Abstract This paper treats a class of combinatorial designs which are essentially partially balanced incomplete block designs with two associate classes and with the additional feature that there are constants s and t so that for any treatment-block pair (x, B), the number of first associates of x in the block B is s or t depending on whether x ϵ B or x ∉ B. We obtain necessary and sufficient conditions that a P.B.I.B.D. should have this feature and obtain generalizations of earlier results on triangular and L2 (n) P.B.I.B.D.'s. Finally, we consider two graphs naturally associated with quasi symmetric P.B.I.B.D.'s having this special feature determine their spectra, and investigate the conditions under which they are strongly regular.