Asymptotics of adaptive designs based on URN models

ASYMPTOTICS OF ADAPTIVE DESIGNS BASED ON URN MODELS YAN XIU-YUAN NATIONAL UNIVERSITY OF SINGAPORE 2004 ASYMPTOTICS OF ADAPTIVE DESIGNS BASED ON URN MODELS Yan Xiu-yuan (Bachelor of Economics, Renmin University of China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2004 i Acknowledgements It’s the 41st month of my stay in Singapore Finally, I get the chance to show my gratitude to many people who are important in my life The cloud covers the sunshine and the rain is dripping outside the window My memory is brought back to the past three and half years In the duration, I’ve enjoyed the thrill of happiness and suffered the hell of sadness However, no matter what happens, my family is the one who always stands by me and never deserts me Without their love and lenience, the completion of the thesis is impossible I wish to thank my supervisor, Prof Bai Zhidong, who is always there to help whenever I have difficuties in either my life or my study It has been a real pleasure to be his student Thanks to Dr Hu Feifang and my seniors, Mr Chen Yuming and Ms Cheng Yu, who led me into the field of research in the very beginning Thanks to Dr Wang Yougan, who gave me many valuable suggestions on the thesis I wish to thank my fiends, Ms Luo Xiaorong, Ms Lv Qing, Ms Wu Yingjuan and Mr Xing Yuchen I really appreciate their understanding and help I also wish to express my appreciation to my friend, Ms Zeng xiaohua, who cared about me in my life throughout the past years Finally, I’d like to show my special thanks to Mr Mao Bo-ying, whose optimistic and positive attitude towards life is influential to i ii me and lead me out of the difficulty I thank him for his kindness and patience and I enjoy every talk between us Thanks to those who love me and whom I love, with whom my life becomes rich and enjoyable iii Contents Preliminaries 1.1 1 Motivation of Adaptive Designs Literature Review and Current Research 1.2.1 Historical Development of Adaptive Designs 1.2.2 GPU Models and Existing Results 10 Some Preliminary Results 22 1.3.1 An Identity and a Limit 22 1.3.2 Preliminary Results on Matrices 22 1.3.3 Preliminary Theorems on RPW Rule 23 1.3.4 1.4 Background 1.1.2 1.3 1.1.1 1.2 Introduction Preliminary Results on Martingales 24 Organization of the Thesis 25 A Type of Adaptive Design with Delayed Responses 26 2.1 Introduction 26 2.2 Formulation of the Model 27 iii CONTENTS iv Asymptotic Properties of Yn 30 2.3.1 Strong Consistency 30 2.3.2 Asymptotic Normality 37 Asymptotic Properties of Nn 45 2.4.1 Strong Consistency 46 2.4.2 Asymptotic Normality 47 2.5 Monte-Carlo Simulation 53 2.6 Estimation Efficiency 70 2.7 Conclusions 73 2.3 2.4 Adaptive Design with Missing Responses 75 3.1 Formulation of the Model 75 3.2 Asymptotic Properties of Yn 78 3.2.1 Strong Consistency 78 3.2.2 Asymptotic Normality 80 Asymptotic Properties of Nn 84 3.3.1 Strong Consistency 84 3.3.2 Asymptotic Normality 85 3.3 Adaptive Design with Two Alternating Generating Matrices 4.1 88 Adaptive Designs with Two Alternating Generating Matrices For Two Treatments 89 4.1.1 Strong Consistency 91 4.1.2 The asymptotic variance 102 CONTENTS v 4.1.3 Asymptotic normality 106 4.2 General Case 108 4.2.1 Formulation of the Model 108 4.2.2 Strong Consistency 111 4.2.3 Asymptotic Normality 117 4.3 Monte Carlo Simulation and Results 123 4.4 Conclusions 135 Asymptotic Properties of a Linear Combination of Yn and Nn 138 5.1 Introduction 138 5.2 Formulation of the Model 141 5.3 Asymptotic Properties of Yn ξ 143 5.3.1 5.3.2 5.4 Asymptotic Expectation 143 Asymptotic normality 146 Asymptotic Properties of Nn ξ 149 5.4.1 Asymptotic Expectation 149 5.4.2 Asymptotic Normality 152 5.5 Applications 156 5.6 Comments and Conclusions 160 Future Research 162 Appendix 165 Bibliography 173 vi Summary In clinical trials, due to ethical considerations, adaptive designs are adopted as an improvement to the standard 50-50 randomization In a trial, a patient’s response may be delayed for several stages or may not occur at all However, due to the scarcity of resources, it may be impossible to trace each patient’s response if it is delayed for too long Hence, we propose a model in which those responses that are delayed for more than M stages are discarded, where M is some finite constant defined for each trial Under this setting, we have proved that the strong consistency and asymptotics of both Yn and Nn still hold, where Yn is the urn composition and Nn is the number of patients assigned to each treatment in n trials Some applications are also discussed In addition, when there are missing responses, we also establish the strong consistency and asymptotic normality of Yn and Nn In the application of the Generalized P´lya urn (GPU) model to adaptive deo signs, the standard way is to use the urn models associated with a homogeneous generating matrix However, it is more reasonable to employ nonhomogeneous generating matrices, especially when the patients’ responses show a time trend In the thesis, we propose a kind of design using nonconvergent generating matrices vi CONTENTS vii Explicitly, two alternating generating matrices, H1 and H2 , are used In this case, the generating matrices not converge, but have two different limiting points After thorough investigation, we can show that the urn composition will stabilize as the number of patients increases The convergence corresponds to the mean of H1 and H2 Moreover, the asymptotic variance has the same order as in the homogeneous case and asymptotic normality still holds In addition, Monte-Carlo simulation is carried out and the