We develop two analytic approaches to solve D-optimal approximate designsunder generalized linear models. The first approach provides analytic D-optimalallocations for generalized linear models with two factors, which include as aspecial case the $2^2$ main-effects model considered by Yang, Mandal andMajumdar (2012). The second approach leads to explicit solutions for a class ofgeneralized linear models with more than two factors. With the aid of theanalytic solutions, we provide a necessary and sufficient condition under whicha D-optimal design with two quantitative factors could be constructed on theboundary points only. It bridges the gap between D-optimal factorial designsand D-optimal designs with continuous factors.
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