Construction of Congruent Classes of Pairwise Balanced Design Using Lotto Design - Osuolale Peter Popoola (ZBW – Leibniz Information Centre for Economics, 2019)
Balanced Incomplete Block Designs (BIBDs) are commonly used when the number of experimental treatments is less than the block size. However, there are cases when the block sizes available for an experiment are not the same. Hence the use of a Pairwise Balanced Design (PBD). A PBD (n, K, λ) is a block design where n is the number of treatments, K= k1, k2…, kb is the set of sizes of a block, and λ is the number of times a pair of treatments appears together within blocks. Also, little is known about the construction of PBDs using Lotto Designs (LDs). Methods of constructing PBDs in literature are complex. The aim of this study, therefore, was to provide a simple method for constructing PBDs when K= 3, 4 and 3, 4, 5. The specific objectives were to: (i) investigate various methods of constructing PBDs (ii) establish conditions for the identification of LDs that qualify as PBDs, and (iii) provide a simple method for constructing two classes of PBDs from LDs; The study utilized Li's inequality ⌊pr/(t-1)⌋((t-1)2)+((pr-⌊pr/(t-1)⌋)( 2) (t-1))<(p2)λ to obtain LDs that are PBDs where p is the set of treatments that can intersect relevant k-blocks of an LD, t is the number of treatments in p that match k-blocks, r is the number of time a particular treatment appears in each block and λ is the number of time a pair of treatment appear within the block. Some conditions were imposed on the generated LDs to obtain those that were qualified as PBDs. Hence, two classes of PBDs were constructed by partitioning n -treatments into a set of blocks when K = 3, 4, 5 such that each pair of treatments is contained in precisely one block. A program to do this was written using C++. The following results were obtained: Various methods investigated shows that certain PBDs is difficult to construct especially when the number of treatment is large also, there is no standard technique to generate block sizes of the PBDs. Any LD(n, k, p, t) satisfying the Li inequality, k = 3, 4, 5, and n = p qualified as PBDs. (a) 2-LDs(n, 3, p, 3), (n, 4, p, 4) can be used to construct PBD(n, 3, 4) provided n ≡ 0, 1(mod 3) (b) 3-LDs(n, 3, p, 3), (n, 4, p, 4), (n, 5, p, 5) can be used to construct PBD(n, 3, 4, 5) provided n ≡ 2, 3 (mod4) by partitioning n-treatments into a block of sets K = 3, 4, 5 such that each pair of treatments is contained in precisely one block. This study concluded that congruence classes of PBDs(n, 3, 4, 1) and (n, 3, 4, 5, 1) could be constructed from the appropriate LDs if the specified conditions hold. The study recommends the use of PBDs for constructing other important designs, and the use of LDs to construct PBDs.
Keywords: 330; Balanced Incomplete Block Designs; Exeperiental Designs; Block Designs; Pairwise Balanced Deisgns