ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 4 - 2016
A Bayesian Response-Adaptive Randomization Design for Clinical Trials with Survival Endpoints
Jianchang Lin*, Li An Lin and Serap Sankoh
Takeda Pharmaceuticals, USA
Received:March 11, 2016 | Published: April 16, 2016
*Corresponding author: Jianchang Lin, Takeda Pharmaceuticals, 35 Landsdowne Street, Cambridge, MA 02139, USA, Tel: 8502287421; Email:
Citation: Lin J, Lin LA, Sankoh S (2016) A Bayesian Response-Adaptive Randomization Design for Clinical Trials with Survival Endpoints. Biom Biostat Int J 3(4): 00073. DOI:10.15406/bbij.2016.03.00073

Abstract

Accordingly to FDA draft guidance (2010), adaptive randomization (e.g. response-adaptive (RA) randomization) has become popular in clinical research because of its flexibility and efficiency, which also have the advantage of assigning fewer patients to inferior treatment arms. The RA design based on binary outcome is commonly used in clinical trial where “success” is defined as the desired (or undesired) event occurring within (or beyond) a clinical relevant time. As patients entering into trial sequentially, only part of patients have sufficient follow-up during interim analysis. This results in a loss of information as it is unclear how patients without sufficient follow-up should be handled. Alternatively, adaptive design for survival trial was proposed for this type of trial. However, most of current practice assumes the event times following a pre-specified parametric distribution. We adopt a nonparametric model of survival outcome which is robust to model of event time distribution, and then apply it to response-adaptive design. The operating characteristics of the proposed design along with parametric design are compared by simulation studies, including their robustness properties with respect to model misspecifications.

Keywords: Adaptive Design; Clinical Trials; Bayesian Adaptive Design; Survival Analysis

Introduction

Response-adaptive (RA) randomization scheme has become popular in clinical research because of its flexibility and efficiency. Based on the accruing history of patients’ responses to treatment, the RA randomization scheme adjusts the future allocation probabilities, thereby allowing more patients to be assigned to the superior treatment as the trial progresses. As a result, RA randomization can offer significant ethical and cost advantages over equal randomization.

The RA design based on binary outcome is commonly used in clinical trial where “success” is defined as the desired (or undesired) event occurring within (or beyond) a clinical relevant time. As patients entering into trial sequentially, only part of patients have sufficient follow-up during interim analysis. This results in a loss of information as it is unclear how patients without sufficient follow-up should be handled. Alternatively, adaptive design for survival trial was proposed for this type of trial.

However, most of current practice assumes the event times following a pre-specified parametric distribution. We adopt a nonparametric model of survival outcome which is robust to model of event time distribution, and then apply it to response-adaptive design. The operating characteristics of the proposed design along with parametric design are compared by simulation studies, including their robustness properties with respect to model misspecifications.

Method

A nonparametric survival model

Patients are enrolled in sequential groups of size {Nj }, j =1, . . . , J , where Nj is the sample size of the sequential group j . Typically, before conducting the trial, researchers have little prior information regarding the superiority of the treatment arms. Therefore, initially, for the first j’ groups, e.g. j’=1, patients are allocated to K treatment arms with an equal probability 1/K. As patients accurate, the number of current patients increased. Let Ti be the event time for patient i and τ be the clinical relevant time where θ=Pr(T>τ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCcqGH9aqpciGGqbGaaiOCaiaacIcacaWGubGaeyOp a4JaeqiXdqNaaiykaaaa@4022@ is the event of interest. For example, a trial is conducted to assess the progression-free survival probability at 9 months. During the trial, the number of current patients increased as patients accurate. Let N(t) denote the current number of patients who have been accrued and treated at a given calendar time t during the trial. Without censoring, θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCaaa@385A@ can be modeled by binomial model where the likelihood function evaluated at time t is

