## Electronic Journal of Statistics

### Analytic solutions for D-optimal factorial designs under generalized linear models

#### Abstract

We develop two analytic approaches to solve D-optimal approximate designs under generalized linear models. The first approach provides analytic D-optimal allocations for generalized linear models with two factors, which include as a special case the $2^{2}$ main-effects model considered by Yang, Mandal and Majumdar [19]. The second approach leads to explicit solutions for a class of generalized linear models with more than two factors. With the aid of the analytic solutions, we provide a necessary and sufficient condition under which a D-optimal design with two quantitative factors could be constructed on the boundary points only. It bridges the gap between D-optimal factorial designs and D-optimal designs with continuous factors.

#### Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 1322-1344.

Dates
First available in Project Euclid: 20 August 2014

https://projecteuclid.org/euclid.ejs/1408540289

Digital Object Identifier
doi:10.1214/14-EJS926

Mathematical Reviews number (MathSciNet)
MR3263124

Zentralblatt MATH identifier
1298.62136

Subjects
Primary: 62K05: Optimal designs
Secondary: 62K15: Factorial designs

#### Citation

Tong, Liping; Volkmer, Hans W.; Yang, Jie. Analytic solutions for D-optimal factorial designs under generalized linear models. Electron. J. Statist. 8 (2014), no. 1, 1322--1344. doi:10.1214/14-EJS926. https://projecteuclid.org/euclid.ejs/1408540289

#### References

• [1] Atkinson, A. C., Donev, A. N. and Tobias, R. D. (2007). Optimum Experimental Designs, with SAS, Oxford University Press.
• [2] Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM J. Scientific Computing 16, 1190–1208.
• [3] Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: a review. Statistical Science 10, 273–304.
• [4] Chernoff, H. (1953). Locally optimal designs for estimating parameters. Annals of Mathematical Statistics 24, 586–602.
• [5] Collett, D. (1991). Modelling Binary Data. Chapman & Hall/CRC, New York.
• [6] Cox, D., Little, J. and O’Shea, D. (2005). Using Algebraic Geometry, 2nd Edition. Springer-Verlag, New York.
• [7] Dobson, A. J. and Barnett, A. (2008). An Introduction to Generalized Linear Models, 3rd edition. Chapman and Hall/CRC, New York.
• [8] Ford, I., Titterington, D. M. and Kitsos, C. P. (1989). Recent advances in nonlinear experimental design. Technometrics 31, 49–60.
• [9] Imhof, L. A. (2001). Maximin designs for exponential growth models and heteroscedastic polynomial models. Annals of Statistics 29, 561–576.
• [10] Karush, W. (1939). Minima of functions of several variables with inequalities as side constraints. M.Sc. Dissertation. Dept. of Mathematics, Univ. of Chicago.
• [11] Khuri, A. I., Mukherjee, B., Sinha, B. K. and Ghosh, M. (2006). Design issues for generalized linear models: a review. Statistical Science 21, 376–399.
• [12] Kuhn, H. W. and Tucker, A. W. (1951). Nonlinear programming. Proceedings of 2nd Berkeley Symposium. University of California Press, Berkeley, 481–492.
• [13] McCullagh, P. and Nelder, J. (1989). Generalized Linear Models, 2nd Edition. Chapman and Hall/CRC, New York.
• [14] Pronzato, L. and Walter, E. (1988). Robust experiment design via maximin optimization. Mathematical Biosciences 89, 161–176.
• [15] Sitter, R. R. and Torsney, B. (1995). Optimal designs for binary response experiments with two design variables. Statistica Sinica 5, 405–419.
• [16] Sitter, R. R. and Wu, C. F. J. (1993). Optimal designs for binary response experiments: Fieller, D, and A criteria. Scandinavian Journal of Statistics 20, 329–341.
• [17] Stufken, J. and Yang, M. (2012). Optimal designs for generalized linear models. In: Design and Analysis of Experiments, Volume 3: Special Designs and Applications, K. Hinkelmann (ed.). Wiley, New York.
• [18] Woods, D. C., Lewis, S. M., Eccleston, J. A. and Russell, K. G. (2006). Designs for generalized linear models with several variables and model uncertainty. Technometrics 48, 284–292.
• [19] Yang, J., Mandal, A. and Majumdar, D. (2012). Optimal designs for two-level factorial experiments with binary response. Statistica Sinica 22, 885–907.
• [20] Yang, J., Mandal, A. and Majumdar, D. (2013). Optimal designs for $2^{k}$ factorial experiments with binary response, Technical Report, available at http://arxiv.org/pdf/1109.5320v4.pdf.
• [21] Yang, J. and Mandal, A. (2013). D-optimal factorial designs under generalized linear models. To appear in Communications in Statistics – Simulation and Computation, available also via http://arxiv.org/pdf/1301.3581v2.pdf.