Integral approximations for computing optimum designs in random effects logistic regression models

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.