L( data )= Π i=1 N(t) θ I( T i >τ) ( 1θ ) I( T i τ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaeitamaabmaapaqaa8qacaWGKbGaamyyaiaadshacaWGHbGa aeiFaiaabI7aaiaawIcacaGLPaaacqGH9aqpcqqHGoaudaqhaaqcfa saaiaadMgacqGH9aqpcaaIXaaabaGaamOtaiaacIcacaWG0bGaaiyk aaaajuaGcaqG4oWdamaaCaaabeqcfasaa8qacaqGjbGaaiikaiaabs fajuaGpaWaaSbaaKqbGeaapeGaaeyAaaWdaeqaa8qacqGH+aGpcqaH epaDcaGGPaaaaKqbaoaabmaapaqaa8qacaaIXaGaeyOeI0IaaeiUda GaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaaeysaKqbaoaabmaa juaipaqaa8qacaqGubqcfa4damaaBaaajuaibaWdbiaabMgaa8aabe aapeGaeyizImQaaeiXdaGaayjkaiaawMcaaaaaaaa@6046@  (1)

However, censoring is not avoidable in clinical practice. As patients enter into the trial sequentially, the follow-up time for certain patients may less than τ when we evaluate θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCaaa@385A@  at any calendar time t. Other reason for censoring, including, but not limited to, patient drop out, failure to measure the outcome of interest, and so on. If we ignore the censoring, substantial information will be lost. Cheung & Chappell [1] introduced a simple model for dose-finding trial. Later, Cheung & Thall [2] adopted this model to continuous monitoring for phase II clinical trials. With censoring, the likelihood function (1) can be rewritten as

L( data|θ )= Π i=1 N(t) PrPr { T i £minmin( x i ,τ ) } Y( x i ) Pr { T i >minmin( x i ,τ )} 1-Y( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGmbqcfa4aaeWaaOWdaeaajugib8qacaWGKbGaamyyaiaa dshacaWGHbGaaiiFaiabeI7aXbGccaGLOaGaayzkaaqcLbsacqGH9a qpcqqHGoaujuaGdaqhaaqcbasaaKqzadGaamyAaiabg2da9iaaigda aKqaGeaajugWaiaad6eacaGGOaGaamiDaiaacMcaaaqcLbsaciGGqb GaaiOCaiGaccfacaGGYbqcfa4aaiWaaOWdaeaajugib8qacaWGubqc fa4damaaBaaajeaibaqcLbmapeGaamyAaaWcpaqabaqcLbsapeGaai 4OaiGac2gacaGGPbGaaiOBaiGac2gacaGGPbGaaiOBaKqbaoaabmaa k8aabaqcLbsapeGaamiEaKqba+aadaWgaaqcbasaaKqzadWdbiaadM gaaSWdaeqaaKqzGeWdbiaacYcacqaHepaDaOGaayjkaiaawMcaaaGa ay5Eaiaaw2haaKqba+aadaahaaWcbeqcbasaaKqzadWdbiaadMfalm aabmaajeaipaqaaKqzadWdbiaadIhal8aadaWgaaqccasaaKqzadWd biaadMgaaKGaG8aabeaaaKqaG8qacaGLOaGaayzkaaaaaKqzGeGaci iuaiaackhacaGG7bGaamivaKqba+aadaWgaaqcbasaaKqzadWdbiaa dMgaaSWdaeqaaKqzGeWdbiabg6da+iGac2gacaGGPbGaaiOBaiGac2 gacaGGPbGaaiOBaKqbaoaabmaak8aabaqcLbsapeGaamiEaKqba+aa daWgaaqcbasaaKqzadWdbiaadMgaaSWdaeqaaKqzGeWdbiaacYcacq aHepaDaOGaayjkaiaawMcaaKqzGeGaaiyFaKqba+aadaahaaWcbeqc basaaKqzadWdbiaaigdacaGGTaGaamywaSWaaeWaaKqaG8aabaqcLb mapeGaamiEaSWdamaaBaaajiaibaqcLbmapeGaamyAaaqccaYdaeqa aaqcbaYdbiaawIcacaGLPaaaaaaaaa@9A12@  (2)

