Crossing, nesting, marginality and Hasse diagrams in designed experimentsRosemary Bailey (University of St Andrews)
John Nelder introduced simple orthogonal block structures (SOBS) in one of his famous 1965 papers. They provide a compact description of many of the structures in common use in experiments, so much so that some people find it hard to understand a structure that cannot be expressed in this way. Graham Wilkinson used these ideas to develop software for ANOVA before it was easy to invert matrices, and his algorithms underlie much software used today. Building on work by Kempthorne, Zyskind and their co-authors, Terry Speed and I later generalized SOBS to poset block structures.
But there are still misunderstandings. If there are 5 blocks of 4 plots each, should the plot factor have 4 levels or 20? What is the difference between nesting and marginality? What is the difference between a factor, the effect of that factor (this effect may be called an interaction in some cases), and the smallest model which includes that factor whilst respecting marginality? My take on this is that there are three different partial orders involved, and each can be represented by a Hasse diagram. I will try to explain the differences.