Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt

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1 Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt Joachim Gerß Institute of Biostatistics and Clinical Research

2 Overview 1. The Retinoblastoma Trial 2. Long-Term Binary Endpoints a. Suspend Recruitment: Simon Design b. Continue Recruitment: Bayesian Approach 3. Simulation 4. Summary and Conclusion J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 2

3 Overview 1. The Retinoblastoma Trial 2. Long-Term Binary Endpoints a. Suspend Recruitment: Simon Design b. Continue Recruitment: Bayesian Approach 3. Simulation 4. Summary and Conclusion J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 3

4 1. The Retinoblastoma Trial Retinoblastoma: rare, rapidly developing malignant tumor of the eye in children 10 patients per year Single-arm Phase II trial: Efficacy of intra-arterial chemotherapy with carboplatin Primary outcome: 6 months event-free eye survival (EFES) H 0 : unacceptable response EFES 0.4 event rate p 0.6 α = β = 0.1 versus H 1 : desirable response EFES 0.6 event rate p 0.4 Interim analysis with futility stop J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 4

5 Overview 1. The Retinoblastoma Trial 2. Long-Term Binary Endpoints a. Suspend Recruitment: Simon Design b. Continue Recruitment: Bayesian Approach 3. Simulation 4. Summary and Conclusion J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 5

6 2. Long-Term Binary Endpoint a. Suspend Recruitment: Simon Design H 0 : unacceptable response EFES 0.4 event rate p 0.6 H 1 : desirable response EFES 0.6 event rate p 0.4 Simon Optimal Design Pat. 1 Pat. 2 Pat. 3 Pat. 18 Pat. 19 Pat. 20 Pat. 21 Pat st stage: 18 patients If 11/18 events => Futility stop, Reject drug If 10/18 events => Continue to 2 nd stage 2 nd stage: +28 patients (=> Σ=46 patients) If 24/46 events => Reject drug If 23/46 events => Accept drug, Reject H 0 J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 6

7 2. Long-Term Binary Endpoint a. Suspend Recruitment: Bayesian Approach Bayesian Beta-Binomial Model Pat. 1 Pat. 2 Pat. 3 Pat. 18 Pat. 19 Pat. 20 Pat. 21 Pat. 46 Interim analysis: n = 18 patients x i ϵ{0,1} event indicator Data model: Prob(X i =1 p) = p Prob(X i =0 p) = 1-p Noninformative prior: f(p) = I [0,1] (p) => Posterior f(p x 1,,x n ) = = f(x 1,,x n,p) / f(x 1,,x n ) = f(x 1,,x n p) f(p) / f(x) Π Prob(X i =x i p) I [0,1] (p) = Π Prob(X i =1 p) Π Prob(X i =0 p) I [0,1] (p) with event = p Σxi (1-p) n-σxi I [0,1] (p) without event during 6 mo follow-up J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 7

8 Overview 1. The Retinoblastoma Trial 2. Long-Term Binary Endpoints a. Suspend Recruitment: Simon Design b. Continue Recruitment: Bayesian Approach 3. Simulation 4. Summary and Conclusion J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 8

9 2. Long-Term Binary Endpoint b. Continue Recruitment: Bayesian Approach Bayesian Model Pat. 1 Pat. 2 Pat. 3 Pat. 18 Pat. 19 Pat. 20 Pat. 21 Pat. 46 Interim analysis: n = patients x i ϵ{0,1} event indicator with event: Prob(X i =1 p) = p Pat w/o event during 6 mo follow-up: Prob(X i =0 p) = 1-p Noninformative prior: f(p) = I [0,1] (p) => Posterior f(p x 1,,x n ) Π Prob(X i =x i p) I [0,1] (p) = Π Prob(X i =1 p) Π Prob(X i =0 p) I [0,1] (p) with event = p Σxi (1-p) n-σxi I [0,1] (p) without event during 6 mo follow-up J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 9

10 2. Long-Term Binary Endpoint b. Continue Recruitment: Bayesian Approach Bayesian Model Pat. 1 Pat. 2 Pat. 3 Pat. 18 Pat. 19 Pat. 20 Pat. 21 Pat. 46 f i Interim analysis: n = patients x i ϵ{0,1} event indicator with event: Prob(X i =1 p) = p Pat w/o event during 6 mo follow-up: Prob(X i =0 p) = 1-p Pat w/o event, follow-up f i < 6mo: Prob(X i =0 p) = 1 - p f i /6 Noninformative prior: f(p) = I [0,1] (p) => Posterior f(p x 1,,x n ) Π Prob(X i =x i p) I [0,1] (p) = Π Prob(X i =1 p) Π Prob(X i =0 p) Π Prob(X i =0 p) I [0,1] (p) with event without event during 6 mo follow-up without event during follow-up f i < 6mo = p Σxi (1-p) #no event in 6mo fu Π (1 - p f i /6) I [0,1] (p) without event during follow-up f i < 6mo J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 10

