ABSTRACT
In Chapter 5, the analyses of completely randomized designs, randomized complete block designs, and row-column designs were discussed. These designs have experimental units that are unstructured or have one of the two simplest structures obtained by crossing two factors or nesting one in the other. Nesting and crossing can be combined iteratively to form more complicated block structures. Such block structures and the more general orthogonal block structures, including their properties, Hasse diagrams, and appropriate statistical models, are discussed in this chapter. For the designs presented in Chapter 5, the eigenspaces of the patterned covariance matrix of the randomization model depend only on the block structure. There is one eigenspace, called a stratum, for each of the factors that define the block structure. These results are extended to more general block structures. Algorithms for determining the strata, including their degrees of freedom and orthogonal projections, are derived. The strata play an important role in the design and analysis of randomized experiments. We follow the mathematical framework presented in Bailey (1996, 2004, 2008) and include some results from these sources. Readers are advised to review the material in Chapters 3, 4, and 5.
