D-optimal designs for the two-variable binary logistic regression model with interaction

The analytical construction of optimal designs for generalized linear models with interaction between the explanatory variables is challenging and the optimal designs reported in the statistical literature for such models are invariably constructed numerically. In the present paper, the two-variable binary logistic model with interaction is introduced and the problem of constructing approximate globally D-optimal designs for the model within the context of drug combination studies is considered. The requisite designs are found by a blend of analytical and numeric techniques and are shown to depend sensitively on the value of the intercept parameter. More specifically, for settings in which the intercept parameter is greater than or equal to a specified bound, the globally D-optimal designs are shown to be based on four equally-weighted points. In contrast, for those settings in which the intercept parameter is less than the specified bound, the globally D-optimal designs are shown to comprise either four, five or six points, with only the 5-point designs obtained explicitly. The results of this study are illustrated throughout by means of examples involving a range of parameter settings. In addition, a real world example is introduced and the practical advantages which accrue from implementing exact designs based on approximate D-optimal and near-D-optimal designs are highlighted.