Approximate and exact optimal designs for $$2^k$$ 2 k factorial experiments for generalized linear models via second o
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Approximate and exact optimal designs for 2k factorial experiments for generalized linear models via second order cone programming Belmiro P. M. Duarte1,2
· Guillaume Sagnol3
Received: 19 February 2018 / Revised: 5 December 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
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 2k 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. Keywords D-optimal designs · 2k Factorial experiments · Exact designs · Second order cone programming · Generalized linear models · Quasi-Monte Carlo sampling Mathematics Subject Classification 62K05 · 90C47
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Belmiro P. M. Duarte [email protected] Guillaume Sagnol [email protected]
1
Department of Chemical and Biological Engineering, Instituto Politécnico de Coimbra, Instituto Superior de Engenharia de Coimbra, Rua Pedro Nunes, Quinta da Nora, 3030-199 Coimbra, Portugal
2
CIEPQPF, Department of Chemical Engineering, University of Coimbra, Coimbra, Portugal
3
Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
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B. P. M. Duarte, G. Sagnol
1 Motivation We consider the problem of determining model-based A- and D-optimal designs of 2k factorial experiments for generalized linear models where k is the number of factors considered in the study. Our setup is that we have a given Generalized Linear Model defined on a design space formed by combinations of {−1, +1} of n p covariates, and a given total number of observations, N , available for the study. The design problem is to find the number of replicates (if any) at each of these design points, n ∈ N0 , subject to the requirement that they sum to N , and N0 is the set of non-negative integers. A standard approach to deal with the problem is to compute approximate designs, which can be seen as a continuous relaxation of the M-bODE problem, and can be interpreted as the optimal proportions wi of trials to perform on the design points x i ∈ {−1, +1}k . The practical implementation of the approximate design r
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