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 DD-, AA-, cc-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 DD-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.
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 the Fisher information matrix of the random effects logistic regression model depends on some integrals which need to be evaluated. In addition, the algorithms to compute optimum designs depend on these integrals as well. If numerical integral approximations are used instead of the proposed algebraic approximations then the algorithms become very slow. In Section 4, the univariate logistic regression model with normally distributed random coefficients is deeply studied. In this context it is shown, through an example and some graphics of the integrals, that the two integral approximations (numerical and algebraic) are consistent, but the algebraic one is faster to be computed. In Section 5, DD-, AA- and cc-optimum designs for this model are derived while Section 6 deals with the important problem of precise estimation of a percentile. Finally, Section 7 describes how the efficiency of the locally DD-optimum designs changes if the chosen nominal value for the parameter vector differs from the true value. A prior distribution for the parameter vector is also specified and the corresponding Bayesian DD-optimum design is computed. Differently from the locally optimal designs, the efficiency of the Bayesian DD-optimum design is always “good” (greater than 97%).
In the context of binary regression models with random effects, the information matrix of a design depends on some integrals of specific functions of the random effects. In this work some algebraic integral approximations are proposed, which are proved to be very useful to compute optimal designs for the logistic regression model with Gaussian random effects. If numerical techniques are applied to evaluate the integrals, then the computation of optimum designs becomes very slow. More specifically, the evaluation of vector View the MathML sourceg(x;θ) given in Section 4 is essential to find the information matrix (see Eq. (3)) and also to apply the first order algorithm in order to find optimum designs. Using, for instance, the software Wolfram Mathematica 7.0, four seconds are necessary to compute numerically each component gj(x;θ),j=1,…,4gj(x;θ),j=1,…,4 of the vector View the MathML sourceg(x;θ), while, with the proposed algebraic approximation, the execution time is practically zero seconds. This is for a specific set of prior values of the parameters; when there are many possible combinations of such sets, there may be a great difference in computing time.
Different criteria of optimality (DD-, AA- and cc-criteria) and the problem of a percentile estimation are studied, and examples of computation for different values of the parameter vector are provided. A sensitivity analysis shows the strong dependence of locally optimum designs on the chosen nominal value for the parameter vector. Therefore, a Bayesian DD-optimum design is computed, whose efficiency with respect to the DD-optimum design at the true value of the parameters is always “good”. Computing Bayesian optimum designs requires a much larger computational effort than obtaining locally optimal designs; therefore, the proposed algebraic integral approximations become even more useful in this context.
