Penalized designs of multi-response experiments Valerii Fedorov Acknowledgements: Yuehui Wu GlaxoSmithKline, Research Statistics Unit
Outline IR approach Reminder Model Information, utility functions, criteria Design Analysis of scenarios
IR approach: recycling of ideas
Observations and penalty Reminder Observations and penalty
Likelihood and information matrix Reminder Likelihood and information matrix
Cost constrained design Reminder Cost constrained design
Reminder Design transform
Reminder Equivalence theorem
Reminder First order algorithm
Reminder Adaptive design
Model Binary observations
Two drugs: typical responses Model Two drugs: typical responses
Model Probit model
More about probit model can be found in:
Diagram from Pearson’s paper
Linear predictor Model In what follows = {-1.26, 2.0, 0.9, 14.8; -1.13, 0.94, 0.36, 4} , = 0.5
Information Information Matrix for a Single Observation or What Can be Learned from One Patient
Model 2D Penalty function Penalty = { Probability of efficacy} – 1 {Probability of no toxicity} – 1.
Utility and optimality criterion Utility (or what we want): Probability of efficacy without toxicity, i.e. p10(x) Distance from desirability point: (x*) All unknown parameters Optimality criteria: Var {estimated p10(x)} Var {estimated max[p10(x)]} + Var {estimated location of max[p10(x)]} Var {estimated x*} Var {estimated }
Utility - I Utility and optimality criterion arg max p10(x,) ={ 0.18, 0.68 }
Utility and optimality criterion Utility - II
Analysis of scenarios Design scenarios 91 subjects were assigned to three doses per drug in initial design (7 support points) Another 90 subjects to be assigned Total sample size: 181
Penalized locally D-optimal Analysis of scenarios Penalized locally D-optimal
Penalized composite locally D-optimal Analysis of scenarios Penalized composite locally D-optimal Initial design Added design points
Design comparison for D-criterion initial design included Analysis of scenarios Design comparison for D-criterion initial design included Design type Information per patient Penalty per patient Information per penalty u. Locally D-optimal 1 2.07 0.48 Penalized locally optimal 0.79 Penalized composite locally optimal 0.77 1.23 0.62 Penalized composite (median, 500) 0.70 1.36 0.52 Penalized adaptive (median, 500) 0.75 1.50 0.50 “Do your best” adaptive (median, 500) 0.46 1.19 0.39 Restricted “Uniform ” , 12 points 0.67 1.34
Simulation results (500 runs): the estimated best combination Analysis of scenarios Simulation results (500 runs): the estimated best combination Composite D-optimal Adaptive D-optimal Best dose Do your best
Simulation results (500 runs): the estimated best response Analysis of scenarios Simulation results (500 runs): the estimated best response Composite D-adaptive Do your best
CONCLUSIONS Recycling helps Select carefully whom you collaborate with
References I [1] Box, G.E.P. and Hunter, W.G. (1963). Sequential design of experiments for nonlinear models. In "Proceedings of IBM Scientic Computing Symposium (Statistics)". 113-137. [2] Cook, D. and Fedorov, V. (1995). Constrained optimization of experimental design. Statistics 26: 129-178. [3] Dragalin, V. and Fedorov, V. (2006) Adaptive model-based designs for dose-finding studies. JSPI, 136, 1800-1823. Dragalin, V., Fedorov, V., and Wu, Y. (2006). Optimal Designs for Bivariate Probit Model. GSK Technical Report 2006-01. http://www.biometrics.com/8D97129158B901878025714D00389BEB.html Dragalin, V., Fedorov, V., and Wu, Y. (2007) . Adaptive designs for selecting drug combinations based on efficacy-toxicity responses. JSPI, early view. [4] Fan, S.K. and Chaloner, K. (2001). Optimal designs for a continuation-ratio model. In MODA 6 Advances in Model-Oriented Design and Analysis. A.Atkinson, P.Hackl, and W.G.Müller (eds), 77-85. Heidelberg: Physica-Verlag. [6] Fedorov, V.V. and Atkinson, A.C. (1988). The optimum design of experiments in the presence of uncontrolled variability and prior information. In Optimal Design and Analysis of Experiments. Eds. Dodge Y., Fedorov V.V. and Wynn H.P. New York: North-Holland. pp 327- 344. [7] Fedorov, V.V. and Hackl, P. (1997). Model-Oriented Design of Experiments. Lecture Notes in Statistics, pp.125. Springer. [8] Fedorov, V., Gagnon, R., Leonov, S., Wu, Y. (2007). Optimal design of experiments in pharmaceutical applications. In: Dmitrienko, A. et al. (eds), Pharmaceutical Statistics, SAS Press. [9] Fedorov V, Wu Y. (2007). Generalized probit model in design of dose finding experiments. In: Lopez-Fidalgo J, Rodriguez-Diaz JM, Torsney B (Eds), mODa 8 – Advances in Model-Oriented Design and Analysis, Physica-Verlag, 67–74.
References II [10] Ford, I., Titterington, D.M. and Wu, C.F.J. (1985). Inference and sequential design. Biometrika 72: 545-551. [11] Heise, M.A. and Myers, R.H. (1996). Optimal designs for bivariate logistic regression. Biometrics 52: 613-624. [13] Hu, I. (1998). On sequential designs in nonlinear problems. Biometrika 85: 446-503. [14] Hu, F. and Rosenberger, W. (2006), The theory of response-adaptive randomization in clinical trials, Wiley. [15] Murtaugh, P.A. and Fisher, L.D. (1990). Bivariate binary models of efficacy and toxicity in dose-ranging trials. Commun. Statist. {Theory Meth.} 19: 2003-2020. [16] O'Quigley, J., Hughes, M., and Fenton, T. (2001). Dose finding designs for HIV studies. Biometrics 57: 1018 -1029. [17] Rabie, H.S. and Flournoy, N. (2004). Optimal designs for contingent response models. In MODA 7 | Advances in Model-Oriented Design and Analysis. A. Di Bucchianico, H. Läuter and H.P. Wynn (eds), 133-142. Heidelberg: Physica-Verlag. [18] Thall, P. and Russell, K. (1998). A strategy for dose-finding and safety monitoring based on efficacy and adverse outcomes in phase I/II clinical trials. Biometrics 54: 251-264. [19] Whitehead, J. and Williamson, D. (1998). An evaluation of Bayesian decision procedures for dose-finding studies. J. Biopharma.Stat. 8: 445-467. [20] Ying, Z. and Wu, C.F.J. (1997). An asymptotical theory of sequential designs based on maximum likelihood recursions. Statistica Sinica 7: 75-91.
Estimation of the best dose with and without dichotomization I Model Estimation of the best dose with and without dichotomization I Dichotomized Continuous
Estimation of the best dose with and without dichotomization II Model Estimation of the best dose with and without dichotomization II
2D binary Model Responders without toxicity - Responders with toxicity Non-responders without toxicity Non-responders with toxicity