NPBBBdesigns: Construction and a-Efficiency of Nested Partially Balanced Bipartite Block Designs

Construction and evaluation of nested partially balanced bipartite block (NPBBB) designs for comparing a set of test treatments with a set of control treatments under a nested (blocks within blocks) structure. Six systematic construction methods are provided: composing partially balanced bipartite block designs with nested balanced incomplete block designs; augmenting nested partially balanced incomplete block designs with controls; merging rows of group-divisible nested designs; direct construction from group-divisible schemes; and expansion of partially balanced incomplete block designs (Vinayaka et al. 2026: In press). The A-efficiencies of the block and sub-block classifications are computed against the A-optimal completely symmetric reference design, following the test-versus-control optimality framework of Gupta and Parsad (1996) and Vinayaka et al. (2024) . These designs are particularly suited to agricultural, animal husbandry, industrial, and clinical trials involving multiple standard checks under nested experimental conditions, such as multi-environment trials where field heterogeneity (blocks) and within-field variation (sub-blocks) must be controlled simultaneously.