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

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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