On the construction of balanced repeated measurements designs with good circular properties

Abstract

Several fields, such as biological, medical, public health, agricultural sciences, etc., require circular balanced repeated measurement designs with fewer unequal number of repeated measurements than the number of treatments. Also, the availability and high cost of experimental subjects in these fields prefer the design in fewer experimental units. However, balancing the carryover effects of the treatments in minimal experimental subjects is one of the problems in this case. In this paper, several new series of minimal circular nearly strongly balanced RMDs in periods of two and three different sizes are constructed. The proposed construction of designs has high efficiency and, therefore, can save the cost of experimentations due to a fewer experimental subjects. Most of the designs are very useful because of the unavailability of strongly balanced RMDs for these combinations of parameters. A list of sets of shifts for the construction of minimal circular nearly SBRMDs has also been mentioned in the Appendix.

KEYWORDS:

• Carryover effects
• repeated measurement designs
• balanced repeated measurement designs
• strongly balanced repeated measurement designs
• circular nearly strongly balanced repeated measurement designs

1. Introduction

Experimental design deals with the arrangement of experimental units and the assignment of treatments to them in such a way that the comparisons among the treatments are unbiased and as precise as possible. The precision of the experiment depends to a large extent on the size of the experiment and the variations present in the experimental units. In designing experiments, keeping in view the limitation on experimental resources, efforts are made to minimize standard errors of estimated contrasts of interest. Mostly error variation arises because of the variation in experimental units under the same treatments. Therefore, it is suggested to use the same experimental unit for different treatments in an experiment but in different periods. This results in a type of design known as repeated measurements design (RMD) which makes them different from other designs presented in the literature which make use of different experimental units for different treatments. Repeated measurements design (RMD) is the type of experimental design in which experimental subjects are repeatedly measured by giving a sequence of treatments. RMDs are a special kind of carryover design in which each subject is influenced by a treatment applied in a current period called treatment effects and treatment applied in a previous period called carryover effects. Cross-over designs are RMDs in which each unit receives the sequence of each treatment over different periods. RMDs in unequal period sizes are very useful if there is a restriction on the total number of treatments, and some experimental units can receive on the total length of time, while some experimental units can remain in the trial which results in RMDs of different or unequal period sizes. RMDs balanced in treatments and carryover effects are useful in several fields such as medicine, pharmacology, animal sciences and psychology where carryover effects are natural.

2. Literature review

Cheng and Wu (1980) proposed the construction of BRMDs and CSBRMDs, especially for unequal block sizes. With the help of advanced technology in computer systems, the uses of block designs of unequal sizes k are beneficial, and hence it is very useful in industrial and agricultural experiments. The utilization of different block sizes in biological experiments was mentioned by Pearce (1964). Afsarinejad (1994) constructed minimal CSBRMDs for different period sizes. Iqbal and Jones (1994) proposed the construction of efficient RMDs and CSBs for two different period sizes. Iqbal and Tahir (2009), Iqbal et al. (2010) and Bashir et al. (2018) constructed CSBs for some classes. Rasheed et al. (2018) recently developed some generators to obtain MCSBs in a period of three different sizes for some cases of t, where 2 ≤ k₃< k₂ ≤ 10. However, there do not exist MCSBs for certain combinations of treatments and periods when several treatments are less than the number of periods. Here, minimal circular nearly CSBs are useful. A design is said to be CNBs (Circular nearly strongly balanced repeated measurement designs) if every treatment is immediately followed by every other treatment including itself except the treatment (labelled as t − 1) which is not preceded by itself. In this article, some generators are developed to construct CNBs in periods of (i) two different sizes and (ii) three different sizes. These designs have their own importance in extended scientific and biological experimentations with different block sizes which consist of clinical experimentations of human and also animal behaviour responses for the comparison of several non-curative treatments for their effectiveness. With the advancement of technological computer systems, block designs with unequal sizes p have become very useful in huge industrial and agricultural experiments. In biological experiments, the utilization of different block sizes has been studied significantly. RCTs become prone to the unravelling of sizes of the block when it remains the same throughout the trial, so investigators use different block sizes, i.e. they randomly vary the block size to lower the chances of unbiasedness (Schulz & Grimes, 2002; Schulz, 1995). There exist different scientific research areas such as animal husbandry and genetic experimentation. RCTs, where these designs are applied as within-subject treatment comparisons, are more efficient than between-subject treatment comparisons. These proposed designs are additions to the literature as they are incorporating different period sizes and also possess good efficiency in estimating direct and residual effects.

