Generalized Probit Model in Design of Dose Finding Experiments Yuehui Wu Valerii V. Fedorov RSU, GlaxoSmithKline, US
Outline Motivation Generalized probit model Utility function Locally optimal designs Simulation
Motivation In dose-finding studies, one popular goal is to locate the best dose What does best mean? Balance between efficacy and toxicity Need to model for multiple endpoints Observe continuous responses Report results based on dichotomized responses
Background Model Continuous responses (may not be observed) Some function of Z is reported (used), e.g. dichotomization
Bivariate Probit Model Both responses are dichotomized: Y 1 for toxicity, Y 2 for efficacy Note: if c k is unknown, in general, σ k is not estimable. Here we assume σ k =1 and known.
2D dichotomization
Utility Function Goal: locate an efficacious while non-toxic dose A regulatory agency is interested only in function no matter the observed responses are continuous or not Toxicity responses are binary and efficacy responses are continuous
Dose-Response Curves and Utility Function dose
Dichotomize or not? Utility function is based on dichotomized responses When the continuous observations are available, do not use dichotomized responses in the analysis because dichotomization leads to loss of information If one needs to report the results based on dichotomized responses, use utility function
Locally Optimal Design Most popular goal: locate the dose which achieve the maximum value of utility function Locally optimal design Design criterion: L(Θ)- optimality: D-optimality:
Numerical Algorithm Sensitivity function L(Θ)-optimal D-optimal First order exchange algorithm Forward step Backward step
Design with Cost Constrains (no example) It s unethical to assign patients to non-efficacious or potentially toxic doses May introduce penalty function Corresponding design criterion Corresponding sensitivity function
Information Matrix Notation Design: Information matrix: Both responses are continuous
Information Matrix (2) Both responses are binary, ρ is known When ρ is unknown, see GSK Technical Report 2006-01
Example Design region: [0,1] Parameter values: (1.5; 2.7; 0.05; 2.2) For the same utility function with cut off value c1=0.5, c2=0.1 Assume ρ=0.5 and known Locally optimal designs: Using continuous responses Using binary responses
Locally optimal designs L(Θ)-optimal D-optimal dose
Impact of dichotomization Assume continuous responses are available, binary utility function is used Sample size n=200 Will using dichotomized responses in the analysis have negative impact on the precision of parameter estimations?
Precision comparison (1) Number of subjects needed to get the same precision of
Precision comparison (2) Number of subjects needed to get the same precision of
Simulation Generate 200 continuous observations according to four designs: L and D optimal designs under bivariate probit model and bivariate linear regression model respectively Fit bivariate linear regression model to obtain parameter estimation, calculate corresponding utility function and locate the target doses Dichotomize the same continuous observations and fit bivariate probit model with cutoff values c1 = 0.5 and c2 = 0.1 and estimate the target dose Repeat the procedure 1000 times to compare the distributions of the estimated target doses under the two models.
L(Θ)-optimal design results Target dose under true parameters
L(Θ)-optimal design results (2)
D-optimal design results
D-optimal design results (2)
Conclusion Use continuous responses in the analysis whenever they are available The utility function can be based on either continuous or dichotomized response by whatever is expatiate L(Θ)- optimal design is more complicated to derive D-optimal is a robust choice for different study goals
References Fedorov, V.V. Theory of Optimal Experiments, 1972, New York: Academic Press. Fedorov, V.V. and Hackl, P. Model-Oriented Design of Experiments; Lecture Notes in Statistics 125; Springer-Verlag, New York, 1997. Dragalin, V., Fedorov, V., and Wu, Y. (2006). Optimal Designs for Bivariate Probit Model. GSK Technical Report 2005-07. 62 pages. http://www.biometrics.com/downloads/tr_2006_01.p df. Fedorov, V. and Wu, Y. Dose Finding for Binary Utility. Submitted to Journal of Biopharmaceutical Statistics.