Partially Neighbor Balanced Designs for Circular Blocks

Abstract

A partially neighbor balanced design is a design in which for any fixed treatment, other treatments occur as neighbor λ i times. This paper generates infinite series of one-dimensional partially neighbor balanced designs for v = n treatments. The blocks used in these designs are considered circular. Designs given here are partially balanced in terms of nearest neighbors and not necessarily in terms of variance. Binary and non-binary concepts have been used for the construction of designs. Theorem 1 generates binary generalized 2-neighbor designs and theorem 2 generates non-binary generalized 3-neighbor designs. These theorems generate designs for v = n treatments i.e., for odd and even number of treatments simultaneously. This concept remains relatively under-explored in the literature. The objective is to decrease error variance due to neighbor effect and reduce computational cost.

Some New Generators to Produce Minimal Circular Strongly Balanced Generalized Neighbor Designs of Class-I

“…S * = [1, 6, 7, 0, 3,4,5,12,13,8,7,11,10,9,2], Here each of 0, 1, 2, • • • , 13 appears once except 7 which appear twice. Hence S 1 = [1,6,7] and S 2 = [3,4,5] produce MCSBGND-I for v = 14, k = 4.…”

“…2 appears twice with v even then these sets will produce MCSBGND-I, where [1,6,7] and S 2 = [3,4,5] produce MCSBGND-I for v = 14 and k = 4.…”

“…Proof. S * = [3,4,6,8,21,2,9,11,0,7,10,17,19,18,16,14,1,20,13,11,15,12,5], Here each of 0, 1, 2, • • • , 21 appears once except 11 which appear twice. Hence S 1 = [3,4,6,8], S 2 = [2,9,11], S 3 = [7,10] produce MCSBGNDs-I for v = 22, k 1 = 5, k 2 = 4 and k 3 = 3.…”

“…MCSBGNDs-I for v = 2ik 1 + 4k 2 + 2k 3 -2 with m (mod 4) ≡ 2. See Table 11 6.4 MCSBGNDs-I can be constructed in Three Different Block Sizes for v = 2ik 1 + 4k 2 + 2k 3 -2 when m (mod 4) ≡ 3 with [6,13,14,22,27,46,47,48,52,53]+ [17,19,23,24,25,30,37,40] + [4,5,21,30,37,39,40,43]+ [20,21,23,28] (v)…”

“…Hamad (2014) [6], Hamad and Hanif (2016) [7] constructed some partially balanced NDs. Nadeem et al (2021) [19] constructed economical classes of MCGNDs.…”