Efficiency of Promising Zone Designs
Cyrus Mehta
Joint with Sam Hsiao & Lingyun Liu
Cytel Inc
FDA and Industry Workshop 2017FDA and Industry, 9/26/17 1
Outline
• Motivating example from oncology trial
• Proposed promising zone adaptive design
• Efficiency comparisons with:
• Optimal adaptive design (Jennison & Turnbull 2015)
• Constrained optimal design
• Conclusions
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Pivotal Trial in Oncology at a Small Biotech
• Indication - Advanced pancreatic cancer
• Endpoint – Progression free survival
• Effect size – HR from 0.67 (expected) to 0.75 (still clinically meaningful)
• Power
FDA and Industry, 9/26/17 3
𝜽 = 𝟎. 𝟐𝟗 𝜽 = 𝟎. 𝟒
N = 280 68% 92%
N = 500 90% 99%
HR = 0.75
𝜽 = -ln(HR)=0.29
HR = 0.67
𝜽 = -ln(HR)=0.4
Considerations for Adaptive Design (AD)
• Difficult to get up-front investment to ensure 90% power at 𝜽 = 𝟎. 𝟐𝟗 (requires 500 PFS events)
• Stakeholders expressing conditional utility:
o Initial investment sufficient for 𝜽 = 𝟎.4 (HR=0.67)
o Additional investment linked to interim milestone
o Must attain ≥ 80% conditional power at 𝜽 = 𝟎. 𝟐𝟗(HR=0.75) to justify additional investment
• Early efficacy stopping not of much interest, need adequate drug profile and credibility
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Promising Zone Adaptive Design (AD)
• Plan 𝑛2 = 𝟐𝟖𝟎; interim at 𝑛1 = 𝟏𝟒𝟎; maximum 𝑛𝑚𝑎𝑥 = 𝟒𝟐𝟎
• Construct a SSR rule consisting of a promising zone and a sample size 𝒏𝟐
∗ (𝒛𝟏) for every z1 inside the promising zone:
Maximize 𝐶𝑃0.29 𝑧1, 𝑛2∗ in promising zone subject to
Constraint 1: 𝑛2 ≤ 𝑛2∗ ≤ 𝑛𝑚𝑎𝑥 = 420
Constraint 2: 𝐶𝑃0.29 𝑧1, 𝑛2∗ ≥ 80%
Constraint 3: 𝐶𝑃0.29 𝑧1, 𝑛2∗ ≤ 90%
• No sample size modification outside of promising zone
• Testing uses CHW combination statistic
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Conditional Power if no SSR
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Promising
Zone
Conditional Power if SSR up to nmax=420
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Conditional Power and Prob(zone): nmax=420
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Prob(zone)=43%Prob(zone)=30% Prob(zone)=27%
Conditional Power and Prob(zone): nmax=500
Prob(zone)=20% Prob(zone)=26%
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• Sponsor likes this SSR rule
• Even in the pessimistic case, where q=0.29:
• If SSR is invoked, ≥ 80% power is guaranteed
• But how does this rule compare with the “Optimal” SSR rule as defined by J&T?
Optimal SSR Rule: Has the best unconditional power among
all SSR rules with the same average sample size
But is this SSR rule optimal?
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• SSR Rule: Sample size rule 𝑛2∗(𝑧1) satisfies:
Objective: Maximize 𝐶𝑃𝜃 𝑧1, 𝑛2∗ − 𝛾𝑛2
∗ for every z1
Constraint: 𝑛2 ≤ 𝑛2∗ ≤ 𝑛𝑚𝑎𝑥
where 𝛾 is a constant “exchange rate” between CP and N
• Optimality property: By integrating the objective with respect to fq(z1):
• = Pwr(q) – gE(N) is maximized
• Highest possible unconditional power among SSR rules with matching nmax and E(N)
• Benchmarking tool for adaptive designs
Jennison Turnbull (JT) Optimal SSR Rule
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* *
θ 1 2 2 θ 1 1max CP (z ,n )-γn f ( )dzz
Efficiency Comparison with JT Optimal Design
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• Method: For each 𝜃, compare unconditional power of AD against JT design with 𝛾 chosen so expected sample size matches AD
Efficiency Comparison with JT Optimal Design
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• Comparison at 𝜃 = 0.29
Pr(zone)=0.54
Pr(zone)=0.43
Does it meet the sponsor’s objectives?
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Conclusions
• JT Optimal Design gains 2-3% unconditional power
• Requirement of high CP at lowest meaningful 𝜃 is not met by JT Design
Constrained JT Rule
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• Impose an additional CP constraint on the JT SSR rule.
• Constrained SSR Rule: Final sample size 𝑛2∗ determined by:
Objective: Maximize 𝐶𝑃𝜃 𝑧1, 𝑛2∗ − 𝛾𝑛2
∗
Constraint 1: 𝑛2 ≤ 𝑛2∗ ≤ 𝑛𝑚𝑎𝑥
Constraint 2: 𝐶𝑃0.29 𝑧1, 𝑛2∗ ≥ 80%
• Optimality property: Highest unconditional power among promising zone designs satisfying same constraints and matching E(N)
Comparison of AD and Constrained JT
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• Method: For each 𝜃, compare unconditional power of AD against constrained JT Design with 𝛾 chosen so expected sample size matches AD
Comparison of AD and Constrained JT
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• Comparison at 𝜃 = 0.29
Comparison AD and Constrained JT
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Conclusions
• Equally efficient in terms of unconditional power
• Similar conditional power profiles
Conclusions
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• We considered a promising zone AD for an oncology trial
• Maximize CP
• Require sufficiently high CP to justify sample size increase
• Provide method for objective efficiency comparison
• 2-3% loss of unconditional power compared to optimal JT design which has wider SSR zone and recommends increasing N at lower 𝑧1 values
• No loss of efficiency compared to optimal constrained JT design which requires 𝐶𝑃0.29 𝑧1, 𝑛2
∗ > 80%
• Sponsor’s utility will determine whether a CP constraint makes sense, at the cost of some efficiency loss compared to JT
References
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• Liu, Lingyun, Sam Hsiao, and Cyrus R. Mehta. 2017. “Efficiency Considerations for Group Sequential Designs with Adaptive Unblinded Sample Size Re-Assessment.” Statistics in Biosciences, March.
• Mehta, Cyrus, and Lingyun Liu. 2016. “An Objective Re-Evaluation of Adaptive Sample Size Re-Estimation: Commentary on ‘Twenty-Five Years of Confirmatory Adaptive Designs.’” Statistics in Medicine 35: 350–58.
• Jennison, Christopher, and Bruce W. Turnbull. 2015. “Adaptive Sample Size Modification in Clinical Trials: Start Small Then Ask for More?” Statistics in Medicine 34 (29): 3793–3810.
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