Important definitions with abbreviations used that will be discussed throughout the article are discussed below briefly.

BRMDs	=	Balanced repeated measurement designs
CSBs	=	Circular strongly balanced repeated measurement designs
CNBs	=	Circular nearly strongly balanced repeated measurement designs

2.1. BRMDs

In an RMD each experimental subject or unit receives a series of several treatments in successive periods.

2.2. CSBs

If every treatment is directly preceded by every other treatment (including itself) then it is called CSBs.

2.3. CNBs

In this type of design, every treatment is immediately followed by every other treatment exactly once as well as with itself except the treatment labelled as $v$−1, which is not preceded by itself.

In Section 3, the model and information matrix are described along with the formula for the computation of the efficiency for CNBs. In Section 4, the Method of Cyclic Shifts (MOCS) is described to generate CNBs in different period sizes. A series of generators is proposed in Section 5 to obtain CNBs in two different sizes of periods. Also, a series of generators for the construction of CNBs in three different sizes of periods are proposed in Section 6. Section 7 contains the concluding remarks about the proposed designs. A list of sets of shifts for the construction of CNBs in two and three different sizes of periods with many treatments smaller than thirty is provided in the Appendix.

3. Model and efficiency for CNBs

The model used for repeated measurements designs in the literature is the conventional model proposed by Magda (1980) (1) $y_{ijk}=\mu +{\tau }_{d(k,j)}+{\gamma }_{d(k-1,j)}+{\pi }_k+{\xi }_{ij}+{ϵ}_{ijk},$(1) where $y_{ijk}$ is the $i\text{th}$ observation from the subject $j\text{th}$ with the sequence i in period k for which treatment d(k, j) is given.

$\mu$ = Overall mean, ${\tau }_{d(k,j)}$ =  treatment d(k, j) effect, ${\gamma }_{d(k-1,j)}$ =  treatment d(k − 1, j) effect in the period k which was applied in the period k−1 to the same experimental subject, ${\pi }_k$ = $k\text{th}$ period effect, ${\xi }_{ij}$  = $j$th subject effect of sequence i, and ${ϵ}_{ijk}$ is i.i.d normally distributed residual term having mean 0 and fixed variance ${\sigma }^2$.

The joint information matrix of treatment and carryover effects for the CNB is expressed by $A_{(\tau ,\gamma )}=\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{{}_{12}}^{\mathrm{\prime }}& A_{22}\end{array}\right),$where $A_{11}$, $A_{12}$ and $A_{22}$ in two different period sizes are expressed by $\begin{array}{rl}{A}_{11}& =R-M-\frac{1}{k_1}{N}_1{N}_1^{\mathrm{\prime }}-\frac{1}{k_2}{N}_2{N}_2^{\mathrm{\prime }}+\frac{\underset{\_}{r}{\underset{\_}{r}}^{\mathrm{\prime }}}{n_1{k}_1+n_2{k}_2}J,\\ M& =\left[\begin{array}{cc}\left({\displaystyle \frac{s^2(k_2-1)+{(s-1)}^2}{n_1+n_2}}+{\displaystyle \frac{(k_1-k_2)}{n_1}}\right)J_{t-1}& {\displaystyle \frac{(t-1)(s-1)}{n_1+n_2}}{\underset{\_}{1}}_{t-1}\\ {\displaystyle \frac{(t-1)(s-1)}{n_1+n_2}}{\underset{\_}{1}}_{{}_{t-1}}^{\mathrm{\prime }}& {\displaystyle \frac{{(t-1)}^2}{n_1+n_2}}\end{array}\right],\end{array}$ $\begin{array}{rl}{A}_{12}& =Z-\overline{M}-\frac{1}{k_1}{N}_1{\overline{N}}_1^{\mathrm{\prime }}-\frac{1}{k_2}{N}_2{\overline{N}}_2^{\mathrm{\prime }}+\frac{\underset{\_}{r}{\underset{\_}{\overline{r}}}^{\mathrm{\prime }}}{n_1{k}_1+n_2{k}_2}J,\end{array}$ $\begin{array}{rl}\overline{M}& =\left[\begin{array}{cc}\left({\displaystyle \frac{s^2(k_2-2)+2s(s-1)}{n_1+n_2}}+{\displaystyle \frac{(k_1-k_2)}{n_1}}\right)J_{t-1}& {\displaystyle \frac{s(t-1)}{n_1+n_2}}{\underset{\_}{1}}_{t-1}\\ {\displaystyle \frac{s(t-1)}{n_1+n_2}}{\underset{\_}{1}}_{t-1}& 0\end{array}\right],\end{array}$ $\begin{array}{rl}{A}_{22}& =\overline{R}-M-\frac{1}{k_1}{\overline{N}}_1{\overline{N}}_1^{\mathrm{\prime }}-\frac{1}{k_2}{\overline{N}}_2{\overline{N}}_2^{\mathrm{\prime }}+\frac{\underset{\_}{\overline{r}}{\underset{\_}{\overline{r}}}^{\mathrm{\prime }}}{n_1{k}_1+n_2{k}_2}J,\end{array}$

