Estimation efficiency in continual reassessment method Tian Tian University of Illinois at Chicago Background Introduction Finding an accurate maximum tolerated dose (MTD), the most efficacious dose whose risk of toxicity is tolerable, is a critical step in development of medicine. There are two widely used approaches for identifying MTD. – The conventional 3 + 3 approach. – The continual reassessment method (CRM), which has been shown to be more efficient that the 3 + 3 approach through many simulation studies. Motivation for our design Why is CRM more efficient? Can we prove it theoretically? Is there a way to further improve on the performance of CRM? Description of a dose-finding study The target toxicity rate is set to be p 0 . We have m available dose levels and n patients in total in total. The binary response Y at dose level x is modeled as: Prob (Y = 1|x )= φ(x ,θ ) Design problem: How to assign dose levels to patients at each stage in a trial such that we can identify MTD (x * : φ(x * ,θ )= p 0 ) accurately? Remark: Remember that in a standard CRM procedure, at each stage, we assign each patient the dose level with corresponding toxicity rate p 0 .—-So could this assignment fashion be improved? Combining (locally) optimal design Consider MTD, the parameter of interest, as a function of θ , i.e., MTD = b (θ ). The asymptotic variance of the estimator of MTD could be expressed as V ( [ MTD )= ∂ b (θ ) ∂θ ( n X j =1 I j (θ, d ) ) -1 ∂ b (θ ) ∂θ T . Then a locally optimal design ξ * based on ˆ θ is the one that minimizes V ( [ MTD ). Simple power model problem Model Dose-response model: p = φ(x ,α)= x α , α> 0 and 0 < x < 1. (1) Under design ξ = {(x i , w i ), i = 1, ..., k }, the asymptotic variance for [ MTD is: V ( [ MTD )= h ( p 1/α 0 )( ln p 0 )( - 1 α 2 ] 2 k X i =1 w i x α i (log x i ) 2 1 - x α i -1 . Optimal design for simple power model problem Theorem: Under simple power model (1), regardless of the target toxicity rate p 0 set in the trial, the optimal design always choose the next dose level with corresponding toxicity rate ˜ p , where ˜ p is the solution to equation log p - 2p + 2 = 0. Remark: Numerical approximation: ˜ p ≈ 0.203. Interpretation (I). Theoretically optimal! Recall that most clinical studies set p 0 at 0.2. Thus our optimality result here further confirms their choices of design to be not only medically reasonable, but also statistically optimal. Moreover, as long as p 0 is chosen from a reasonable range (say, from 0.1 to 0.35), the standard CRM procedure will generate an optimal design, or at least a nearly optimal design with negligible efficiency loss. Table: Relative efficiency under different target toxicity rate p 0 0.1 0.15 0.2 0.25 0.3 0.35 Relative efficiency 0.910 0.981 1 0.990 0.960 0.916 (II). Counter-intuitive! No matter what p 0 is, optimal design always collect data at dose level with toxicity rate 0.2! Simulation comparison Table: Comparison of perforemance of standard CRM and optimal CRM Panel 1: α = 0.5 p 0 0.15 0.25 0.3 standard CRM (0.505,0.217) (0.391,0.307) (0.422,0.348) optimal CRM (0.504,0.263) (0.433,0.262) (0.388,0.262) p 0 0.7 0.8 standard CRM (0.578,0.743) (0.363,0.833) optimal CRM (0.783,0.262) (0.580,0.272) Panel 2: α = 2 p 0 0.15 0.25 0.3 standard CRM (0.567,0.134) (0.461,0.214) (0.464,0.260) optimal CRM (0.593,0.180) (0.532,0.173) (0.408,0.174) p 0 0.7 0.8 standard CRM (0.786,0.641) (0.458,0.754) optimal CRM (0.857,0.186) (0.578,0.218) – First value of each entry represents the percentage of accurately identifying the MTD while the second represents the percentage of toxicity occurrence. Guidance to design choices Practical situations (when p 0 is around 0.2) – When p 0 < 0.2, performance of the standard CRM is nearly as good as that of the optimal CRM; but with lower toxicity occurrence. – When p 0 > 0.2, performance of the optimal CRM is similar to that of the standard CRM; but with lower toxicity occurrence. High target toxicity cases (e.g., p 0 = 0.7, 0.8) – Optimal CRM is significantly better from the perspective of estimation efficiency as well as toxicity occurrence. Two-parameter logistic model problem Model Dose-response model: p = φ(x , a)= e α+β x 1 + e α+β x (2) where β> 0, α< log p 0 1 - p 0 and x > 0. (3) We write MTD as a function of θ , i.e., MTD def = η = b (θ )= 1 β ( log p 0 1 - p 0 - α ) . Optimal design for logistic model problem Since MTD is a scalar function of θ , the c -optimality, which is designed to optimize the estimation of linear combination of parameters, is an appropriate choice here. We adopt the geometric approach provided by Elfving (1952) for identifying c -optimal design. Figure: Elfving set of model (2) with parameter (α, β )=(-2, 2) - 0.4 - 0.2 0.2 0.4 h 1 - 0.6 - 0.4 - 0.2 0.2 0.4 0.6 h 2 A B C D R P Q Θ 1 Θ 2 Theorem: Under logistic model (2) with the assumptions (3), for any 0 < p 0 < 0.5, the optimal design selects the next dose level with target toxicity rate p 0 . Conclusion: Under two-parameter logistic model, when p 0 is set in a reasonable range (0, 0.5), the standard CRM algorithm is exactly optimal! Future work – To incorporate model uncertainty problem into optimal design theory by considering model averaging/selecting techniques. – To incorporate delayed-response problem into optimal design theory by considering the use of EM algorithm or other missing-data handling methods like Bayesian data-augmentation. Acknowledgements This project is supported in part by NSF Grants DMS-13-22797 and DMS-14-07518. The author would like to thank Professor Min Yang for many inspiring discussions and insightful assistance. Email: [email protected] Midwest Biophar. Stat. Workshop 2015 MSCS, UIC