Where x i =min( c i , t i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWG4bqcfa4damaaBaaajeaibaqcLbmapeGaamyAaaWcpaqa baqcLbsapeGaeyypa0JaciyBaiaacMgacaGGUbqcfa4aaeWaaOWdae aajugib8qacaWGJbqcfa4damaaBaaajeaibaqcLbmapeGaamyAaaWc paqabaqcLbsapeGaaiilaiaadshajuaGpaWaaSbaaKqaGeaajugWa8 qacaWGPbaal8aabeaaaOWdbiaawIcacaGLPaaaaaa@4BB2@ is the observed event time, ci is the censoring time, and Y( x i )=I{ T i minmin( x i ,τ )} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamywamaabmaapaqaa8qacaWG4bWdamaaBaaajuaibaWdbiaa dMgaaKqba+aabeaaa8qacaGLOaGaayzkaaGaeyypa0JaamysaiaacU hacaWGubWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaapeGaeyiz ImQaciyBaiaacMgacaGGUbGaciyBaiaacMgacaGGUbWaaeWaa8aaba WdbiaadIhapaWaaSbaaKqbGeaacaWGPbaajuaGbeaapeGaaiilaiab es8a0bGaayjkaiaawMcaaiaac2haaaa@5139@ is the censoring indicates function.

Furthermore, the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCaaa@385A@ will be plug into the likelihood function through probability transformation,

PrPr( T i t )=PrPr( T i t, T i τ )+PrPr( T i t, T i >τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaciiuaiaackhaciGGqbGaaiOCamaabmaapaqaa8qacaqGubWd amaaBaaajuaibaWdbiaabMgaaKqba+aabeaapeGaeyizImQaaeiDaa GaayjkaiaawMcaaiabg2da9iGaccfacaGGYbGaciiuaiaackhadaqa daWdaeaapeGaaeiva8aadaWgaaqcfasaa8qacaqGPbaajuaGpaqaba WdbiabgsMiJkaabshacaGGSaGaaeiva8aadaWgaaqcfasaa8qacaqG PbaajuaGpaqabaWdbiabgsMiJkaabs8aaiaawIcacaGLPaaacqGHRa WkciGGqbGaaiOCaiGaccfacaGGYbWaaeWaa8aabaWdbiaabsfapaWa aSbaaKqbGeaapeGaaeyAaaqcfa4daeqaa8qacqGHKjYOcaqG0bGaai ilaiaabsfapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqaa8qacqGH +aGpcqaHepaDaiaawIcacaGLPaaaaaa@65C8@

=PrPr( T i t| T i τ )PrPr( T i τ )f( T i τ )+PrPr( T i t| T i τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeyypa0JaciiuaiaackhaciGGqbGaaiOCamaabmaapaqaa8qa caqGubWdamaaBaaajuaibaWdbiaabMgaaKqba+aabeaapeGaeyizIm QaaeiDaiaacYhacaqGubWdamaaBaaajuaibaWdbiaabMgaaKqba+aa beaapeGaeyizImQaaeiXdaGaayjkaiaawMcaaiGaccfacaGGYbGaci iuaiaackhadaqadaWdaeaapeGaaeiva8aadaWgaaqcfasaa8qacaqG PbaajuaGpaqabaWdbiabgsMiJkaabs8aaiaawIcacaGLPaaacaqGMb WaaeWaa8aabaWdbiaabsfapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4d aeqaa8qacqGHKjYOcaqGepaacaGLOaGaayzkaaGaey4kaSIaciiuai aackhaciGGqbGaaiOCamaabmaapaqaa8qacaqGubWdamaaBaaajuai baWdbiaabMgaaKqba+aabeaapeGaeyizImQaaeiDamaaEiaabeWdae aapeGaaeiva8aadaWgaaqcfasaa8qacaqGPbaajuaGpaqabaaapeGa ay5bSlaawQYiaiabes8a0bGaayjkaiaawMcaaaaa@6FD6@ Pr() MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaadkhapaGaaiikaiaacMcaaaa@39CD@ =w( t )( 1 θ )   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeyypa0Jaam4Da8aadaqadaqaa8qacaWG0baapaGaayjkaiaa wMcaamaabmaabaWdbiaaigdacqGHsislcaqGGaGaeqiUdehapaGaay jkaiaawMcaa8qacaGGGcGaaiiOaaaa@434C@  (3)

