Optimal Designs for Two-Level Factorial Experemients

We consider the problem of obtaining locally D-optimal designs for factorial experiments with qualitative factors at two levels each with binary response. For the 2^2 factorial experiment with main effects model we obtain optimal designs analytically in special cases and demonstrate how to obtain a solution in the general case using cylindrical algebraic decomposition. The optimal designs are shown to be robust to the choice of the assumed values of the prior and when there is no basis to make an informed choice of the assumed values we recommend the use of the uniform design, i.e., the design that assigns equal number of observations to each of the four points. For the general 2^k case we show that the uniform design has a maximin property.