zbMATH — the first resource for mathematics

On designs with a non-orthogonal row-column-structure. (English) Zbl 0878.62055
Müller, W. G. (ed.) et al., Model-oriented data analysis. Proceedings of the 3rd international workshop, MODA 3, held in Petrodvorets, Russia, May 25-30, 1992. Heidelberg: Physica-Verlag. Contributions to Statistics. 105-112 (1993).
Summary: When searching for optimal row-column designs, it is a good strategy to try to find a design which is a balanced incomplete block design when columns are considered as blocks, and for which the treatments are orthogonal to rows. In the usual setting where each row intersects with each column exactly once, the treatments are orthogonal to rows if and only if they are proportional, i.e. each treatment appears in each row equally often. In that case the orthogonality condition does not depend on how the treatments are distributed over the columns, see also V. Kurotschka [Commun. Stat., Theory Methods A 7, 1363-1378 (1978; Zbl 0392.62057)].
Recently, there has been considerable interest in a non-orthogonal row column setting, where not each row intersects with each column. It was observed by F. P. Steward and R. A. Bradley [Biometrika 78, 337-348 (1991)] and others that then the orthogonality condition can be fulfilled by designs which are not proportional. We treat this orthogonality condition in detail and show that it can be fulfilled by designs which are clearly non-optimal in the model without column effects. We also show that the usual two way block model for non-orthogonal row and column structures in general cannot be justified with the help of randomization arguments.
For the entire collection see [Zbl 0840.00042].

62K05 Optimal statistical designs
62K10 Statistical block designs