In this paper, new association schemes with  associated classes are described and called rectangular right angular association schemes  and their parameter expressions are given. Moreover, the partially balanced incomplete block (PBIB) designs based on those association schemes are constructed and their parameters are specified, using an original method of construction of rectangular partially balanced incomplete block (PBIB) designs of size    from rectangular association schemes of  treatments, and the special case  provides singular group divisible designs.

rectangular association scheme, right angular association scheme, partially balanced incomplete block design, group divisible design, nested group divisible design.