Abstract
Model-based optimal designs of experiments (M-bODE) for nonlinear models are typically hard to compute. The literature on the computation of M-bODE for nonlinear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Second Order Programming (MISOCP) formulations to find, respectively, approximate and exact A- and D-optimal designs for \(2^k\) factorial experiments for Generalized Linear Models (GLMs). First, locally optimal (approximate and exact) designs for GLMs are addressed using the formulation of Sagnol (J Stat Plan Inference 141(5):1684–1708, 2011). Next, we consider the scenario where the parameters are uncertain, and new formulations are proposed to find Bayesian optimal designs using the A- and log det D-optimality criteria. A quasi Monte-Carlo sampling procedure based on the Hammersley sequence is used for computing the expectation in the parametric region of interest. We demonstrate the application of the algorithm with the logistic, probit and complementary log–log models and consider full and fractional factorial designs.
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References
Alizadeh F, Goldfarb D (2001) Second-order cone programming. Math Progr 95:3–51
Atkinson AC, Donev AN, Tobias RD (2007) Optimum experimental designs, with SAS. Oxford University Press, Oxford
Ben-Tal A, Nemirovski AS (2001) Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Society for Industrial and Applied Mathematics, Philadelphia
Berger MPF, Wong WK (2005) Applied optimal designs. Wiley, New York
Bisetti F, Kim D, Knio O, Long Q, Tempone R (2016) Optimal Bayesian experimental design for priors of compact support with application to shock-tube experiments for combustion kinetics. Int J Num Methods Eng 108(2):136–155
Caflisch RE (1998) Monte Carlo and quasi-Monte Carlo methods. Acta Num 7:1–49
Chaloner K, Larntz K (1989) Optimal Bayesian design applied to logistic regression experiments. J Stat Plan Inference 59:191–208
Chaloner K, Verdinelli I (1995) Bayesian experimental design: a review. Stat Sci 10:273–304
Diwekar UM, Kalagnanam JR (1997) An efficient sampling technique for optimization under uncertainty. AIChE J 43:440–449
Dorta-Guerra R, González-Dávila E, Ginebra J (2008) Two-level experiments for binary response data. Comput Stat Data Anal 53(1):196–208
Dror HA, Steinberg DM (2008) Sequential experimental designs for generalized linear models. J Am Stat Assoc 103:288–298
Drovandi CC, Tran MN (2018) Improving the efficiency of fully Bayesian optimal design of experiments using randomised quasi-Monte Carlo. Bayesian Anal 13(1):139–162
Duarte BPM, Wong WK (2015) Finding Bayesian optimal designs for nonlinear models: a semidefinite programming-based approach. Int Stat Rev 83(2):239–262
Duarte BPM, Wong WK, Oliveira NMC (2016) Model-based optimal design of experiments—semidefinite and nonlinear programming formulations. Chemom Intell Lab Syst 151:153–163
Firth D, Hinde JP (1997) On bayesian \(D\)-optimum design criteria and the equivalence theorem in non-linear models. J R Stat Soc 59(4):793–797
GAMS Development Corporation (2013) GAMS—A User’s Guide, GAMS Release 24.2.1. GAMS Development Corporation, Washington, DC, USA
Gotwalt CM, Jones BA, Steinberg DM (2009) Fast computation of designs robust to parameter uncertainty for nonlinear settings. Technometrics 51(1):88–95
Graßhoff U, Schwabe R (2008) Optimal design for the Bradley–Terry paired comparison model. Stat Methods Appl 17(3):275–289
Hammersley JM, Handscomb DC (1964) Monte Carlo methods. Methuen, London
Harman R, Bachratá A, Filová L (2016) Construction of efficient experimental designs under multiple resource constraints. Appl Stoch Models Bus Ind 32(1):3–17
Harman R, Filová L (2016) Package “OptimalDesign”. https://cran.r-project.org/web/packages/OptimalDesign/OptimalDesign.pdf
Huan X, Marzouk YM (2013) Simulation-based optimal Bayesian experimental design for nonlinear systems. J Comput Phys 232(1):288–317
IBM (2015) IBM ILOG CPLEX Optimization Studio—CPLEX User’s Manual. Version 12, Release 6
Li G, Majumdar D (2008) \(D\)-optimal designs for logistic models with three and four parameters. J Stat Plan Inference 138(7):1950–1959
Lindley DV (1956) On a measure of the information provided by an experiment. Ann Math Statist 27(4):986–1005
Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applications of second-order cone programming. Linear Algebr Appl 284(1):193–228
Overstall AM, Woods DC (2017) Bayesian design of experiments using approximate coordinate exchange. Technometrics 59(4):458–470
Overstall AM, Woods DC, Adamou M (2017) acebayes: An R package for bayesian optimal design of experiments via approximate coordinate exchange. CoRR arXiv:1705.08096 (1705.08096v2)
Reznik YA (2008) Continued fractions, diophantine approximations, and design of color transforms. Proc SPIE Appl Digit Image Process 7073:707309. https://doi.org/10.1117/12.797245
Sagnol G (2011) Computing optimal designs of multiresponse experiments reduces to second-order cone programming. J Stat Plan Inference 141(5):1684–1708
Sagnol G, Harman R (2015) Computing exact \(D-\)optimal designs by mixed integer second order cone programming. Ann Stat 43(5):2198–2224
Ueberhuber C (1997) Numerical computation 1: methods, software, and analysis. numerical computation 1, vol XVI. Springer, Berlin
Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev 8:49–95
Wang Y, Myers RH, Smith EP, Ye K (2006) \(D\)-optimal designs for poisson regression models. J Stat Plan Inference 136(8):2831–2845
Waterhouse T, Woods D, Eccleston J, Lewis S (2008) Design selection criteria for discrimination/estimation for nested models and a binomial response. J Stat Plan Inference 138(1):132–144
Wong WK (1992) A unified approach to the construction of minimax designs. Biometrika 79:611–620
Woods DC, Lewis SM, Eccleston JA, Russell KG (2006) Designs for generalized linear models with several variables and model uncertainty. Technometrics 48(2):284–292
Woods DC, van de Ven PM (2011) Blocked designs for experiments with correlated non-normal response. Technometrics 53(2):173–182
Yang J, Mandal A, Majumdar D (2012) Optimal design for two-level factorial experiments with binary response. Statistica Sinica 22(2):885–907
Yang J, Mandal A, Majumdar D (2016) Optimal designs for 2\(^k\) factorial experiments with binary response. Stat Sinica 26:381–411
Yang M, Stufken J (2009) Support points of locally optimal designs for nonlinear models with two parameters. Ann Stat 37(1):518–541
Zhang Y, Ye K (2014) Bayesian \(D-\)optimal designs for Poisson regression models. Commun Stat 43(6):1234–1247
Acknowledgements
The authors thank Radoslav Harman of Comenius University in Bratislava for valuable comments and advice on an earlier draft of the manuscript. We also thank two anonymous reviewers whose comments allowed to undoubtedly improving the quality of the paper.
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Duarte, B.P.M., Sagnol, G. Approximate and exact optimal designs for \(2^k\) factorial experiments for generalized linear models via second order cone programming. Stat Papers 61, 2737–2767 (2020). https://doi.org/10.1007/s00362-018-01075-7
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DOI: https://doi.org/10.1007/s00362-018-01075-7
Keywords
- D-optimal designs
- \(2^k\) Factorial experiments
- Exact designs
- Second order cone programming
- Generalized linear models
- Quasi-Monte Carlo sampling