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Penalized designs of multi-response experiments

Valerii FedorovAcknowledgements: Yuehui Wu

GlaxoSmithKline, Research Statistics Unit

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OutlineRI approach

Reminder

Model

Information, utility functions, criteria

Design

Analysis of scenarios

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RI approach: recycling of ideas

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¾ came from A&F, 1988

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Twenty years ago

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Thirty years ago

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Observations and penaltyReminder

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Likelihood and information matrixReminder

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Cost constrained designReminder

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Design transformReminder

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Equivalence theoremReminder

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Specific cases from 1988, A&F

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Bayesian* and composite designs

Stage Conservative Optimistic

Design Use prior Use prior

Estimation No prior Use prior

*

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First order algorithmReminder

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Adaptive designReminder

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Binary observationsModel

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Two drugs: typical responsesModel

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Probit modelModel

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More about probit model can be found in:

Model

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Diagram from Pearson’s paper

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Linear predictorModel

In what follows θ = {-1.26, 2.0, 0.9, 14.8; -1.13, 0.94, 0.36, 4} , ρ = 0.5

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Information Matrix for a Single Observationor What Can be Learned from One Patient

Information

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2D Penalty functionPenalty = { Probability of efficacy} – 1 ∗ {Probability of no toxicity} – 1.

Model

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Utility and optimality criterion

Utility (or what we want): Probability of efficacy without toxicity, i.e. p10(x)Distance from desirability point: δ (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 θ }

Utility and optimality criterion

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Utility - I

arg max p10(x,θ) ={ 0.18, 0.68 }

Utility and optimality criterion

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Utility - IIUtility and optimality criterion

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Design scenarios

91 subjects were assigned to three doses per drug in initial design (7 support points)Another 90 subjects to be assignedTotal sample size: 181

Analysis of scenarios

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Penalized locally D-optimalAnalysis of scenarios

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Penalized composite locally D-optimal

Initial design

Added design points

Analysis of scenarios

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Design comparison for D-criterioninitial design included

Design type Information per patient

Penalty per patient

Information per penalty u.

Locally D-optimal 1 2.07 0.48

0.79

0.62

0.52

0.50

0.39

0.50

Penalized locally optimal 0.79 1

Penalized composite locally optimal 0.77 1.23

Penalized composite (median, 500) 0.70 1.36

Penalized adaptive (median, 500) 0.75 1.50

“Do your best” adaptive (median, 500) 0.46 1.19

Restricted “Uniform ” , 12 points 0.67 1.34

Analysis of scenarios

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Simulation results (500 runs): the estimated best combination

Analysis of scenarios

Adaptive D-optimalComposite D-optimal

Do your best

Best dose

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Simulation results (500 runs): the estimated best response

Composite D-adaptive Do your best

Analysis of scenarios

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CONCLUSION

Recycling helps

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References Atkinson, A.C. and Fedorov, V.V. (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.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.Cook, D. and Fedorov, V. (1995). Constrained optimization of experimental design. Statistics 26: 129-178.Dragalin, V., Fedorov, V., and Wu, Y. (2006). Optimal Designs for BivariateProbit Model. GSK Technical Report 2006-01. http://www.biometrics.com/8D97129158B901878025714D00389BEB.htmlFedorov, V.V. and Hackl, P. (1997). Model-Oriented Design of Experiments. Lecture Notes in Statistics, pp.125. Springer.Ford, I., Titterington, D.M. and Wu, C.F.J. (1985). Inference and sequential design. Biometrika 72: 545-551.Hu, I. (1998). On sequential designs in nonlinear problems. Biometrika 85: 446-503.Hu, F. and Rosenberger, W. (2006), The theory of response-adaptive randomization in clinical trials, Wiley.Ying, Z. and Wu, C.F.J. (1997). An asymptotical theory of sequential designs based on maximum likelihood recursions. Statistica Sinica 7: 75-91.

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Appendix

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Estimation of the best dose with and without dichotomization I

Dichotomized Continuous

Model

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Estimation of the best dose with and without dichotomization II

Model

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2D binaryResponders without toxicity -

Responders with toxicity

Non-responders without toxicity

Non-responders with toxicity

Model