Generators to Construct an Efficient Generalized Class of Minimal Circular Neighbor Designs
Sajid Hussain 1 *, Jamshaid ul Hassan 1, Abdul Salam 1, Hurria Ali 1, and Muhammad Rasheed 1
1Department of Statistics, The Islamia University of Bahawalpur, Pakistan
ABSTRACT
A design that is robust to neighbor effects is one that can protect against the neighbor effects. In situations where minimal circular balanced neighbor designs (MCBNDs) cannot be constructed, minimal circular weakly balanced neighbor designs-I (MCWBNDs-I) are preferred, which is an efficient generalized class. MCWBNDs-I are neighbor designs in which v/2 pairs of distinct treatments appear twice as neighbors, while the remaining pairs appear once. New generators are developed in this study to obtain MCWBNDs-I in blocks of three different sizes.
1. INTRODUCTION
In field experiments, competition or interference between adjacent units can increase the variability of the results and may reduce efficiency. In experiments where neighboring unit treatments influence the effect of the current unit treatment, balanced neighbor designs (BNDs) are used to control neighbor effects. BNDs reduce bias caused by neighbor effect [1-3].
- If each treatment appears λ́́ times with all other treatments as neighbor, then it is neighbor balanced. For λ́́ = 1, design would be minimal.
- A neighbor design is known as generalized neighbor design (GND) in which the constancy of λ́́ is relaxed. GN2-design refers to GND with only two possible values of λ́́ .
- Minimal weakly BNDs are GN2-designs in which λ́́ 1 = 1 and λ́́ 2 = 2.
- MCWBNDs come with v/2 unordered pairs appearing twice as neighbors and the remaining pairs appearing once are known as MCWBNDs-I.
Rees [4] applied CBNDs in serology (virus research) and presented minimal CBNDs for some odd value of v (number of treatments). Azais [5] suggested that BNDs or partial BNDs control neighbor effects. Hwang [6], Cheng [7], Iqbal [8], Akhtar [9], Shehzad [10], Ahmed [11], Akhtar [12], Shahid [13], and Shahid [14] constructed BNDs for some cases. Misra [15], Chaure and Misra [16], Mishra [17], and Kedia and Misra [18] constructed GNDs for some specific cases. Ahmed [19], Zafaryab [20], Iqbal [21], Ahmed and Akhtar [22] developed some series to generate circular partial BNDs. Noreen [23] developed some series for MCWBNDs-I and MCWBNDs-II in equal block sizes. In this study, some generators have been developed to generate sets of cyclic shifts for MCWBNDs-I in blocks of three different sizes.
2. METHOD OF CYCLIC SHIFTS
For neighbor designs, Iqbal [24] introduced the method of cyclic shifts. Its Rule I is described here only for MCWBNDs-I.
If Sj = [qj1, qj2, …,qj(k-1)], where j = 1, 2, …, l and 1 ≤ qji ≤ v-1. For v, even if each of 1, 2, …, v-1 appears once in S* but v/2 appears twice, then these sets produce MCBND, where S* contains
- All values in each of sets Sj, along with their sum (mod v), and
- Complements of all values present in (i). Complement of ‘d’ is ‘v-d’ in Rule I.
Example 2.1. S1 = [2,4,5,12], S2 = [9,10,21], S3 = [7,11] produce MCWBND-I for v = 24, k1 = 5, k2 = 4, and k3= 3.
Proof: S* = [2,4,5,12,23,9,10,21,8,7,11,18,22,20,19,12,1,15,14,3,16,17, 13,6]. Here, each of 1, 2, …, 23 appears once except 12 which appears twice. Hence, S1 = [2,4,5,12], S2 = [9,10,21], S3 = [7,11] produce MCWBND-I for v = 24, k1 = 5, k2 = 4, and k3 = 3.