Where w( t )={ Pr( T i t| T i τ) 1 ,tτ t>τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Damaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyyp a0Zaaiqaa8aabaqbaeqabiqaaaqaa8qaciGGqbGaaiOCaiaacIcaca WGubWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaapeGaeyizImQa amiDaiaacYhacaWGubWdamaaBaaajuaibaWdbiaadMgaaKqba+aabe aapeGaeyizImQaeqiXdqNaaiykaaWdaeaapeGaaGymaaaaaiaawUha a8aafaqabeGabaaabaWdbiaacYcacaWG0bGaeyizImQaeqiXdqhapa qaa8qacaWG0bGaeyOpa4JaeqiXdqhaaaaa@56A2@ , is a weight function

Finally, we can obtain a working likelihood with unbiased estimation of w(t).

L( data|θ )= i=1 N( t ) w ˜ ( x i ) ( 1θ ) Y( x i ) { 1 w ˜ ( x i )( 1θ ) } 1Y( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGmbWaaeWaa8aabaWdbiaabsgacaqGHbGaaeiDaiaabgga caqG8bGaaeiUdaGaayjkaiaawMcaaiabg2da9maawahabeqcfaYdae aapeGaaeyAaiabg2da9iaaigdaa8aabaWdbiaab6eajuaGdaqadaqc faYdaeaapeGaaeiDaaGaayjkaiaawMcaaaqcfa4daeaapeGaey4dIu naaiqadEhagaacamaabmaapaqaa8qacaqG4bWdamaaBaaajuaibaWd biaabMgaaKqba+aabeaaa8qacaGLOaGaayzkaaWaaeWaa8aabaWdbi aaigdacqGHsislcaqG4oaacaGLOaGaayzkaaWdamaaCaaabeqcfasa a8qacaqGzbqcfa4aaeWaaKqbG8aabaWdbiaabIhajuaGpaWaaSbaaK qbGeaapeGaaeyAaaWdaeqaaaWdbiaawIcacaGLPaaaaaqcfa4aaiWa a8aabaWdbiaaigdacqGHsislceWG3bGbaGaadaqadaWdaeaapeGaae iEa8aadaWgaaqcfasaa8qacaqGPbaajuaGpaqabaaapeGaayjkaiaa wMcaamaabmaapaqaa8qacaaIXaGaeyOeI0IaaeiUdaGaayjkaiaawM caaaGaay5Eaiaaw2haa8aadaahaaqabKqbGeaapeGaaGymaiabgkHi TiaabMfajuaGdaqadaqcfaYdaeaapeGaaeiEaKqba+aadaWgaaqcfa saa8qacaqGPbaapaqabaaapeGaayjkaiaawMcaaaaaaaa@722D@ (4)

Theorem: if w ˜ ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaGaadaqadaWdaeaapeGaaeiEa8aadaWgaaqcfasa a8qacaqGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaaa@3C5A@ converges almost surely to w ˜ ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaGaadaqadaWdaeaapeGaaeiEa8aadaWgaaqcfasa a8qacaqGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaaa@3C5A@  for all I as N( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobWdamaabmaabaWdbiaadshaa8aacaGLOaGaayzkaaGa eyOKH4QaeyOhIukaaa@3D85@ , then θ ^ =argmaxL(data|θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacuaH4oqCgaqcaiabg2da9iGacggacaGGYbGaai4zaiGac2ga caGGHbGaaiiEaiaadYeacaGGOaGaaeizaiaabggacaqG0bGaaeyyai aacYhacqaH4oqCcaGGPaaaaa@4793@ is strongly consistent for true survival probability θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCaaa@385A@ .