and $A_{11}$, $A_{12}$ and $A_{22}$ in periods of three different sizes are expressed by $\begin{array}{rl}{A}_{11}& =R-M-\frac{1}{k_1}{N}_1{N}_1^{\mathrm{\prime }}-\frac{1}{k_2}{N}_2{N}_2^{\mathrm{\prime }}-\frac{1}{k_3}{N}_3{N}_3^{\mathrm{\prime }}+\frac{\underset{\_}{r}{\underset{\_}{r}}^{\mathrm{\prime }}}{n_1{k}_1+n_2{k}_2+n_3{k}_3}J,\end{array}$ $\begin{array}{rl}M& =\left[\begin{array}{cc}\left({\displaystyle \frac{s^2(k_3-1)+{(s-1)}^2}{n_1+n_2+n_3}}+{\displaystyle \frac{{(s-1)}^2(k_2-k_3)}{n_1+n_2}}+{\displaystyle \frac{{(s-2)}^2(k_1-k_2)}{n_1}}\right)J_{t-1}& {\displaystyle \frac{n_3(s-1)}{n_1+n_2+n_3}}{\underset{\_}{1}}_{t-1}\\ {\displaystyle \frac{n_3(s-1)}{n_1+n_2+n_3}}{\underset{\_}{1}}_{{}_{t-1}}^{\mathrm{\prime }}& {\displaystyle \frac{n_3^2}{n_1+n_2+n_3}}\end{array}\right],\end{array}$ $\begin{array}{rl}{A}_{12}& =Z-\overline{M}-\frac{1}{k_1}{N}_1{\overline{N}}_1^{\mathrm{\prime }}-\frac{1}{k_2}{N}_2{\overline{N}}_2^{\mathrm{\prime }}-\frac{1}{k_3}{N}_3{\overline{N}}_3^{\mathrm{\prime }}+\frac{\underset{\_}{r}{\underset{\_}{\overline{r}}}^{\mathrm{\prime }}}{n_1{k}_1+n_2{k}_2+n_3{k}_3}J,\end{array}$ $\begin{array}{rl}\overline{M}& =\left[\begin{array}{cc}\left({\displaystyle \frac{s^2(k_3-2)+2s(s-1)}{n_1+n_2+n_3}}+{\displaystyle \frac{{(s-1)}^2(k_2-k_3)}{n_1+n_2}}+{\displaystyle \frac{{(s-2)}^2(k_1-p{k}_2)}{n_1}}\right)J_{t-1}& {\displaystyle \frac{s{n}_3}{n_1+n_2+n_3}}{\underset{\_}{1}}_{t-1}\\ {\displaystyle \frac{s{n}_3}{n_1+n_2+n_3}}{\underset{\_}{1}}_{{}_{t-1}}^{\mathrm{\prime }}& 0\end{array}\right],\end{array}$ $\begin{array}{rl}{A}_{22}& =\overline{R}-M-\frac{1}{k_1}{\overline{N}}_1{\overline{N}}_1^{\mathrm{\prime }}-\frac{1}{k_2}{\overline{N}}_2{\overline{N}}_2^{\mathrm{\prime }}-\frac{1}{k_3}{\overline{N}}_3{\overline{N}}_3^{\mathrm{\prime }}+\frac{\underset{\_}{\overline{r}}{\underset{\_}{\overline{r}}}^{\mathrm{\prime }}}{n_1{k}_1+n_2{k}_2+n_3{k}_3}J.\end{array}$Here,