To generate the design, use v blocks for S1. Write 0, 1, …, v-1 in Row 1. Add the 1st value of S1 (mod v) to Row 1 to get Row 2. Similarly, add the 2nd value of S1 (mod v) to Row 2 to get Row 3, and so on, see Table 1.
Table 1. Blocks Obtained from S1 = [2,4,5,12]
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
|
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
|
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
|
23 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
|
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
|
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
0 |
1 |
|
18 |
19 |
20 |
21 |
22 |
23 |
0 |
1 |
2 |
3 |
4 |
5 |
|
23 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
v more blocks are taken for S2, see Table 2.
Table 2. Blocks Obtained from S2 = [9,10,21]
|
25 |
26 |
27 |
28 |
29 |
30 |
31 |
32 |
33 |
34 |
35 |
36 |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
|
19 |
20 |
21 |
22 |
23 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
0 |
1 |
2 |
3 |
|
37 |
38 |
39 |
40 |
41 |
42 |
43 |
44 |
45 |
46 |
47 |
48 |
|
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
|
21 |
22 |
23 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
|
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
v more blocks are taken for S3, see Table 3.
Table 3. Blocks Obtained from S3 = [7,11]
|
49 |
50 |
51 |
52 |
53 |
54 |
55 |
56 |
57 |
58 |
59 |
60 |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
|
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
|
18 |
19 |
20 |
21 |
22 |
23 |
0 |
1 |
2 |
3 |
4 |
5 |
|
61 |
62 |
63 |
64 |
65 |
66 |
67 |
68 |
69 |
70 |
71 |
72 |
|
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
|
19 |
20 |
21 |
22 |
23 |
0 |
1 |
2 |
3 |
4 |
5 |
|
|
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
Here, pairs (0,12), (1,13), (2,14), (3,15), (4,16), (5,17), (6,18), (7,19), (8,20), (9,21), (10,22), and (11,23) appear twice as neighbors.
3. MCWBNDS-I FOR V (EVEN) = 2IK1+2K2+2K3 IN THREE DIFFERENT BLOCK SIZES
For v even, let m = (m-2)/2. The procedure is described to generate sets of shifts from constructor A or B to obtain MCWBNDs-I in three different block sizes.
- Constructor A = [1, 2,…, m, (m+1)] for m (mod 4) ≡ 2.
- Constructor B = [1, 2, …, (m-3)/4, (m+5)/4, (m+9)/4, …, m, (m+1), and 7(m+1)/4] for m (mod 4) ≡ 3.
Divide values of the selected constructor (C) into i groups, each of k1 values, one of k2 values, and one of k3 values, such that the sum of each group is divisible by v. The required i sets of shifts for k1 and one each for k2 and k3 would be obtained by discarding any one value from each group. Here, a (mod b) ≡ c, which means ‘c’ is remainder if ‘a’ is divided by ‘b’.
Example 3.1. Constructor B = [1, 2, 4, 5, …, 12, 21] for v = 24 can be divided into the following three groups.
G-I = (1,2,4,5,12), G-II = (8,9,10,21), G-III = (6,7,11)
Deleting the smallest value of each group, following are the sets of shifts to obtain MCWBND-I for v = 14, k1 = 5, k2 = 4, k3 = 3, Es = 0.7410, and En = 0.7772.
