-and-down design cannot target morethan one ed level. These results provide strong support for the proposeddesign. Second, because in practice the true dose–response relation neverfully conforms to the model, optimal designs do not perform at their peaklevels in general. We therefore use simulation studies to evaluate the effectof model misspecification. The ED–design still has the best performance interms of the RMSE. We also provide some simulation evidence for the lim-iting ED-design when the sample size n goes to infinity. It appears that asa distribution over the dose range, the design has a limit with two supportpoints.Another apparent approach to reduce the risk of model misspecificationis to apply a more flexible and hence more complex dose-response model.The choice of such a model reflects a trade-off between the model flexibilityand inference efficiency. Commonly used logistic or probit models are simpleand have good mathematical and statistical properties. They are satisfac-tory in many applications. Nevertheless, their model assumptions imposesome severe restriction on the dose-response relationship. Hence, a mildlymore complex model can be useful to lower the risk of model misspecifica-tion if it does not complicate the issues related to optimal designs and dataanalyses as well as maintaining good efficiency in estimating the ed levels.In this dissertation, we introduced the three–parameter logistic model.Some details of the ED-design under the three–parameter logistic regres-1517.2. Future Researchsion model are given. We investigate the effectiveness of the sequential ED-design, the D-optimal design, and the up-and-down design under this model,and develop an effective model fitting strategy. We develop an easy way toimplement an iterative numerical algorithm with guaranteed convergencefor computing the maximum likelihood estimation of the model parameters.The sequential ED-design can be implemented after some laborious but sim-ple mathematical derivations. Although we have yet to generate any theoryon its D-optimal design, a numerical procedure via the well-developed vertexdirection method (VDM) works well. Simulations show that the combina-tion of the proposed model and the data analysis strategy performs well.When the logistic model is correct, using the more complex model suffershardly any efficiency loss. When the three-parameter model holds but thelogistic model is violated, the new approach can be more efficient.In addition to these achievements, we discuss the use of the ED-designwhen experimental subjects become available in groups. We introduce thegroup sequential ED–design, and show how to construct this design.7.2 Future ResearchIn this dissertation, the property of the ED-design is studied numericallyand analytically. The theoretical aspect of the proposed design has notbeen fully explored. We will focus on the theoretical aspect such as theasymptotic properties of the ED-design in future research. For example,simulation studies show that for sample size n = 1000, the doses generatedfrom the ED-design cluster around two ed levels of the underlying dose–1527.2. Future Researchresponse curve. In the future, we will investigate if doses generated fromthe ED-design converge to these two specific ed levels as n tends to infinity.In this dissertation, we apply the vertex direction method (VDM) to nu-merically compute the D-optimal designs under the three-parameter logisticand probit models. The resulting D-optimal designs are uniform distribu-tions on three support points. Under these two models, design points anddesign weights change with different λ values. We do not have a compara-ble theory for the D-optimal design under new models but point out that avertex direction method remains effective for numerical solutions. In futureresearch, we plan to generate a theory on the D-optimal design under morecomplicated models.The proposed two-stage ED-design is very easy to implement in medicalresearch. To utilize this design, the responses of experimental subjects needto be observed quickly, such as in anesthesia research, a subject’s responseto anesthetic drugs, i.e., being anesthetized or not, is observed immediately.However, in most clinical practice, especially in cancer trials, a subject’sresponse is not immediately obtained. The ED-design may delay the assign-ment of the subsequent subjects and lead to long trial duration. Thus, it isinteresting to modify our design to incorporate delayed responses.We may also generalize our proposed design in many ways. In thisdissertation, we mainly focus on studying the ED-design under the logisticresponse model. As a starting point, we have investigated the performanceof our proposed design under the three–parameter logistic model. We believethat the same results can be generalized to other popular response models,such as the double exponential model.1537.2. Future ResearchWe also believe that the proposed design can be extended to estimateother sets of ed levels, including sets of two or more ed levels of the un-derlying dose-response curve, for example, estimating ed25 and ed75 si-multaneously. More simulations will be carried out to confirm the aboveclaims.More studies will be conducted to modify and extend our proposed designin the future. Our group ED-design has a natural extension to more complexmodels and can also be used for any ed levels, and satisfy a broad range ofdemands that may arise in applications. We feel that our proposed designreveals some interesting directions and provides great potential for furtherresearch.154BibliographyAbdelbasit, K. M. and Plackett, R. (1983). Experimental design for binarydata. Journal of the American Statistical Association, 78(381):90–98.Albert, A. and Anderson, J. 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Sequential ED-design for binary dose-response experiments Yu, Xiaoli 2017
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Title | Sequential ED-design for binary dose-response experiments |
Creator |
Yu, Xiaoli |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | Dose-response experiments and subsequent data analyses are often carried out according to optimal designs for the purpose of accurately determining a specific effective dose (ED) level. If the interest is the dose-response relationship over a range of ED levels, many existing optimal designs are not accurate. In this dissertation, we propose a new design procedure, called two-stage sequential ED-design, which directly and simultaneously targets several ED levels. We use a small number of trials to provide a tentative estimation of the model parameters. The doses of the subsequent trials are then selected sequentially, based on the latest model information, to maximize the efficiency of the ED estimation over several ED levels. Although the commonly used logistic and probit models are convenient summaries of the dose-response relationship, they can be too restrictive. We introduce and study a more flexible albeit slightly more complex three-parameter logistic dose-response model. We explore the effectiveness of the sequential ED-design and the D-optimal design under this model, and develop an effective model fitting strategy. We develop a two-step iterative algorithm to compute the maximum likelihood estimate of the model parameters. We prove that the algorithm iteration increases the likelihood value, and therefore will lead to at least a local maximum of the likelihood function. We also study the numerical solution to the D-optimal design for the three-parameter logistic model. Interestingly, all our numerical solutions to the D-optimal design are three-point-support distributions. We also discuss the use of the ED-design when experimental subjects become available in groups. We introduce the group sequential ED-design, and demonstrate how to construct this design. The ED-design has a natural extension to more complex model and can satisfy a broad range of the demands that may arise in applications. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-10-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0357352 |
URI | http://hdl.handle.net/2429/63447 |
Degree |
Doctor of Philosophy - PhD |
Program |
Statistics |
Affiliation |
Science, Faculty of Statistics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2018-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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