Analysis of Partially Balanced Incomplete Block Designs Illustrated on the Simple Square and Rectangular Lattices

Abstract

E statistical analysis appropriate for partially balanced incomplete block (p. b. i. b.) designs was discussed by Bose and Nair (1939) taking into account only intra-block information. Methods for using inter-block, in addition to intra-block, information have been given by Nair (1944) and Rao (1947). Although the methods of statistical analysis of p.b.i.b. designs discussed by these authors apply to the general case of m associate classes, detailed formulae for the variances of estimates of treatment differences and for the efficiency factor of the designs were worked out only for the cases m = 2 and 3, as it was thought at the time that for practically useful designs m should not exceed 3. Accordingly, the many illustrative examples of p.b.i.b. designs given by them include only cases for which m is either 2 or 3. It was shown by Bose and Nair (1939) that the simple and triple square lattices were special cases of p.b.i.b. designs for which m = 2. The combinatorial conditions satisfied by p.b.i.b. designs as originally developed by Bose and Nair (1939) required that the m associate classes with respect to each treatment could be distinguished by the X-criterion as Xl, .. * Xm were assumed to be unequal. Nair and Rao (1942) found, however, that this assumption of unequal X's was not necessary. By this amendment of the definition of p.b.i.b. designs they were able to show that the cubic lattice was a special case of p.b.i.b. design for which m = 3.