S1 = [2,4,5,12], & S2 = [9,10,21], S3 = [7,11]
4. GENERATORS TO OBTAIN MCWBNDS-I FROM CONSTRUCTOR A (M (MOD 4) ≡ 2)
4.1. MCWBNDs-I for k1 = 4l
MCWBNDs-I can be generated for k1 = 4l and
- k2 = k1-1, k3 = 4, i integer and l > 1
- k2 = k1-2, k3 = k1-3, l > 1 and i integer
- k2 = k1-2, k3 = 5, l > 2 and i integer
Table 4. MCWBNDs-I obtained from Generator 4.1
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
54 |
12 |
11 |
4 |
1 |
A |
[4,5,6,7,8,9,10,11,12,16,17],[14,15,18,19,20,21,22,23,25,26], [2,4,27] |
0.87 |
0.90 |
|
38 |
8 |
6 |
5 |
1 |
A |
[2,3,4,5,6,8,9],[19,11,12,13,14],[15,16,17,18] |
0.84 |
0.86 |
|
54 |
12 |
10 |
5 |
1 |
A |
[2,4,8,12,13,16,17,19,21,23,26],[9,10,11,14,18,20,22,25,27], [5,7,15,24] |
0.85 |
0.90 |
4.2. MCWBNDs-I for k1 = 4l+2
MCWBNDs-I can be generated for k1 = 4l+2 and
- k2 = k1-1,k3 = k1-2 and i odd
- k2 = k1-1, k3 = 4 and i odd
- k2 = k1-2, k3 = k1-3 and i even
- k2 = k1-2, k3 = 3, l > 1 and i even
- k2 = k1-2 and k3 = 5, l > 1 and i odd
Table 5. MCWBNDs-I obtained from Generator 4.2
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
30 |
6 |
5 |
4 |
1 |
A |
[6,10,12,13,15], [3,7,8,11],[5,9,14] |
0.82 |
0.82 |
|
46 |
10 |
9 |
4 |
1 |
A |
[4,5,6,7,8,9,10,17,23],[12,13,14,15,16,18,19,20],[2,21,22] |
0.85 |
0.89 |
|
38 |
6 |
4 |
3 |
1 |
A |
[3,4,5,6,18],[10,11,12,19,15],[8,13,16],[14,17] |
0.84 |
0.81 |
|
62 |
10 |
8 |
3 |
2 |
A |
[4,8,9,10,14,16,17,21,23],[7,12,18,19,26,27,28,31], [3,6,15,20,24,25,30],[22,29] |
0.87 |
0.88 |
|
46 |
10 |
8 |
5 |
1 |
A |
[4,5,6,7,8,9,10,17,23],[14,15,16,18,19,21,22],[2,11,12,20] |
0.85 |
0.89 |
4.3. MCWBNDs-I for k1 odd
MCWBNDs-I can be generated for k1 odd and
- k2 = k1-1, k3 = k1-2, i(mod 4) ≡ 0
- k2 = k1-1, k3 = 4 and i(mod 4) ≡ 3
- k2 = k1-2, k3 = k1-3 and i(mod 4) ≡ 2
- k2 = k1-2, k3 = 3 and i(mod 4) ≡ 1
- k2 = k1-2, k3 = 5 and i(mod 4) ≡ 3
Table 6. MCWBNDs-I obtained from Generator 4.3
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
54 |
5 |
4 |
3 |
4 |
A |
[5,14,15,16],[8,9,13,17],[2,6,22,23],[19,20,24,27],[11,12,21], [25,26] |
0.87 |
0.79 |
|
78 |
9 |
8 |
4 |
3 |
A |
[3,4,5,6,7,8,9,34],[14,15,16,17,18,20,21,22], [23,24,25,26,29,27,30,31],[12,28, 33,36,37,38,39],[10,32,35] |
0.89 |
0.88 |
|
46 |
7 |
5 |
4 |
2 |
A |
[3,4,5,6,7,19],[11,12,13,14,15,17],[18,20,22,23],[8,16,21] |
0.85 |
0.76 |
|
30 |
7 |
5 |
3 |
1 |
A |
[2,3,4,5,7,8],[11,12,13,15],[10,14] |
0.82 |
0.82 |
|
78 |
9 |
7 |
5 |
3 |
A |