$n_1$	=	subjects measured repeatedly up to $k_1$ size
$n_3$	=	subjects measured repeatedly up to $k_3$ size
$s$	=	number of sets of shifts
$N_1$	=	t incidence matrix vs 1 to $n_1$
$N_2$	=	t incidence matrix vs $n_1+1$ to $n_1+n_2$
$N_3$	=	t incidence matrix vs $n_1+n_2+1$ to $n_1+n_2+n_3$
${\overline{N}}_1$	=	residual incidence matrix vs 1 to $n_1$
${\overline{N}}_2$	=	residual incidence matrix vs $n_1+1$ to $n_1+n_2$
${\overline{N}}_3$	=	residual incidence matrix vs $n_1+n_2+1$ to $n_1+n_2+n_3$
$\underset{\_}{r}$	=	vector of replication of treatments
$\underset{\_}{\overline{r}}$	=	vector of replication of residual
$R$	=	diagonal matrix of replication of t
$\overline{R}$	=	diagonal matrix of replication of residual
$J$	=	matrix of 1s

Then the information matrix of the treatment and carryover effects are $A_{\tau }=A_{11}-A_{12}{A}_{22}^{-}{A}_{21}$ and $A_{\gamma }=A_{22}-A_{21}{A}_{11}^{-}{A}_{12}$, respectively.

RMDs should be analysed for their capability of discriminating the effects of treatment from carryover effects. Hanford (2005) has given the criteria to compare the RMDs based on this ability. Divecha and Gondaliya (2014) gave a convenient method for calculating the efficiency of separability (ES) for the BRMDs. This formula considering the provided constraints of our proposed type of RMDs is given by (2) $\text{ES}=\left[\frac{t\sqrt{t-1}-1}{t\sqrt{t-1}}\right]\times 100\mathrm{\%}.$(2)

4. Construction methodology

Construction of CNBs from all the series of generators requires an understanding of rule II of an MOCS which was proposed by Iqbal and Jones (1994). MOCS (Rule II) is defined here briefly for constructing CNBs in periods of different sizes.

Let S₁ = [$q_{11}$, $q_{12}$, … , $q_{p_1-1}$], S₂ = [$q_{21}$, $q_{22}$, … , $q_{p_2-1}$] and S₃ = [$q_{31}$, $q_{32}$, … , $q_{p_3-2}$]t be the sets of shifts, where 0 ≤$q_{ij}$ ≤ v − 2. If every element 0, 1, … , v − 2 appears exactly once in S*, it is CNB in periods of sizes p₁, p₂ and p₃, where S* = [$q_{11}$, $q_{12}$, … , $q_{p_1-1}$, $q_{21}$, $q_{22}$, … , $q_{p_2-1}$, $q_{31}$, $q_{32}$, … , $q_{p_3-1}$, v − 1 − ($q_{11}$+ $q_{12}$+ ··· + $q_{p_1-1}$) mod (v − 1), v − 1 − ($q_{21}$+ $q_{22}$+ ··· + $q_{p_2-1}$) mod (v − 1)]. In these sets of shifts, the sum of any 2, 3, … , or (p − 3) successive elements should not be 0 mod (v − 1). If so, rearrange them.

Consider here the Rule II of an MOCS briefly for the construction of CNBs with the help of an example as t = 13, S₁ = [3, 4, 5, 9, 8] and S₂ = [6]t.

Take t − 1 experimental subjects for one set of shifts [3,4,9,5,8]. Write 0, 1, … , t − 2 in the first period of t − 1 experimental subjects, respectively. For the second period of every subject, add 3 mod (t − 1) to every element of the first period of each subject. Then add 4 mod (t − 1) to every element of the second period which gives the treatment number of the third period in each subject. Similarly add 9, 5 and 8, respectively.

