For BIB designs $N_i$ and their complements $N_i^\ast (i = 1,2, \cdots, n)$, Kageyama (1972) gave necessary and sufficient conditions for a PBIB design $N = N_1 \otimes N_2 + N_1^\ast \otimes N_2^\ast$ with at most three associate classes having the rectangular association scheme to be reducible to a PBIB design with only two distinct associate classes having the $L_2$ association scheme. In this paper similar results for the PBIB design $N_1 \otimes N_2 \otimes \cdots \otimes N_n + N_1^\ast \otimes N_2^\ast \otimes \cdots \otimes N_n^\ast$, which is in a sense a generalization of the Kronecker products of the above type, are described.