This paper generalizes certain results which are known for balanced incomplete block designs and group divisible designs to partially balanced incomplete block (PBIB) designs with $m$ associate classes. Some of the results are for general $m$ but others are for $m =$ 2, 3, or 4.
W. S. Connor. W. H. Clatworthy. "Some Theorems for Partially Balanced Designs." Ann. Math. Statist. 25 (1) 100 - 112, March, 1954. https://doi.org/10.1214/aoms/1177728849