The gamma model is a generalized linear model for gamma-distributed outcomes. The model is widely applied in psychology, ecology or medicine. Recently, Gaffke et al. (J Stat Plan Inference 203:199–214, 2019) established a complete class and an essentially complete class of designs for gamma models to obtain locally optimal designs in particular when the linear predictor includes an intercept term. In this paper we extend this approach to gamma models having linear predictors without intercept. For a specific scenario sets of locally D- and A-optimal designs are established. It turns out that the optimality problem can be transformed to one under gamma models with intercept leading to a reduction in the dimension of the experimental region. On that basis optimality results can be transferred from one model to the other and vice versa. Additionally by means of the general equivalence theorem optimality can be characterized for multiple regression by a system of polynomial inequalities which can be solved analytically or by computer algebra. Thus necessary and sufficient conditions can be obtained on the parameter values for the local D-optimality of specific designs. The robustness of the derived designs with respect to misspecification of the initial parameter values is examined by means of their local D-efficiencies.

中文翻译：

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具有无截距线性预测器的伽马模型局部优化设计的解析解

伽马模型是伽马分布结果的广义线性模型。该模型广泛应用于心理学、生态学或医学。最近，Gaffke 等人。(J Stat Plan Inference 203:199–214, 2019) 为伽玛模型建立了一个完整的类和一个基本完整的设计类，以获得局部最优设计，特别是当线性预测变量包括截距项时。在本文中，我们将这种方法扩展到具有无截距的线性预测器的伽马模型。对于特定场景，建立了局部 D 和 A 最优设计集。事实证明，最优性问题可以在伽马模型下转换为一个具有截距的伽马模型，从而导致实验区域的维数减少。在此基础上，优化结果可以从一个模型转移到另一个模型，反之亦然。此外，通过一般等价定理，最优性可以通过多项式不等式系统来表征为多元回归，该系统可以通过分析或计算机代数解决。因此，可以在特定设计的局部 D 最优性的参数值上获得充分必要条件。派生设计在初始参数值指定错误方面的稳健性通过其局部 D 效率进行检查。