Circular neighbor-balanced designs using cyclic shifts

Abstract

In agriculture experiments, the response on a given plot may be affected by the treatments on neighboring plots as well as by the treatments applied to that plot. In this paper we consider such type of situations and construct circular neighbor-balanced designs (CNBDs) by the method of cyclic shifts or sets of shifts. An important feature of this method is that the properties of a design can be easily obtained from the sets of shifts instead of constructing the actual blocks of the design. That is, the off-diagonal elements of the concurrence matrix can be easily obtained from the sets of shifts. Since the suggested designs are circular, balanced and binary, so they are universally optimal.

Efficient Neighbor Designs Weakly Balanced in Circular Blocks of Three Different Sizes

“…Example 1. [3,7,10,11,16] + [4,6,12] + [9,10] produce following MCWBND-II for v = 24, k 1 = 6, k 2 = 4, k 3 = 3.…”

“…See Table 6. [7,8,9,11,12], [16,18,25,26], [19,20,21,23], 0.821 [13,14,22] 50 5 4 3 i (mod 4) ≡ 3 [3,4,16,26], [20,21,22,23], [7,9,13,15], [8,18,19], [11,12,17], 0.781 [23,25] • k 1 = 4l + 2, i even.…”

The Construction of Some New Quasi Rees Neighbor Designs Using Cyclic Shifts

“…Here, (0, 1), (1, 2), (v-2, v-1) and (0, v-1) pairs appear two times as neighbors while all other pairs appear only once as neighbors. (0,1),(0,10),(1,2),(2,3), (3,4), (4,5), (5,6),(6,7), (7,8), (8,9), (9,10). 17 3 [1,2]+ [4,5]+ [7,11] (0,1),(0,16),(1,2),(2,3), (3,4), (4,5), (5,6),(6,7), (7,8), (8,9), (9,10), (10 ,11), (11,12), (12,13), (13,14), (14,15), (15,16).…”

“…(0,1),(0,10),(1,2),(2,3), (3,4), (4,5), (5,6),(6,7), (7,8), (8,9), (9,10). 17 3 [1,2]+ [4,5]+ [7,11] (0,1),(0,16),(1,2),(2,3), (3,4), (4,5), (5,6),(6,7), (7,8), (8,9), (9,10), (10 ,11), (11,12), (12,13), (13,14), (14,15), (15,16).…”

An Easy Construction of Generalized Neighbor Designs in Minimal Circular Blocks

“…Cheng (1983) constructed neighbor designs for block size (k) equal to v. NBDs reduce bias due to neighbor effects, see Azais (1987), Langton (1987), Azais et al (1993) and Kunert (2000). Iqbal et al (2009) constructed some series of NBDs through method of cyclic shifts. Akhtar et al (2010) constructed minimal circular blocks neighbor designs for some limited cases.…”