The Use of Treatment Concurrences to Assess Robustness of Binary Block Designs Against the Loss of Whole Blocks

Abstract

SummaryCriteria are proposed for assessing the robustness of a binary block design against the loss of whole blocks, based on summing elements of selected upper non-principal sections of the concurrence matrix, which improve on the minimal concurrence concept that has been used previously and provide new conditions for measuring the robustness status of a design. The robustness properties of two-associate partially balanced designs are considered and it is shown that two categories of group divisible designs are maximally robust. These results expand a classic result in the literature, obtained by Ghosh, which established maximal robustness for the class of balanced block designs.

A note on the robustness of PBIBD(2)s against breakdown in the event of observation loss

“…Every design in a with υ <2 k has t * = r . See Theorem 1 of Godolphin & Godolphin () for details. Therefore, it is assumed throughout that υ ≥2 k .…”

“…By an argument based on the number of pairs of first associates in treatment subsets, Godolphin & Godolphin () established that t * = r for all singular designs, for all semi‐regular designs and for regular designs with λ 1 < λ 2 . The regular designs with λ 1 > λ 2 do not all have optimal block breakdown number.…”

“…Bose & Connor (1952) categorised the group divisible designs into three exhaustive and mutually exclusive types: singular if r − 1 = 0; semi-regular if r − 1 > 0 and rk − 2 = 0; regular if r − 1 > 0 and rk − 2 > 0. By an argument based on the number of pairs of first associates in treatment subsets, Godolphin & Godolphin (2015) established that t Å = r for all singular designs, for all semiregular designs and for regular designs with 1 < 2 . The regular designs with 1 > 2 do not all have optimal block breakdown number.…”

“…The regular designs with 1 > 2 do not all have optimal block breakdown number. Clatworthy (1973) listed twelve regular designs with 1 > 2 which have t Å < r. Godolphin & Godolphin (2015) gave a lower bound for t Å for regular designs with 1 > 2 . This is useful for assessing robustness on a design by design basis, but the design specific nature of the bound has not enabled its use in the development of general robustness conditions.…”

“…Note that Corollary 2 does not apply to designs with = 2 and m = 2. In examinaton of the regular designs in Clatworthy (1973), Godolphin & Godolphin (2015) identified one design with = 2 and m = 2 which does not have t Å = r, namely R2.…”