Cheung & Chappell [1] assumed the nuisance parameter w ˜ ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaGaadaqadaWdaeaapeGaaeiEa8aadaWgaaqcfasa a8qacaqGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaaa@3C5A@  as a linear function w ˜ ( x i )= x i /τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaGaadaqadaWdaeaapeGaaeiEa8aadaWgaaqcfasa a8qacaqGPbaajuaGpaqabaaapeGaayjkaiaawMcaaiabg2da9iaadI hadaWgaaqcfasaaiaadMgaaKqbagqaaiaac+cacqaHepaDaaa@42A0@ . Ji & Bekele [3] shows that these estimated weights are based on strong assumption of linearity and independence; it may leads to biased results when the assumptions are violated. We propose to estimate w ˜ ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaGaadaqadaWdaeaapeGaaeiEa8aadaWgaaqcfasa a8qacaqGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaaa@3C5A@  with KM estimation, where

w ˜ ( x i )= 1 S ˜ ( x i ) 1 S ˜ ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaGaadaqadaWdaeaapeGaaeiEa8aadaWgaaqcfasa a8qacaqGPbaajuaGpaqabaaapeGaayjkaiaawMcaaiabg2da9maala aapaqaa8qacaaIXaGaeyOeI0Iabm4uayaaiaWaaeWaa8aabaWdbiaa bIhapaWaaSbaaKqbGeaapeGaaeyAaaqcfa4daeqaaaWdbiaawIcaca GLPaaaa8aabaWdbiaaigdacqGHsislceWGtbGbaGaadaqadaWdaeaa peGaaeiXdaGaayjkaiaawMcaaaaaaaa@4A68@

It’s easy to show that w ˜ ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG3bGbaGaadaqadaWdaeaapeGaamiEa8aadaWgaaqcfasa a8qacaWGPbaajuaGpaqabaaapeGaayjkaiaawMcaaaaa@3C5E@  is a unbiased estimation of ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaamiEa8aadaWgaaqcfasaa8qacaWGPbaa juaGpaqabaaapeGaayjkaiaawMcaaaaa@3B53@ .

Adaptive randomization

Under model (4), the survival probability evaluated at time τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@37BB@ , is used as a conventional measure of treatment efficacy. However, such a survival probability at τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@37BB@  ignores the entire path of survival curve. While, particular interest in a clinical trial is the estimation of the difference between survival probability for the treatment groups at several points in time. As shown in Figure 1, the survival curve under treatment B declines faster than that under treatment A, although both treatments have the same survival probability at τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@37BB@ . In the renal cancer trial, this indicates that patients under treatment B would experience disease progression much faster than those under treatment A. Because delayed disease progression typically leads to a better quality of life, treatment A would be preferred in this situation [4,5]. Another example is showed in Figure 2. The survival curves are almost identical between two treatments before time 20. If we compare the survival probability between two treatments at the time before 20, the treatment effect is inconclusive. To provide a comprehensive measure of efficacy by accounting for the shape of the survival curve, we propose to evaluate survival probability at several points in time. Let θkj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCcaWGRbGaamOAaaaa@3A39@  be the survival probability at time τj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHepaDcaWGQbaaaa@3958@  for treatment k where j=1,, J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGQbGaeyypa0JaaGymaiaacYcacqGHMacVcaGGSaGaaeii aiaadQeaaaa@3DB4@ . The treatment allocation probability for treatment k is defined as,

