Computational Statistics and Data Analysis 71 (2014) 1208–1220 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Integral approximations for computing optimum designs in random effects logistic regression models C. Tommasi a , J.M. Rodríguez-Díaz b , M.T. Santos-Martín b,∗ a Department of Economics, Business and Statistics, University of Milan, Italy b Department of Statistics, University of Salamanca, Spain article info Article history: Received 10 March 2011 Received in revised form 23 May 2012 Accepted 28 May 2012 Available online 2 June 2012 Keywords: Binary regression model Fisher information matrix Information matrix Optimal design of experiments Influence function abstract In the context of nonlinear models, the analytical expression of the Fisher information matrix is essential to compute optimum designs. The Fisher information matrix of the random effects logistic regression model is proved to be equivalent to the information matrix of the linearized model, which depends on some integrals. Some algebraic approximations for these integrals are proposed, which are consistent with numerical integral approximations but much faster to be evaluated. Therefore, these algebraic integral approximations are very useful from a computational point of view. Locally D-, A-, c - optimum designs and the optimum design to estimate a percentile are computed for the univariate logistic regression model with Gaussian random effects. Since locally optimum designs depend on a chosen nominal value for the parameter vector, a Bayesian D-optimum design is also computed. In order to find Bayesian optimum designs it is essential to apply the proposed integral approximations, because the use of numerical approximations makes the computation of these optimum designs very slow. © 2012 Elsevier B.V. All rights reserved. 1. Introduction The interest in finding optimum designs in the context of regression models with random effects is steadily increasing. See for instance, Mentré et al. (1997), Patan and Bogacka (2007), Schmelter et al. (2007), Graßhoff et al. (2009), Holland-Letz et al. (2011) and Debusho and Haines (2011). Another setting where optimal designs have been extensively studied is the context of (fixed effects) binary regression models. See Abdelbasit and Plackett (1983), Minkin (1987), Ford et al. (1992), Sitter and Wu (1993), Biedermann et al. (2006) and Sitter and Fainaru (1997), among many others. Recently, Ouwens et al. (2006) have studied optimum designs for logistic models with random intercept. In this paper, different optimum designs are derived for the logistic regression model where not only the intercept but all the coefficients are random. When the interest is to estimate as precisely as possible the parameters of the model (or some function of them), optimum designs can be computed minimizing some convex criterion function of the Fisher information matrix. Therefore, in order to find such designs, the explicit representation of this matrix is a very valuable tool. The expression of the Fisher information matrix for the random effects logistic regression model is given in Section 2, where the equivalence with the information matrix of the linearized model is also proved. The main contribution, however, is given in Section 3, where some algebraic integral approximations are provided. These approximations are very useful from a computational point of view, because ∗ Correspondence to: Faculty of Sciences, Plaza de los Caídos, 37008 Salamanca, Spain. Tel.: +34 923294458; fax: +34 923294514. E-mail addresses: chiara.tommasi@unimi.it (C. Tommasi), juanmrod@usal.es (J.M. Rodríguez-Díaz), maysam@usal.es (M.T. Santos-Martín). 0167-9473/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2012.05.024