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Generalized Probit Model in Design of Dose Finding Experiments - - PowerPoint PPT Presentation
Generalized Probit Model in Design of Dose Finding Experiments - - PowerPoint PPT Presentation
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
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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
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Background Model
Continuous responses (may not
be observed)
Some function of Z is reported (used), e.g.
dichotomization
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Bivariate Probit Model
Both responses are dichotomized: Y1 for toxicity, Y2
for efficacy
Note: if ck is unknown, in general, σk is not estimable.
Here we assume σk =1 and known.
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2D dichotomization
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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
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Dose-Response Curves and Utility Function
dose
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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
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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:
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Numerical Algorithm
Sensitivity function
L(Θ)-optimal D-optimal
First order exchange algorithm
Forward step Backward step
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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
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Information Matrix
Notation
Design: Information matrix:
Both responses are continuous
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Information Matrix (2)
Both responses are binary, ρ is known When ρ is unknown, see GSK Technical Report 2006-01
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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
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Locally optimal designs
L(Θ)-optimal D-optimal
dose
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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?
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Precision comparison (1)
Number of subjects needed to get the same
precision of
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Precision comparison (2)
Number of subjects needed to get the same
precision of
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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.
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L(Θ)-optimal design results
Target dose under true parameters
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L(Θ)-optimal design results (2)
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D-optimal design results
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D-optimal design results (2)
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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
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References
Fedorov, V.V. Theory of Optimal Experiments, 1972,
New York: Academic Press.
Fedorov, V.V. and Hackl, P. Model-Oriented Design
- f 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.