The multivariate generalized linear model is considered. Each univariate response follows a generalized linear model. In this situation, the linear predictors and the link functions are not necessarily the same. The quasi-Fisher information matrix is obtained by using the method of generalized estimating equations. Then locally optimal designs for multivariate generalized linear models are investigated under the D- and A-optimality criteria. It turns out that under certain assumptions the optimality problem can be reduced to the marginal models. More precisely, a locally optimal saturated design for the univariate generalized linear models remains optimal for the multivariate structure in the set of all saturated designs. Moreover, the general equivalence theorem provides a necessary and sufficient condition under which the saturated design is locally D-optimal in the set of all designs. The results are applied for multivariate models with gamma-distributed responses. Furthermore, we consider a multivariate model with univariate gamma models having seemingly unrelated linear predictors. Under this constraint, locally D- and A-optimal designs are found as product of all D- and A-optimal designs, respectively for the marginal counterparts.

中文翻译：

---

多元广义线性模型的局部最优设计

考虑多元广义线性模型。每个单变量响应都遵循广义线性模型。在这种情况下，线性预测变量和链接函数不一定相同。采用广义估计方程的方法得到拟Fisher信息矩阵。然后在 D 和 A 最优性标准下研究多元广义线性模型的局部最优设计。事实证明，在某些假设下，最优性问题可以简化为边际模型。更准确地说，单变量广义线性模型的局部最优饱和设计对于所有饱和设计的集合中的多元结构仍然是最优的。而且，一般等价定理提供了一个充分必要条件，在该条件下，饱和设计在所有设计的集合中是局部 D 最优的。结果适用于具有伽马分布响应的多变量模型。此外，我们考虑了一个多变量模型，其中单变量 gamma 模型具有看似无关的线性预测变量。在此约束下，局部 D 和 A 最优设计被发现为所有 D 和 A 最优设计的乘积，分别用于边缘对应物。