π k = j=1 J w j Pr( θ kj =max{ θ lj , 1lJ }|data) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaqGapWdamaaBaaajuaibaWdbiaabUgaaKqba+aabeaapeGa eyypa0ZaaybCaeqajuaipaqaa8qacaqGQbGaeyypa0JaaGymaaWdae aapeGaaeOsaaqcfa4daeaapeGaeyyeIuoaaiaabEhapaWaaSbaaKqb GeaapeGaaeOAaaqcfa4daeqaa8qacaqGqbGaaeOCaiaacIcacaqG4o WdamaaBaaajuaibaWdbiaabUgacaqGQbaajuaGpaqabaWdbiabg2da 9iaab2gacaqGHbGaaeiEamaacmaapaqaa8qacaqG4oWdamaaBaaaju aibaWdbiaabYgacaqGQbaajuaGpaqabaWdbiaacYcacaqGGcGaaGym aiabgsMiJkaabYgacqGHKjYOcaqGkbaacaGL7bGaayzFaaGaaiiFai aabsgacaqGHbGaaeiDaiaabggacaGGPaaaaa@62CB@

Where wj is the prespecified weight. Currently, we use equal weight with wj =1/J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bGaamOAaiaabccacqGH9aqpcaaIXaGaai4laiaadQea aaa@3C75@ .

Figure 1: Survival curves of the time to disease progression, where the two survival curves have the same survival probability at the follow-up time τ=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq Naeyypa0JaaGymaiaac6cacaaI1aaaaa@3B7B@ months, but different areas under the survival curves until τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiXdq haaa@3849@ .

Figure 2: Survival curves of the time to disease progression, where the two survival curves have the same survival probability before week 20, but gradually show difference as time increase.

During the trial, we continuously monitor posterior probability of π k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHapaCpaWaaSbaaKqbGeaapeGaam4Aaaqcfa4daeqaaaaa @3A5D@ . When the efficacy of π k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHapaCpaWaaSbaaKqbGeaapeGaam4Aaaqcfa4daeqaaaaa @3A5D@  is lower than the prespecified lower limit pl, then the treatment arm k will be terminated early due to futility. When π k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHapaCpaWaaSbaaKqbGeaapeGaam4Aaaqcfa4daeqaaaaa @3A5D@  is higher than pu, the treatment arm k will be selected as promising treatment. In practice, the values of pl and pu are chosen by simulation studies in order to achieve desirable operating characteristics for the trial.

Simulation Study

We simulate a single arm trial where the event times follow weibull distribution with α=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIYaaaaa@3A05@  and λ=50 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBcqGH9aqpcaaI1aGaaGimaaaa@3AD7@ . And, patients enter into the trial sequentially with accrual rate of one per week. At week 50, we stop enrol the patients and continue to follow the trial for additional 30 weeks. The parameter of interesting is θ=Pr( T>40 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH4oqCcqGH9aqpcaWGqbGaamOCa8aadaqadaqaa8qacaWG ubGaeyOpa4JaaGinaiaaicdaa8aacaGLOaGaayzkaaaaaa@403C@ .

The purpose of this simulation study is to compare the performance of estimation with different methods and to show whether the estimation at different trial monitoring time is consistent. Four estimation method will be evaluated, including proposed method, true parametric method (weibull distribution), misspecified parametric method (exponential distribution), and original method ( w ˜ ( x i )= x i /τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai qacEhagaacaabaaaaaaaaapeWaaeWaa8aabaWdbiaadIhapaWaaSba aKqbGeaapeGaamyAaaqcfa4daeqaaaWdbiaawIcacaGLPaaacqGH9a qpcaWG4bWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaapeGaai4l aiabes8a09aacaGGPaaaaa@4448@ . Trial monitoring starts at time 40 and continue until the end of study. Figure 3 shows the estimated θ at different monitoring time. The results show that the true parametric method and proposed method can provide unbiased estimation over monitoring time while the original method and misspecified parametric method give large bias. Its worth to note that the original method gives small bias at the end of trial because the number of censored observed decreased as follow-up time increase. In Figure 4, we present the coverage probability along the monitoring times. The figure shows that the proposed method and true parametric method provide constant coverage probability over the monitoring time which is close to the nominal value of 95%. While the original method and misspecified parametric method provides low coverage probability.

Figure 3: Estimated θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  with different method.

Figure 4: Coverage probability of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  with different method.

