Construction of Optimal and Efficient Designs of Experiments for Individualized Prediction in Hierarchical Models

本课题的研究对象是层次随机系数回归模型以及广义线性和非线性混合模型。这些模型最初是在植物和动物育种的生物科学中引入的，现在在越来越多的统计应用领域中使用。该项目的目的是开发分析方法，以确定这些模型中预测问题的最佳设计。对于实验设计，最有效的分析结果具有等价定理意义上的最优性条件的形式。只有在某些特殊情况下，解才被明确地给出。优化设计的计算方法是本项目的重要组成部分。确定最优设计的分析方法常常成功地基于近似设计的概念。虽然近似设计不能直接实现，但可以使用合适的舍入算法确定最优或至少有效的精确设计，其中近似设计可以作为获得的精确设计效率的基准。在本项目中，将在总体实验条件（平衡纵向、横截面、稀疏、多因素或随机块设计）造成的现实实验限制下详细研究最佳设计的构建和特征。必须注意的是，所得到的最优设计只是局部最优的，因为它们依赖于随机效应的色散矩阵。如果色散矩阵是未知的，这个问题将通过使用所谓的鲁棒设计准则来解决，该准则对色散参数的灵敏度很低。在项目的最后一部分，将线性随机系数回归模型的结果推广到更复杂的（广义线性和非线性混合）模型。

The object of the present project is hierarchical random coefficient regression models as well as generalized linear and nonlinear mixed models. Such models were initially introduced in biosciences for plant and animal breeding and are nowadays utilized in an increasing number of fields in statistical applications. The aim of the project is to develop analytical approaches for the determination of optimal designs for the problem of prediction in these models. The most available analytical results for experimental designs have the form of an optimality condition in the sense of an equivalence theorem. Only for some particular cases the solutions are given explicitly. Methods for the computation of optimal designs are a substantial part of this project.Analytical approaches for the determination of optimal designs are often successfully based on the concept of approximate designs. Although approximate designs are not directly realizable, optimal or at least efficient exact designs can be determined using suitable rounding algorithms, in which approximate designs may then serve as a benchmark for the efficiency of the obtained exact designs. Within this project the construction and characterization of optimal designs will be investigated in detail under realistic experimental restrictions caused by overall experimental conditions (balanced longitudinal, cross sectional, sparse, multi-factor or randomized block designs). It has to be noted that the so obtained optimal designs are only locally optimal in the sense that they depend on the dispersion matrix of the random effects. If the dispersion matrix is unknown, this problem will be attacked by using so-called robust design criteria, which show a low sensitivity with respect to the dispersion parameters. In the final part of the project the results obtained for linear random coefficient regression models will be extended to more complicated (generalized linear and nonlinear mixed) models.