1. Penalized designs of multi-response experiments
Valerii Fedorov
Acknowledgements: Yuehui Wu
GlaxoSmithKline, Research Statistics Unit

2. Outline IR approach Reminder Model
Information, utility functions, criteria
Design
Analysis of scenarios

3. IR approach: recycling of ideas

4. Observations and penalty
Reminder
Observations and penalty

5. Likelihood and information matrix
Reminder
Likelihood and information matrix

6. Cost constrained design
Reminder
Cost constrained design

7. Reminder
Design transform

8. Reminder
Equivalence theorem

9. Reminder
First order algorithm

10. Reminder
Adaptive design

11. Model
Binary observations

12. Two drugs: typical responses
Model
Two drugs: typical responses

13. Model
Probit model

14. More about probit model can be found in:

15. Diagram from Pearson’s paper

16. Linear predictor Model
In what follows  = {-1.26, 2.0, 0.9, 14.8; -1.13, 0.94, 0.36, 4} ,  = 0.5

17. Information
Information Matrix for a Single Observation or What Can be Learned from One Patient

18. Model
2D Penalty function Penalty = { Probability of efficacy} – 1  {Probability of no toxicity} – 1.

19. 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  }

20. Utility - I Utility and optimality criterion
arg max p10(x,) ={ 0.18, 0.68 }

21. Utility and optimality criterion
Utility - II

22. 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

23. Penalized locally D-optimal
Analysis of scenarios
Penalized locally D-optimal

24. Penalized composite locally D-optimal
Analysis of scenarios
Penalized composite locally D-optimal
Initial design
Added design points

25. 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

26. 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

27. 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

28. CONCLUSIONS
Recycling helps
Select carefully whom you collaborate with

29. 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)"
[2] Cook, D. and Fedorov, V. (1995). Constrained optimization of experimental design. Statistics 26:
[3] Dragalin, V. and Fedorov, V. (2006) Adaptive model-based designs for dose-finding studies. JSPI, 136,
Dragalin, V., Fedorov, V., and Wu, Y. (2006). Optimal Designs for Bivariate Probit Model. GSK Technical Report
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), 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
[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.

30. References II
[10] Ford, I., Titterington, D.M. and Wu, C.F.J. (1985). Inference and sequential design. Biometrika 72:
[11] Heise, M.A. and Myers, R.H. (1996). Optimal designs for bivariate logistic regression. Biometrics 52:
[13] Hu, I. (1998). On sequential designs in nonlinear problems. Biometrika 85:
[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:
[16] O'Quigley, J., Hughes, M., and Fenton, T. (2001). Dose finding designs for HIV studies. Biometrics 57:
[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), 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:
[19] Whitehead, J. and Williamson, D. (1998). An evaluation of Bayesian decision procedures for dose-finding studies. J. Biopharma.Stat. 8:
[20] Ying, Z. and Wu, C.F.J. (1997). An asymptotical theory of sequential designs based on maximum likelihood recursions. Statistica Sinica 7:

31. Estimation of the best dose with and without dichotomization I
Model
Estimation of the best dose with and without dichotomization I
Dichotomized
Continuous

32. Estimation of the best dose with and without dichotomization II
Model
Estimation of the best dose with and without dichotomization II

33. 2D binary Model Responders without toxicity - Responders with toxicity
Non-responders without toxicity
Non-responders with toxicity