Split-plot randomization versus composed randomizations
A split-plot experiment is often thought to involve two randomizations. For example, one randomization might randomize one or more treatment factors to main plots while the remaining treatment factors are randomized, in a second randomization, to the subplots. Indeed the split-plot experiment involves two independent randomizations. However, as noted by Brien and Bailey (2006, section 8.5), these two randomizations can be achieved with a single randomization in which there is a single set of objects, say units, to which another set of objects, say treatments, is randomized. Consequently, the randomizations in a split-plot experiment are not innately a multiple randomizations and, where they can be so reduced, we regard it as a single randomization.
Composed randomizations occur in two-phase experiments. Consider a two-phase experiment in which treatments are randomized to plots in the first phase and the plots are randomized to positions in the laboratory phase, in each case using a possibly-different experimental design. Because the two randomizations involve different sets of objects, plots and positions, it is impossible to reduce the two randomizations to a single permutation of one of the sets of objects. That is, composed randomizations are innately multiple randomizations.
The fundamental difference between the randomization for a split-plot experiment and an experiment involving composed randomizations, such as a two-phase experiments, is underlined by the difference between their randomization diagrams. In particular, compare figures 4 and 6 from Brien and Bailey (2006).
