D-optimal designs for two-variable logistic regression model with restricted design space

The problem of constructing locally D-optimal designs for two-variable logistic model with no interaction has been studied in many literature. In Kabera, Haines, and Ndlovu (2015), the model is restricted to have positive slopes and negative intercept for the assumptions that the probability of response increases with doses for both drugs and that the probability of response is less than 0.5 at zero dose level of both drugs. The design space mainly discussed is the set [0, infinity) x [0, infinity), while the finite rectangular design space is presented only in scenarios where the results for the unlimited design space are still appropriate. In this paper, we intend to loose these restrictions and discuss the rectangular design spaces for the model where the D-optimal designs can not be obtained. The result can be extended to the models where drugs have negative or opposite effects, or the models with positive intercept, by using translation and reflection in the first quadrant.