Definitions and Properties for M-Concurrence Designs

Abstract

Summary Incomplete block designs can be regarded as being in some sense approximately partially balanced. In this paper, we show that parameters pjki corresponding to those used for partially balanced designs can be defined for certain designs with m distinct concurrences. These parameters have properties analogous to those for m associate‐class partially balanced incomplete block (PBIB(m)) designs and have proved useful in searching for optimal designs. This leads to a new upper bound for the efficiency factor which, for given pjki values, is attained if a PBIB(2) design exists. Some results on complement and dual designs are also given.

Partially Balanced Designs

“…The regular graph for the design in Table 12. some of the specific results in his paper are incorrect. Jarrett [24] used the term sconcurrence design for Pearce's PBIBD and clarified the situation somewhat. However, he insisted that θ i = θ j unless i = j.…”

Average Efficiency Factors and Adjusted Orthogonality in Multidimensional Designs

“…where k is the block size, b is the number of blocks, k' = k mod v and r' = r mod b. A tighter bound for Ehas been developed by Jarrett (1983) for two-concurrence designs, i.e. designs in which pairs of treatment labels occur together in either Al or A z blocks, where~I and~2 are non-negative integers.…”

The Efficiency Factors of Particular Effects in Designs with Orthogonal Factorial Structure

“…The treatment concurrence graph has been helpful in constructing good incompleteblock designs for unstructured sets of treatments (Paterson and Patterson, 1984;Jarrett, 1983). The extra complication here is the multiplier CX.…”