An experimental design is called a multidimensional (MD) design, if it involves more than one factor; see e.g. Potthoff (1964a, b). A number of two-, three-, and four-dimensional designs are now in common use. For example, the ordinary balanced and partially balanced incomplete block designs are two-dimensional. The Latin squares, Youden squares, and the designs of Shrikhande (1951) are three-dimensional. Finally, the Graeco-Latin square designs are four-dimensional. The construction of multidimensional designs involving three or more factors has been discussed by several authors both when additivity is assumed and when interactions are present. To mention a few, we cite Plackett and Burman (1946), Plackett (1946), Potthoff (1964a, b), Agarwal (1966), Anderson (1968), and Causey (1968). A general class of multidimensional designs, which are partially balanced (PB), has been introduced in Srivastava (1961) and Bose and Srivastava (1964). These designs are called multidimensional partially balanced (MDPB) designs. The (MDPB) designs are useful for economizing on the number of observations to be taken, retaining at the same time a relative ease in analysis. MDPB designs for models containing interaction terms have been considered by Anderson (1968). The purpose of this paper is to obtain a class of necessary combinatorial conditions satisfied by the parameters of MDPB designs, and to provide a relatively easy method of determining whether a given design is "completely connected". This latter concept, also of a combinatorial nature, is a generalization of the concept of "connected" block-treatment designs. It signifies that for every factor included under the design, the "true" difference between any two factor levels possesses a best linear unbiased estimate. In a succeeding communication, Srivastava and Anderson (1968), general methods of construction of MDPB designs are considered.
J. N. Srivastava. Donald A. Anderson. "Some Basic Properties of Multidimensional Partially Balanced Designs." Ann. Math. Statist. 41 (5) 1438 - 1445, October, 1970. https://doi.org/10.1214/aoms/1177696789