simulation results also support the theoretical results The possible reason for the convergence is also studied in the thesis Some of the research, such as Athreya and Karlin (1968) and Bai and Hu (1999), studied the asymptotic properties of a linear combination or linear transformation of Yn on ξi , where ξi is the right eigenvector of the generating matrix H with respect to some eigenvalue, λi , except the maximal one However, in both papers, in the case that τ = 1/2, the calculation of the variance-covariance matrix of Yn ξ is too rough In the thesis, we study the asymptotic properties of both Yn ξ and Nn ξ and give the exact expression of the variance-covariance matrix for any value of τ ≤ 1/2 Moreover, by studying the eigenstructure of the generating matrix H, we present the reason why the elements in the variance-covariance matrix have different rates of convergence in the case τ = 1/2 Chapter Preliminaries 1.1 1.1.1 Introduction Background In pharmaceutical or medical research, clinical trials are designed to compare the effectiveness of K different treatments, where K ≥ In these trials, patients are sequentially recruited and assigned to one of the K competing treatments based on some allocation rules Then the responses are recorded for evaluating the effects of the treatments The rules how to allocate different treatments to the patients play important roles for the resulting data to contain the necessary information for scientific purpose or for ethical reasons to have higher curing rates The allocation rule thus becomes a major focus in adaptive designs Two of the most commonly used designs are 50-50 randomization and adaptive randomization When K = 2, if 50-50 randomization is used, the patients are assigned to each treatment group CHAPTER FUTURE RESEARCH 163 Secondly, in Chapter and Chapter 3, we assume that the patient’s time to response or the time that the response is missing from the trial is independent of the treatment assignment and the response generated However, in some trials, if this assumption is violated, the expected number of balls added upon the occurrence of each response may differ according to the treatment allocation Thus, the row sum of the generating matrix is not constant If this is true, lemma 1.1 will not hold Whether the asymptotic normality still holds in this case is an important and useful topic for further research Thirdly, in Chapter 3, we assume that the patient’s response occurs within M stages, where M is a finite constant Based on this assumption, a natural consideration is to relax the condition on the time to response Under some restriction, ∞ such as P r(t = i) converges to at certain rate, the asymptotic normality can i=n also be established Another point is, in Chapter 5, we have investigated the asymptotic properties of a non-convergent model Although the results are helpful in clinical trials because it allows flexibility of the generating matrices while does affect the efficiency for making inference, there are still some limitations in the model For example, when we make the assumptions, some restrictions are imposed on the generating matrices to make them converge under some arithmetic operations, which limits the usefulness of the model Since, in practice, if the generating matrices have a time trend or depend on some divergent covariates, they will not converge in any format In this sense, the usefulness of the model is limited But, due to the practical importance of the adaptive designs with nonconvergent generating matrices, CHAPTER FUTURE RESEARCH 164 nonconvergent models with less restrictions as a modification of our models should be considered for future research For example, based on the results in the thesis, we can extend easily to adaptive designs where the K-step weighted mean of the generating matrices is constant or the generating matrices have a time trend In addition, the patient allocation under nonconvergent generating matrices is also of interest A thorough study of urn models with more diversified generating matrices should be considered In addition, in the design of trials, the adding rules should not only depend on the outcome of the previous trial, it should also depend on the current cumulative results A case where the adding rule id dependent on the relative success rates of the K competing treatments is considered in Bai et al (2001) Their results can be further generalized to cases that the adding rules are general functions of current successes rates at that stage However, the problem that the adding rules are functions of the urn composition is still open Whether the asymptotic properties still hold in this case is an important question in the future research 165 Appendix In the appendix, we list the S-plus programs for reference Program for the simulation of adaptive design with delayed responses remove(c("n1","n2","y1","y2","Time","I","resp","p1","p2","H")) n1_1000 n2_1000 p1_0.9 p2_0.6 q1_1-p1 q2_1-p2 a_0.7 b_0.3 y1_matrix(0,n1,n2) y2_matrix(0,n1,n2) Nn1_matrix(0,n1,n2) Nn2_matrix(0,n1,n2) Time_matrix(0,n1,n2) I_matrix(0,n1,n2) resp_matrix(0,n1,n2) #number of type balls #number of type balls #number of patients in treatment #number of patients in treatment #time to response #treatment allocation indicator #response #generating matrix H_matrix(c(a*p1+(1-a)*q1,b*p2+(1-b)*q2,a*q1+(1-a)*p1,(1-b)*p2+b*q2),2,2) #lambda_2 165 Appendix 166 eigen(H)$values[2] #vector V eigenvector.H_c((b*p2+(1-b)*q2)/((1-a)*p1+a*q1+b*p2+(1-b)*q2), ((1-a)*p1+a*q1)/((1-a)*p1+a*q1+b*p2+(1-b)*q2)) for ( i in 1:n1){ y1[i,1]_1/2 y2[i,1]_1/2 for ( j in 2:n2){ r1_runif(1) r11_runif(1) if (r1