[3,4,5,6,7,8,9,34],[12,13,14,15,16,17,19,39], [22,23,24,25,26,28,27,38], [31,32,33,35,36,37],[10,18,20,29] |
0.89 |
0.88 |
4.4. MCWBNDs-I for k1(mod 4) ≡ 1
MCWBNDs-I can be generated for k1(mod 4) ≡ 1 and
- k2 = k1-1, k3 = k1-3 and i(mod 4) ≡ 1
- k2 = k1-1, k3 = 3 and i(mod 4) ≡ 0
- k2 = k1-1, k3 = 5 and i(mod 4) ≡ 2
- k2 = k1-2, k3 = 3 and i(mod 4) ≡ 2
- k2 = k1-2, k3 = 4 and i(mod 4) ≡ 0
Table 7. MCWBNDs-I obtained from Generator 4.4
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
46 |
9 |
8 |
6 |
1 |
A |
[2,3,4,5,6,7,8,10],[14,15,16,17,18,22,23],[11,12,19,20,21] |
0.85 |
0.89 |
|
94 |
9 |
8 |
3 |
4 |
A |
[3,4,5,6,8,9,10,47],[13,14,15,16,17,18,37,46], [24,25,26,27,28,41,44,45], [23,31,32,33,35,36,42,43], [11,19,21,29,30,38,39],[34,40] |
0.90 |
0.88 |
|
86 |
13 |
12 |
5 |
2 |
A |
[3,4,5,6,7,8,9,10,11,43,23,41], [14,15,16,17,19,20,21,22,24,25,26,27], [18,28,29,30,31,32,33,34,35,36,37],[38,39,40,42] |
0.89 |
0.92 |
|
56 |
9 |
7 |
3 |
2 |
A |
[3,4,5,6,8,9,26,49],[14,15,17,18,19,23,24,25], [11,12,16,20,21,22],[27,28]. |
0.87 |
0.87 |
|
94 |
9 |
7 |
4 |
4 |
A |
[3,4,5,6,8,9,10,47],[13,14,15,16,17,18,37,46], [24,25,26,27,28,42,43,44], [29,30,31,32,33,34,35,36], [19,20,21,38,39,40],[7,41,45] |
0.90 |
0.88 |
4.5. Generators to Obtain MCWBNDs-I for k1(mod 4) ≡ 3
MCWBNDs-I can be generated for k1(mod 4) ≡ 1 and
- k2 = k1-1, k3 = k1-3 and i(mod 4) ≡ 3
- k2 = k1-1, k3 = 3 and i(mod 4) ≡ 2
- k2 = k1-2, k3 = 3 and i(mod 4) ≡ 0
- k2 = k1-2, k3 = 4 and i(mod 4) ≡ 2
Table 8. MCWBNDs-I obtained from Generator 4.5Example 4.5.
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
62 |
7 |
6 |
4 |
3 |
A |
[4,5,6,7,18,19],[11,13,14,15,30,31],[12,17,20,21,23,29], [16,22,25,24,28],[8,26,29] |
0.87 |
0.85 |
|
46 |
7 |
6 |
3 |
2 |
A |
[4,5,6,9,10,11],[7,12,14,15,19,23],[13,16,18,17,20],[21,22] |
0.85 |
0.84 |
|
72 |
7 |
5 |
3 |
4 |
A |
[10,22,23,24,26,33],[7,11,28,30,31,35],[13,17,18,25,32,34], [3,12,16,20,29,63],[8,14,19,27],[21,36] |
|
|
|
70 |
11 |
9 |
4 |
2 |
A |
[2,3,4,5,7,8,9,10,11,12],[14,15,16,18,19,20,21,22,23,35], [45,29,30,31,32,33,34,28],[17,24,25] |
0.88 |
0.90 |
5. GENERATORS TO OBTAIN MCWBNDS-I FROM CONSTRUCTOR B (M (MOD 4) ≡ 3)
5.1. MCWBNDs-I for k1 = 4l
MCWBNDs-I can be generated for k1 = 4l and
- k2 = k1-1, k3 = k1-3 and i integer
- k2 = k1-1, k3 = 5, l > 1 and i integer
Table 9. MCWBNDs-I obtained from Generator 5.1
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
40 |
8 |
7 |
5 |
1 |
B |
[3,4,6,7,8,35,12],[10,11,12,13,16,17],[14,18,19,20] |