11 2. Long-Term Binary Endpoint b. Continue Recruitment: Bayesian Approach (!) Bayesian Model Pat. 1 Pat. 2 Pat. 3 Pat. 18 Pat. 19 Pat. 20 Pat. 21 Pat. 46 f 20 Interim analysis: n = patients x i ϵ{0,1} event indicator with event: Prob(X i =1 p,p 20 ) = p Pat w/o event during 6 mo follow-up: Prob(X i =0 p,p 20 ) = 1-p Pat w/o event, follow-up f 20 < 6mo: Prob(X 20 =0 p,p 20 ) = 1-p 20 Noninf. prior: f(p,p 20 ) = f(p 20 p) f(p) = 1/p I [0,p] (p 20 ) I [0,1] (p) => Posterior f(p,p 20 x 1,,x n ) Π Prob(X i =x i p,p 20 ) f(p,p 20 ) = Π Prob(X i =1 p) Π Prob(X i =0 p) Prob(X 20 =0 p 20 ) f(p,p 20 ) with event without event during 6 mo follow-up = p Σxi (1-p) n-σxi-1 (1-p 20 ) 1/p I [0,p] (p 20 ) I [0,1] (p) => Posterior f(p x 1,,x n )= f(p,p 20 x 1,,x n ) dp 20 J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 11

12 2. Long-Term Binary Endpoint b. Continue Recruitment: Bayesian Approach (!) Bayesian Model Pat. 1 Pat. 2 Pat. 3 Pat. 18 Pat. 19 Pat. 20 Pat. 21 Pat. 46 f 20 f 21 Interim analysis: n = patients x i ϵ{0,1} event indicator with event: Prb(X i =1 p,p 20,p 21 ) = p Pat w/o event during 6 mo foll-up: Prb(X i =0 p,p 20,p 21 ) = 1-p Pat w/o event, follow-up f 20 < 6mo:Prb(X 20 =0 p,p 20,p 21 ) = 1-p 20 Pat w/o event, follow-up f 21 < 6mo:Prb(X 21 =0 p,p 20,p 21 ) = 1-p 21 Prior: f(p,p 20,p 21 ) = f(p 21 p 20,p) f(p 20 p) f(p) = 1/p 20 I [0,p20] (p 21 ) 1/p I [0,p] (p 20 ) I [0,1] (p) => Posterior f(p,p 20,p 21 x 1,,x n ) Π Prob(X i =x i p,p 20,p 21 ) f(p,p 20,p 21 ) =p Σxi (1-p) n-σxi-2 (1-p 20 ) (1-p 21 ) 1/p 20 I [0,p20] (p 21 ) 1/pI [0,p] (p 20 ) I [0,1] (p) => Posterior f(p x 1,,x n )= f(p,p 20,p 21 x 1,,x n ) dp 21 dp 20 J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 12

13 H 0 : unacceptable response EFES 0.4 event rate p Long-Term Binary Endpoint b. Continue Recruitment: Bayesian Approach (!) H 1 : desirable response EFES 0.6 event rate p 0.4 Bayesian Model Pat. 1 Pat. 2 Pat. 3 Pat. 18 Pat. 19 Pat. 20 Interim analysis: f interim (p x 1,,x n ) If Prob interim (EFES 0.6) => Futility Stop Final analysis: f final (p x 1,,x n ) If Prob final (EFES 0.4)<0.1 => Reject H 0, Accept drug If Prob final (EFES 0.4) 0.1 => Reject drug Pat. 21 Pat. 46 J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 13

14 Overview 1. The Retinoblastoma Trial 2. Long-Term Binary Endpoints a. Suspend Recruitment: Simon Design b. Continue Recruitment: Bayesian Approach 3. Simulation 4. Summary and Conclusion J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 14

15 H 0 : unacceptable response EFES 0.4 event rate p 0.6 H 1 : desirable response EFES 0.6 event rate p Simulation (1) runs Design Futility Stop Prob(Reject H 0 ) H 0 H 1 H 0 H 1 Suspend Recruitment Continue Recruitment Simon Bayes Bayes J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 15

16 H 0 : unacceptable response EFES 0.4 event rate p 0.6 H 1 : desirable response EFES 0.6 event rate p 0.4 Case and Morgan 2003 Case and Morgan. Design of Phase II cancer trials evaluating survival probabilities. BMC Medical Research Methodology 2003, 3:6. Straightforward extension of Simon s Design to time-to-event data Test statistics based on the Nelson-Aalen estimate of the cumulative hazard function evaluated at the interim (Z 1 ) and final analysis (Z 2 ) Assumptions regarding the expected survival distribution and accrual rate (Z 1,Z 2 ) ~ multivariate normal with correlation ρ Determine critical values of futility stop (C 1 ) and final analysis (C 2 ), satisfying type I and type II error constraints Choose design with either minimal sample size or minimal expected total study length under H 0 J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 16

17 H 0 : unacceptable response EFES 0.4 event rate p 0.6 H 1 : desirable response EFES 0.6 event rate p Simulation (2) Exponential survival, runs Design Case and Morgan 2003, assuming exponential survival Futility Stop Prob(Reject H 0 ) H 0 H 1 H 0 H Bayes Weibull survival, runs Design Case and Morgan 2003, assuming exponential survival Futility Stop Prob(Reject H 0 ) H 0 H 1 H 0 H Bayes J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 17

18 Overview 1. The Retinoblastoma Trial 2. Long-Term Binary Endpoints a. Suspend Recruitment: Simon Design b. Continue Recruitment: Bayesian Approach 3. Simulation 4. Summary and Conclusion J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 18

19 4. Summary and Conclusion The presented Bayesian approach is at least as good as other existing approaches does not require parametric model assumptions has transparent decision rules J. Gerß: Ein universelles Bayes-Design für einarmige Phase II-Studien mit binärem zeitlich erfasstem Endpunkt 19