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Goal : Beat the frequentists at their own game in phase III clinical trial design

Bayesian Doubly Optimal Group Sequential Design for Clinical Trials. Goal : Beat the frequentists at their own game in phase III clinical trial design. Requirements: Maintain overall false-positive error rate and targeted power

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Goal : Beat the frequentists at their own game in phase III clinical trial design

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  1. Bayesian Doubly Optimal Group Sequential Design for Clinical Trials Goal: Beat the frequentists at their own game in phase III clinical trial design • Requirements: • Maintain overall false-positive error rate and targeted power • Compare to O’Brien-Fleming, Pocock and Optimal group-sequential designs •  The method must be robust, and hence must not depend on the proportional hazards assumption

  2. Solution: A Bayesian Doubly Optimal Group Sequential (BDOGS) Design (Wathen and Thall, Stat in Medicine, 2008) • A robust Bayesian decision-theoretic approach to designing group sequential clinical trials 2. The focus is on two-arm trials with time-to-failure (TTF) outcomes 3. Uses Bayesian adaptive model selection 4. Maintains overall frequentist size and power

  3. 1) Assume the data come from one of M models (characterized by their hazard functions) 2) Before the trial:Derive the Optimal Decision Bounds for each model,and store them 3) During the trial:At each interim analysis, make decisions using the Optimal Decision Bounds of the Optimal Model 4) The optimal boundaries depend on the model, and the model is optimized adaptively  The decision boundaries may change from one interim evaluation to the next Basic Elements of BDOGS BDOGSillustration

  4. A Doubly Optimal Procedure Step 1 (Before the Trial):For each of M specific models, obtain the Optimal Decision Boundaries using forward simulation. Step 2 (During the Trial):Obtain posterior model probabilities for the set of M possible models using approximate Bayes Factors to determine the OptimalModel. Step 3 (During the Trial):Apply the optimal decision boundaries corresponding to the optimal model at each interim decision based on the most recent data.

  5. d= mE – mS = actual improvement in median failure time of experimental (E) over standard (S), a parameter under the Bayesian model (hence random) d* = fixed desired improvement in median failure time of E over S Expected Utility = ½ Ed = 0(N) + ½ Ed = d*(N)

  6. Decision Boundaries To facilitate computation, for each modelBDOGS uses the two parametric boundary functions PU = aU – bU { N+(Xn)/N } PL = aL + bL { N+(Xn)/N } whereN = maximum sample size, and N+(Xn) = # failure events in data Xn (aU , bU , cU , aL , bL , cL ) characterize the decision boundary for a given model cU cL

  7. Decision Rules Superiority of S over E rS = Pr( d < -d* | x ) > PU Stop and select S Superiority of E over S rE = Pr( d > d* | x ) > PU Stop and select E Futility rS< PLandrE< PLStop for futility Acquire more information PL rS, rE PU  Continue randomizing to obtain more information

  8. Forward Simulation Simulate the entire trial 5000 times assuming d = 0, and 5000 times assumingd = d* : • For each interim analysis, calculaterEandrS, andstorerE, rS, and alsostore • [# of patients], [# events] for each treatment arm. 2. Applythe decision rule, dto obtain the expected utility for a trial usingd 3. Find dthat maximizes the expected utility. (A complex search algorithm is required.)

  9. Examples of Hazard Functions (Models) Hazard function for M1 = exponential distribution is constant

  10. A Metastatic Non-Small Cell Lung Cell Cancer (NSCLC) Trial Median overall survival (OS) in metastatic NSCLCis about 4 months A phase III trial of localized surgery or radiation therapy versus systemic chemotherapy for metastatic NSCLC was designed with the goal to improve median progression-free survival (PFS) from 4 to 8 months Initially, a conventional .05/.90 group sequential design with O’Brien-Fleming boundaries was planned, with up to 3 tests at 30, 60 and 89 events.

  11. Under the “usual” assumptions, accruing 2 to 4 patients/month, a typical O’Brien-Fleming .05/.90 group sequential design will require ~ 100 to 120 patients and take ~ 2 ½ to 4 ½ years to complete

  12. Analysis of Historical Data on PFS time in Metastatic NSCLS A preliminary goodness-of-fit analysis, based on a published Kaplan-Meier plot of PFS times of NSCLC patients with metastatic disease, showed that the Log Normal distribution gave a much better fit than the Weibull or Exponential.  The proportional hazards assumption was very likely invalid. The hazard function was very likely non-monotone.

  13. A BDOGS Design for the NSCLC Trial To test H0: d = 0 versus H1: d 0 Assume med(T) = 4 mos. for std. therapy Type I Error = .05, Power= 0.90 ford*= 4months, improvement to med(T) = 8 mos. Assume 2 patients per month accrual Up to5 interim analyses + 1 final analysis, at 25, 50, 75, 87, 112 and 122 events Five possible models

  14. Possible Models (Hazard Functions) M1 = constant (Exponential model) M2 = increasing M3 = decreasing M4 = initially increasing, then a slight decrease M5 = initially increasing, then a large decrease A priori, the 5 models were assumed to be equally likely: Pr(M1) = …= Pr(M5) = .20.

  15. Non-Constant Hazard Functions (Models)

  16. Simulation Study for the NSCLS Trial • For comparability in the simulations: • An O’Brien-Fleming design was constructed to have the same 6 looks, for both superiority (reject the null) and inferiority (accept the null) decisions. • Both designs had the same maximum sample size N = 122 patients. • For each case (underlying true PFS distribution) studied, the data were simulated ahead of time and each method was presented with the same data.

  17. Non-constant Hazards Used in Simulation Study for S (solid line) and E (dashed line)

  18. Simulation Results: Null Case Lower - Upper Lines = 2.5 - 97.5 Percentiles Line in Box = Median Box = 25 – 75 Percentiles Dot in Box = Mean B = BDOGS, OF = O’Brien-Fleming

  19. Simulation Results: Alternative Case Lower - Upper Lines = 2.5 - 97.5 Percentiles Line in Box = Median Box = 25 – 75 Percentiles Dot in Box = Mean B = BDOGS, OF = O’Brien-Fleming

  20. Simulation Results If the hazard is constant, both BDOGS and OF maintain targeted size and power, but OF requires a much larger sample(33% to 51% more patients)

  21. Simulation Results If the hazard is Log Normal, both BDOGS and OF maintain targeted size and power, but OF requires a much larger sample

  22. Simulation Results If the hazard is Weibull, both BDOGS and OF maintain targeted power, BDOGS has a reduced size = .02, and OF requires a much larger sample

  23. Simulation Results If the hazard is Weibull with decreasing hazard, BDOGS has size .07, OF has reduced power .81, and OF requires a much larger sample

  24. Simulation Results If the hazard is Weibull with increasing hazard, both methods have greatly reduced size .01, OF has greatly increased power .99, and OF has a 61% to 141% larger sample size

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