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

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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.