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

许多农业实验都有不止一个阶段。例如，植物育种实验涉及田间阶段，其中种植不同的基因型，以及实验室阶段，其中分析地块样品的各种质量性状，如小麦的烘烤质量。在田间试验中进行区组化已经是一个标准的程序了大约世纪，但是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，在第一阶段中使用的块结构可以完全转移到第二阶段。然而，通常情况下，区组大小不同的阶段，在这种情况下，设计问题变得non-trivial。在当前的项目中，我们正在考虑设计两阶段的实验时，区组被建模为固定效应。当大多数处理信息是基于块内信息时，这种方法是合适的，但是在两阶段实验中，至少一个块结构可能是这样的，即存在大量的块间信息，需要用随机块进行分析。因此，在设计实验时考虑到同样的假设，这是有利的。同样，我们目前的项目假设了固定的治疗效果。然而，在植物育种实验中，通常优选将处理（即基因型）建模为随机的，特别是当谱系或标记信息可用时，使得可以使用最佳线性无偏预测来利用相关性，如在基因组预测中通常所做的那样。这个项目将扩展我们以前的两阶段实验的最优设计的工作的情况下，区组或治疗或两者都被认为是随机的。这种方法的主要挑战是方差分量通常是未知的，最优设计取决于方差分量值。因此，本项目将探讨使用不同的备选办法来说明先前关于差异组成部分可能值的假设。特别是，我们将定义贝叶斯准则的A-最优。另一个挑战是在数值设计搜索期间有效计算治疗信息矩阵的更新。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.