Kabera, Muregancuro Gaëtan.

### Abstract:

This thesis is fo cused on the construction of optimal designs for detecting drug interaction using the two-variable binary logistic mo del. Two sp eci c mo dels are considered: (1) the binary two-variable logistic mo del without interaction, and (2) the binary two-variable logistic mo del with interaction. The two explanatory variables are assumed to b e doses of two drugs that may or may not interact when jointly administered to sub jects. The main ob jective of the thesis is to algebraically construct the optimal designs. However, numerical computations are used for constructing optimal designs in cumb ersome cases. The problem of constructing optimal designs is to allo cate weights to sp eci c p oints of the design space in such a way that information asso ciated with mo del parameters is maximized and the variances of the mean resp onses are minimized. Sp eci cally, the D-optimality criterion discussed in this thesis minimizes the determinant of the asymptotic variance-covariance matrix of the estimates of the mo del parameters. The numb er of supp ort p oints of the D-optimal designs for the two- variable binary logistic mo del without interaction varies from 3 to 6. Supp ort p oints are equally weighted only in case of the 3-p oint designs and in some sp ecial cases of the 4-p oint designs. The numb er of supp ort p oints of the D-optimal designs for the two-variable binary logistic mo del with interaction varies from 4 to 8. Supp ort p oints are equally weighted only in case of the 4-p oint designs and in some sp ecial cases of 8-p oint designs. Numerous examples are given to illustrate theoretical results.