0.83 |
0.87 |
|
56 |
12 |
11 |
5 |
1 |
B |
[26,5,6,8,9,10,11,49,12,25,4],[2,13,14,15,18,19,20,21,22,23], [17,24,27,28] |
0.87 |
0.91 |
5.2. MCWBNDs-I for k1 = 4l+2
MCWBNDs-I can be generated for k1 = 4l+2 and
- k2 = k1-1, k3 = k1-3 and i even
- k2 = k1-1, k3 = 3, l > 1 and i even
- k2 = k1-1, k3 = 5 and i odd
- k2 = k1-2, k3 = 4 and i even
Table 10. MCWBNDs-I obtained from Generator 5.2
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
40 |
6 |
5 |
3 |
2 |
B |
[4,8,12,35,20],[11,13,14,16,17],[6,7,10,15],[18,19] |
0.84 |
0.82 |
|
64 |
10 |
9 |
3 |
2 |
B |
[3,4,5,6,7,9,10,26,56],[14,15,16,17,18,19,20,28,32], [12,21,23,24, 25,27,30,29],[22,31] |
0.88 |
0.89 |
|
48 |
10 |
9 |
5 |
1 |
B |
[2,3,4,5,8,9,10,42,12],[13,14,15,16,17,18,19,21], [20,22,23,24] |
0.86 |
0.81 |
|
64 |
10 |
8 |
4 |
2 |
B |
[3,4,5,6,7,9,10,26,56],[14,15,16,17,18,19,20,28,32], [21,23,24,25,27,31,29],[11,22,30] |
0.88 |
0.89 |
5.3. MCWBNDs-I for k1 odd
MCWBNDs-I can be generated for k1 odd and
- k2 = k1-1, k3 = k1-3 and i(mod 4) ≡ 2
- k2 = k1-1, k3 = 3 and i(mod 4) ≡ 1
- k2 = k1-1, k3 = 5 and i(mod 4) ≡ 3
- k2 = k1-2, k3 = k1-3 and i(mod 4) ≡ 1
- k2 = k1-2, k3 = 4 and i(mod 4) ≡ 1
Table 11. MCWBNDs-I obtained from Generator 5.3
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
32 |
7 |
6 |
3 |
1 |
B |
[3,5,6,7,13,28],[9,10,11,12,14],[15,16] |
0.83 |
0.83 |
|
80 |
9 |
8 |
5 |
3 |
B |
[3,4,5,6,7,8,9,36],[14,15,16,17,38,18,39,70], [22,23,24,25,26,27,32,40], [35,29,30,31,33,28,34],[11,12,19,37] |
0.89 |
0.88 |
|
32 |
7 |
5 |
4 |
1 |
B |
[3,5,6,7,13,28],[11,12,15,16],[8,9,14] |
0.83 |
0.83 |
|
40 |
9 |
7 |
4 |
1 |
B |
[7,8,9,10,11,18,19,35],[6,12,13,14,15,16],[2,17,20] |
0.84 |
0.87 |
|
48 |
7 |
6 |
4 |
2 |
B |
[4,5,7,17,18,42],[10,11,13,15,14,24],[12,19,20,21,22],[8,16,23] |
0.86 |
0.84 |
5.4. MCWBNDs-I for k1(mod 4) ≡ 1
MCWBNDs-I can be generated for k1(mod 4) ≡ 1 and
- k2 = k1-1, k3 = k1-2 and i(mod 4) ≡ 1
- k2 = k1-1, k3 = 4 and i(mod 4) ≡ 3
- k2 = k1-2, k3 = k1-3 and i(mod 4) ≡ 3
- k2 = k1-2, k3 = 3 and i(mod 4) ≡ 2
- k2 = k1-2, k3 = 5 and i(mod 4) ≡ 0
Table 12. MCWBNDs-I obtained from Generator 5.4
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
24 |
5 |
4 |
3 |
1 |
B |
[2,4,5,12],[9,10,21],[7,11] |
0.80 |
0.78 |
|
98 |
9 |
8 |
4 |
3 |
B |
[3,4,5,7,8,9,10,48],[13,14,15,16,17,18,46,47], [24,25,26,27,34,42,43,44], [29,30,31,32,33,35,37,40], [11,19,20,28,36,38,39],[41,45,84] |
0.87 |
0.88 |
|
80 |
9 |
7 |
6 |
3 |
B |
