## D-optimal designs for drug synergy.

##### Abstract

This thesis is focused on the construction of optimal designs for detecting drug interaction using the two-variable binary logistic model. Two specific models are considered: (1) the binary two-variable logistic model without interaction, and (2) the binary two-variable logistic model with interaction. The two explanatory variables are assumed to be doses of two drugs that may or may not interact when jointly administered to subjects. The main objective of the thesis is to algebraically construct the optimal designs. However, numerical computations are used for constructing optimal designs in cumbersome cases. The problem of constructing optimal designs is to allocate weights to specific points of the design space in such a way that information associated with model parameters is maximized and the variances of the mean responses are minimized. Specifically, the D-optimality criterion discussed in this thesis minimizes the determinant of the asymptotic variance-covariance matrix of the estimates of the model parameters. The number of support points of the D-optimal designs for the two- variable binary logistic model without interaction varies from 3 to 6. Support points are equally weighted only in case of the 3-point designs and in some special cases of the 4-point designs. The number of support points of the D-optimal designs for the two-variable binary logistic model with interaction varies from 4 to 8. Support points are equally weighted only in case of the 4-point designs and in some special cases of 8-point designs. Numerous examples are given to illustrate theoretical results.