Using CVX to Construct Optimal Designs for Biomedical Studies with Multiple Objectives

ABSTRACT

Model-based optimal designs for regression problems with multiple objectives are common in practice. The traditional approach is to construct an optimal design for the most important objective and hope that the design performs well for the other objectives. Analytical approaches are challenging because the objectives are often competitive and their relative importance has to be incorporated at the onset of the design construction. There are also no general and efficient algorithms for searching such designs for user-specified nonlinear models and criteria. We propose a new and effective approach for finding multiple-objective optimal designs via the CVX software and demonstrate it can efficiently find different types of multiple-objective optimal designs after the optimization problems are carefully formulated as convex optimization problems appropriate for CVX use. We provide three biomedical applications and show our MATLAB code producing the same few multiple-objective optimal designs reported in the statistical literature. MATLAB code files are available online in the supplementary materials of this article.

KEYWORDS:

• A-efficiency
• Convex optimization
• D-efficiency
• Maximin design
• Nonlinear models
• Robustness
• Solver

Supplementary Materials

Algorithm 1: MATLAB code is provided to implement Algorithm 1 for Application 2. The code can generate the results in Table 3. To run the code, it is necessary to install CVX solver in MATLAB (Grant and Boyd 2013). The code can be easily modified for finding MECC optimal designs for other regression models and other objective functions. (Application2Table3.m)

Algorithm 2: MATLAB code is provided to implement Algorithm 2 for Application 3. The code is used to generate the D-optimal maximin design results in Table 5. (Application3Table5.m)

Acknowledgements

The authors thank the Editors and referees for their helpful comments and suggestions to improve the presentation of this article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

This research of Zhou was partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.