Model-based optimal designs of experiments (M-bODE) for nonlinear models are typically hard to compute. The literature on the computation of M-bODE for nonlinear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Second Order Programming (MISOCP) formulations to find, respectively, approximate and exact A - and D -optimal designs for $$2^k$$ 2 k factorial experiments for Generalized Linear Models (GLMs). First, locally optimal (approximate and exact) designs for GLMs are addressed using the formulation of Sagnol (J Stat Plan Inference 141(5):1684–1708, 2011). Next, we consider the scenario where the parameters are uncertain, and new formulations are proposed to find Bayesian optimal designs using the A - and log det D -optimality criteria. A quasi Monte-Carlo sampling procedure based on the Hammersley sequence is used for computing the expectation in the parametric region of interest. We demonstrate the application of the algorithm with the logistic, probit and complementary log–log models and consider full and fractional factorial designs.

中文翻译：

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通过二阶锥规划对广义线性模型进行 $$2^k$$2k 因子实验的近似和精确优化设计

非线性模型的基于模型的优化实验设计 (M-bODE) 通常难以计算。当协变量是分类变量时，关于非线性模型的 M-bODE 计算的文献很少，即因子实验。我们提出了二阶锥规划 (SOCP) 和混合整数二阶规划 (MISOCP) 公式，以分别为 $$2^k$$ 2 k 广义线性模型的因子实验找到近似和精确的 A 和 D 最优设计（ GLM）。首先，使用 Sagnol 的公式（J Stat Plan Inference 141(5):1684–1708, 2011）解决了 GLM 的局部最优（近似和精确）设计。接下来，我们考虑参数不确定的情况，并提出新的公式以使用 A - 和 log det D - 最优性标准来寻找贝叶斯最优设计。基于 Hammersley 序列的准蒙特卡罗采样程序用于计算感兴趣参数区域中的期望。我们展示了该算法在逻辑、概率和互补对数-对数模型中的应用，并考虑了完整和部分因子设计。