Adaptive randomization

We conducted simulations to evaluate the performance of the proposed adaptive randomization design under various clinical scenarios (1000 simulations per scenario). For the simulations, we set the accrual rate to two patients per week. The maximum number of patients is 120. After the initial 60 weeks of enrollment time, there is an additional follow-up period of 40 weeks. The event times are simulated from weibull distribution with α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIXaaaaa@3A04@  in scenario I and α=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHXoqycqGH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3B74@ in scenario II. We assigned the first 30 patients equally to two arms (A or B) and started using the adaptive randomization at the 31st patient. The proposed design will be compared with the following designs: parametric design (exponential distribution) and original design ( w ˜ ( x i )= x i /τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai qacEhagaacaabaaaaaaaaapeWaaeWaa8aabaWdbiaadIhapaWaaSba aKqbGeaapeGaamyAaaqcfa4daeqaaaWdbiaawIcacaGLPaaacqGH9a qpcaWG4bWdamaaBaaajuaibaWdbiaadMgaaKqba+aabeaapeGaai4l aiabes8a09aacaGGPaaaaa@4448@ .

Table 1shows the simulation results where event times simulated from exponential distribution. For each design, we list the average number of patients (with percentage of total patients in the trial) assigned to each treatment arm, and the chance of a treatment being selected as promising. Comparing the proposed design and parametric designs, the proposed design provides comparable operational characteristic where both design assign more patients to more promising treatment (69% for proposed design and 70.3% for parametric design) and both design provide the sample level of power (0.978 for proposed design and 0.979 for parametric design). While the original design provides lower power than then proposed design and parametric design.

Table 2 shows simulation results for scenario II. In the presence of event time distribution misspecification, the parametric design provides lower power than proposed design (0.836 vs 0.647). And, the proposed design assigns more patients to more promising treatment. Once again, the original design has lower power than the other two designs.

Arm

λ

Proposed Design

Exponential

Original

# of Patients

Pr(select)

# of Patients

Pr(select)

# of Patients

Pr(select)

A

40

26.3 (31%)

0.005

24.72 (29.6%)

0.003

36.47 (36.1%)

0.003

B

100

58.56 (69%)

0.978

61.36 (70.3%)

0.979

64.49 (63.9%)

0.749

84.86

90.08

100.96

Table 1: Simulation result for scenario 1.

Arm

λ

Proposed Design

Exponential

Original

# of Patients

Pr(select)

# of Patients

Pr(select)

# of Patients

Pr(select)

A

50

27.76 (28.5%)

0.005

32.48 (32.2%)

0.0003

35.6 (33.8%)

0.001

B

200

69.63 (71.5%)

0.836

68.34 (67.8%)

0.647

69.8 (66.2%)

0.51

97.39

100.82

105.4

Table 2: Simulation result for scenario II.

Discussion

We have developed a Bayesian response-adaptive randomization design for survival trial. A nonparametric survival model is applied to estimate the survival probability at a clinical relevant time. The proposed design provides comparable operational characteristics as true parametric design. Whereas, the proposed design perform better than parametric design when the event time distribution is misspecified. The proposed design can be extended to Response-Adaptive Covariate-Adjusted Randomization (RACA) design when we need to control important prognostics among treatment arms. The benefits of adaptive randomization for survival trial depend on the distributions of event times and patient accrual rate as well as on the particular adaptive design under consideration. If there are short-term response quickly available and can predicting the long-term survival, we can used those short-term response to “speed up” adaptive randomization for survival trial.

References

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  3. Ji Y, Bekele BN (2009) Adaptive randomization for multiarm comparative clinical trials based on joint efficacy/toxicity outcomes. Biometrics 65(3): 876-884.
  4. Case LD, Morgan TM (2003) Design of phase II cancer trials evaluating survival probabilities. BMC Med Res Methodol 3(1): 6.
  5. Huang X, Ning J, Li Y, Estey E, Issa JP, et al. (2009) Using short-term response information to facilitate adaptive randomization for survival clinical trials. Stat in Medicine 28(12): 1680-1689.
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