[3,4,5,6,7,8,9,36],[13,14,16,17,22,38,39,70], [23,24,25,26,27,31,32,34], [19,20,28,29,30,33], [15,21,35,37,40] |
0.89 |
0.88 |
|
56 |
9 |
7 |
3 |
2 |
B |
[2,5,9,10,14,19,24,28],[4,6,8,13,15,18,22,23], [16,17,21,26,27,49],[20,25] |
|
|
|
96 |
9 |
7 |
5 |
4 |
B |
[3,5,7,8,9,10,19,33],[13,14,17,30,37,40,47,84], [15,21,22,24,25,26,27,28], [23,29,31,32,34,35,36,48], [11,18,38,39,42,43],[41,44,45,46] |
0.90 |
0.88 |
5.5. Generators to Obtain MCWBNDs-I for k1(mod 4) ≡ 3
MCWBNDs-I can be generated for k1(mod 4) ≡ 3 and
- k2 = k1-1, k3 = k1-2 and i(mod 4) ≡ 3
- k2 = k1-1, k3 = 4 and i(mod 4) ≡ 2
- k2 = k1-2, k3 = 3 and i(mod 4) ≡ 0
- k2 = k1-2, k3 = 5 and i(mod 4) ≡ 2
Table 13. MCWBNDs-I obtained from Generator 5.5
|
v |
k1 |
k2 |
k3 |
i |
C |
Sets of Shifts |
Es |
En |
|
64 |
7 |
6 |
5 |
3 |
B |
[4,5,6,7,19,20],[10,11,13,15,22,56],[16,17,18,21,23,31], [14,24,25,26,27], [28,29,30,32] |
0.88 |
0.85 |
|
48 |
7 |
6 |
4 |
2 |
B |
[5,7,11,12,15,42],[9,10,13,14,18,24],[16,17,19,20,22], [3,21,23] |
0.86 |
0.85 |
|
72 |
7 |
5 |
3 |
4 |
B |
[4,5,6,7,23,24],[11,12,13,14,21,63],[16,17,18,20,22,36], [2,25,26,27,28,35], [29,31,32,33],[30,34] |
0.88 |
0.85 |
|
72 |
11 |
9 |
5 |
2 |
B |
[2,3,4,5,6,7,8,15,30,63],[13,16,17,18,19,20,21,22,23,36], [14,24,25,26,27, 29,28,31],[32,33,34,35] |
0.88 |
0.90 |
EFFICIENCY OF SEPARABILITY AND OF NEIGHBOR EFFECTS
6.1. Efficiency of Neighbor Effects
The efficiency factor for both direct and neighbor effects is the harmonic mean of eigenvalues (non-zero) of the respective information matrix [25, 26]. For a high value of En, the design would be suitable to estimate neighbor effects.
6.2 Efficiency for Separability
[27] developed the following measure of efficiency for separability (Es).
Es=[1-1/(v√(v-1))]×100%
7. CONCLUSION
New generators have been developed to generate sets of shifts in order to obtain MCWBNDs-I in blocks of three different sizes. MCWBNDs-I obtained through these newly developed generators possess high values of Es and En. Therefore, these designs are efficient to control neighbor effects as well as to estimate the direct effect and neighbor effects independently.
CONFLICT OF INTEREST
The authors of the manuscript have no financial or non-financial conflict of interest in the subject matter or materials discussed in this manuscript.
DATA AVALIABILITY STATEMENT
Data availability is not applicable as no new data was created.
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