Optimal design and analysis for two-phase experiments with random block and treatment effects

许多农业试验都不止一个阶段。例如，植物育种实验包括田间阶段和实验室阶段，前者种植不同的基因型，后者分析地块样本的各种品质特征，如小麦的烘烤品质。一个世纪以来，在现场实验中进行分组已经成为标准程序，但很少有研究人员在实验室阶段也使用有效的统计设计。在最简单的情况下，第一阶段使用的块结构可以完全转移到第二阶段。然而，通常情况下，不同阶段的块大小不同，在这种情况下，设计问题变得非常重要。在当前的一个项目中，我们正在考虑设计两阶段的实验，将块建模为固定效果。这种方法适用于大多数治疗信息基于块内信息的情况，但在两阶段实验中，至少有一个块结构可能存在大量块间信息，需要使用随机块进行分析。因此，在设计实验时考虑到做出同样假设的分析是有利的。同样，我们当前的项目假定固定的处理效果。然而，在植物育种实验中，模型处理（即基因型）通常是随机的，特别是当系谱或标记信息可用时，可以使用最佳线性无偏预测来利用相关性，这在基因组预测中通常是这样做的。这个项目将我们之前关于两阶段实验的优化设计的工作扩展到块或处理或两者都被认为是随机的情况。这种方法的主要挑战是方差成分通常是未知的，而最优设计取决于方差成分的值。因此，该项目将探索使用不同的选项来解释关于方差成分可能值的先前假设。特别地，我们将定义a -最优性的贝叶斯准则。在数值设计搜索过程中，处理信息矩阵更新的高效计算是另一个挑战。所开发的方法将被实施到我们为固定效应案例开发的算法框架中。开发的方法将在Julia包中实现。

Many agricultural experiments have more than one phase. For example, plant breeding experiments involve a field phase, in which different genotypes are grown, and a lab phase, in which plot samples are analysed for various quality traits such as baking quality in wheat. Blocking in field experiments has been a standard procedure for about a century but few researchers use an efficient statistical design also in the lab phase. In the simplest case the block structure used in the first phase can be fully transferred to the second phase. Often, however, the block sizes differ between phases, in which case the design problem becomes non-trivial.In a current project, we are considering the design of two-phase experiments when the blocks are modelled as fixed effects. This approach is suitable when most of the treatment information is based on intra-block information but in two-phase experiments at least one of the block structures may be such that there is substantial inter-block information, calling for an analysis with random blocks. It is then advantageous to design the experiment with an analysis in mind making that same assumption.Similarly, our current project assumes fixed treatment effects. In plant breeding experiments, however, it is often preferable to model treatments, i.e. genotypes, as random, especially, when pedigree or marker information is available such that relatedness can be exploited using best linear unbiased prediction as is commonly done in genomic prediction. This project will extend our previous work on optimal design of two-phase experiments to the case where blocks or treatments or both are considered as random. The main challenge with such an approach is that variance components are often unknown and optimal design depends on the variance component values. The project will therefore explore the use of different options to account for prior assumptions about the likely value of variance components. In particular, we will define Bayesian criteria for A-optimality. A further challenge is the efficient computation of the update for the treatment information matrix during numerical design search. The developed approaches will be implemented into the algorithmic framework we developed for the fixed-effects case. The developed methods will be implemented in a Julia package.