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Approximate and exact optimal designs for $$2^k$$ 2 k factorial experiments for generalized linear models via second order cone programming

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  • Belmiro P. M. Duarte

    (Instituto Politécnico de Coimbra, Instituto Superior de Engenharia de Coimbra
    University of Coimbra)

  • Guillaume Sagnol

    (Technische Universität Berlin)

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 $$2^k$$ 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.

Suggested Citation

  • Belmiro P. M. Duarte & Guillaume Sagnol, 2020. "Approximate and exact optimal designs for $$2^k$$ 2 k factorial experiments for generalized linear models via second order cone programming," Statistical Papers, Springer, vol. 61(6), pages 2737-2767, December.
    • Handle: RePEc:spr:stpapr:v:61:y:2020:i:6:d:10.1007_s00362-018-01075-7
    DOI: 10.1007/s00362-018-01075-7
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    References listed on IDEAS

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    1. Dorta-Guerra, Roberto & González-Dávila, Enrique & Ginebra, Josep, 2008. "Two-level experiments for binary response data," Computational Statistics & Data Analysis, Elsevier, vol. 53(1), pages 196-208, September.
    2. Ulrike Graßhoff & Rainer Schwabe, 2008. "Optimal design for the Bradley–Terry paired comparison model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 17(3), pages 275-289, July.
    3. Dror, Hovav A. & Steinberg, David M., 2008. "Sequential Experimental Designs for Generalized Linear Models," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 288-298, March.
    4. Radoslav Harman & Alena Bachratá & Lenka Filová, 2016. "Construction of efficient experimental designs under multiple resource constraints," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 32(1), pages 3-17, January.
    5. Belmiro P. M. Duarte & Weng Kee Wong, 2015. "Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming-Based Approach," International Statistical Review, International Statistical Institute, vol. 83(2), pages 239-262, August.
    6. D. Firth & J. P. Hinde, 1997. "On Bayesian D‐optimum Design Criteria and the Equivalence Theorem in Non‐linear Models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(4), pages 793-797.
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    Cited by:

    1. Belmiro P. M. Duarte, 2023. "Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming," Mathematics, MDPI, vol. 11(4), pages 1-17, February.

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