Table

	Experimental Subjects
Periods	1	2	3	4	5	6	7	8	9	10	11	12
1	0	1	2	3	4	5	6	7	8	9	10	11
2	3	4	5	6	7	8	9	10	11	0	1	2
3	7	8	9	10	11	0	1	2	3	4	5	6
4	4	5	6	7	8	9	10	11	0	1	2	3
5	9	10	11	0	1	2	3	4	5	6	7	8
6	5	6	7	8	9	10	11	0	1	2	3	4

Take t − 1 more subjects for the second set of shifts [6]t. Allocate 0, 1, … , t − 2 to every subject in the first period, respectively. To obtain the elements of the second period for every subject, add 6 mod (t − 1) to every element of the first period for all subjects. Then insert t − 1 (i.e. 12) in each element of the third period.

Table

	Subjects
Periods	13	14	15	16	17	18	19	20	21	22	23	24
1	0	1	2	3	4	5	6	7	8	9	10	11
2	6	7	8	9	10	11	0	1	2	3	4	5
3	12	12	12	12	12	12	12	12	12	12	12	12

The above design is CNB in t = 13, k₁ = 6, k₂ = 3, n₁ = 12, n₂ = 12 and ES = 98%.

5. Generators to obtain CNBs in two different sizes of periods

Let S₁ = [$e_{11}$, $e_{12}$, … , $e_{p_1-1}$] and S₂ = [$e_{21}$, $e_{22}$, … , $e_{p_2-2}$]t be two sets of shifts, where 0 ≤$e_{ij}$ ≤ t − 2. Define, S* = [$e_{11}$, $e_{12}$, … , $e_{p_1-1}$, $e_{21}$, $e_{22}$, … , $e_{p_2-2}$, t − 1 − ($e_{11}$+ $e_{12}$+ ⋅⋅⋅ + $e_{p_1-1}$) mod (t − 1)]. If every element 0, 1, … , t − 2 appears exactly once in the new set of shifts S*, then the design from the set of shifts will be CNB in periods of sizes k₁ and k₂. The sum of any 2, 3, … ,(k − 3) successive elements of any set of shifts should not be 0 mod (t − 1). If so, rearrange the elements.

Example:

Sets of shifts S₁ = [1, 7, 4, 3, 9, 6] and S₂ = [2, 5, 8]t give following CNB for t = 11 in k₁= 7 and k₂= 5 with ES = 97%.

Table

Subjects
1	2	3	4	5	6	7	8	9	10
00	11	22	33	44	55	66	77	88	99
10	21	32	43	54	65	76	87	98	09
81	92	03	14	25	36	47	58	69	70
28	39	40	51	62	73	84	95	06	17
52	63	74	85	96	07	18	29	30	41
45	56	67	78	89	90	01	12	23	34
04	15	26	37	48	59	60	71	82	93

Table

Subjects
1	2	3	4	5	6	7	8	9	10
010	110	210	310	410	510	610	710	810	910
20	31	42	53	64	75	86	97	08	19
72	83	94	05	16	27	38	49	50	61
57	68	79	80	91	02	13	24	35	46
105	106	107	108	109	100	101	102	103	104

Here, 10₆ means that treatment 10 was applied in the current period, while the subscript ‘6’ is the treatment applied in the previous period.

Generator 5.1:

If t = 2zi + 2a − 1, i is an integer, k₁ = 2z, k₂ = 2a, a ≠ z and a, z > 1, then CNBs can be obtained from the sets given below where j = 0, 1, … , i − 1. $\begin{array}{rl}{S_j}_{+1}& =\left[zj+1,zj+2,\dots ,zj+z,t-1-\left(zj+1\right),t-1-\left(zj+2\right),\dots ,t-1-\left(zj+z-1\right)\right];\\ {S_i}_{+1}& =\left[zi+1,zi+2,\dots ,zi+a-1,t-1-\left(zi+1\right),t-1-\left(zi+2\right),\dots ,t-1-\left(zi+a-2\right),0\right]t.\end{array}$

Example 5.1:

CNBs for t = 13, k₁= 8 and k₂ = 6 are constructed from the sets of shifts given below with 98% efficiency. $S_1=\left[1,2,3,4,11,10,9\right],S_2=\left[5,6,7,0\right]t.$

Generator 5.2:

If t = 2zi + 2a, i is an integer, k₁= 2z, k₂= 2a + 1 and a, z > 1, then CNBs can be obtained from the sets given below where j = 0, 1, … , i − 1. $\begin{array}{rl}{S_j}_{+1}& =\left[zj+1,zj+2,\dots ,zj+z,t-1-\left(zj+1\right),t-1-\left(zj+2\right),\dots ,t-1-\left(zj+z-1\right)\right];\\ {S_i}_{+1}& =\left[zi+1,zi+2,\dots ,zi+a-1,t-1-\left(zi+1\right),t-1-\left(zi+2\right),\dots ,t-1-\left(zi+a-1\right),0\right]t.\end{array}$

Example 5.2:

CNBs for t = 16, k₁= 10 and k₂ = 7 are constructed from the sets given below with 98% efficiency. $S_1=\left[1,2,3,4,14,5,13,11,12\right],S_2=\left[6,7,9,8,0\right]t.$

Generator 5.3:

If t = 2zi + 4a − 1, i is an integer, k₁= 2z, k₂= 2a and a ≠ z, a, z > 1, then CNBs can be obtained from the sets given below. $\begin{array}{rl}{S_j}_{+1}& =\left[zj+1,zj+2,\dots ,zj+z,t-1-\left(zj+1\right),t-1-\left(zj+2\right),\dots ,t-1-\left(zj+z-1\right)\right];\\ {S_i}_{+1}& =\left[zi+1,zi+2,\dots ,zi+a-1,t-1-\left(zi+a-1\right),t-1-\left(zi+a-2\right),\dots ,t-1-\left(zi+1\right)\right];\\ {S_i}_{+2}& =[zi+a+1,zi+a+2,\dots ,zi+2a-1,t-1-(zi+a+1),t-1-(zi+a+2),\\ & \dots ,t-1-\left(zi+2a-2\right),0]t.\end{array}$

Example 5.3:

CNBs for t = 25, k₁ = 10 and k₂ = 8 are constructed from the sets given below with 99% efficiency. $S_1=\left[1,2,3,4,5,23,22,21,20\right],S_2=\left[6,7,8,17,9,18,16\right],S_3=\left[10,11,12,14,13,0\right]t.$

Generator 5.4:

If t = 2zi + 4a + 1, i is an integer, k₁= 2z, k₂= 2a + 1 and a, z > 1, then CNBs can be obtained from the sets given below. $\begin{array}{rl}{S_j}_{+1}& =\left[zj+1,zj+2,\dots ,zj+z,t-1-\left(zj+1\right),t-1-\left(zj+2\right),\dots ,t-1-\left(zj+z-1\right)\right];\\ {S_i}_{+1}& =\left[zi+1,zi+2,\dots ,zi+a,t-1-\left(zi+1\right),t-1-\left(zi+2\right),\dots ,t-1-\left(zi+a\right)\right];\\ {S_i}_{+2}& =[zi+a+1,zi+a+2,\\ & \dots ,zi+2a,t-1-\left(zi+a+1\right),t-1-\left(zi+a+2\right),\dots ,t-1-\left(zi+2a-1\right)]t.\end{array}$

Example 5.4:

CNBs for t = 21, k₁ = 8 and k₂ = 7 are constructed from the sets given below with 99% efficiency. $S_1=\left[1,2,3,4,19,18,17\right],S_2=\left[5,6,7,15,14,13\right],S_3=\left[8,9,10,12,11\right]t.$

6. Generators to obtain CNBs in three different sizes of periods

Let S₁ = [$e_{11}$, $e_{12}$, … , $e_{p_1-1}$], S₂ = [$e_{21}$, $e$, … , $e_{p_2-1}$] and S₃ = [$e_{31}$, $e_{32}$, … , $e_{p_3-2}$]t be the sets of shifts, where 0 ≤$e_{ij}$ ≤ t − 2. Define, S* = [$e_{11}$, $e_{12}$, … , $e_{p_1-1}$, $q_{21}$, $e_{22}$, … , $e_{p_2-1}$, $e_{31}$, $e_{32}$, … , $e_{p_3-1}$, t − 1 − ($e_{11}$+ $e_{12}$+ ··· + $e_{p_1-1}$) mod (t − 1), t − 1 − ($e_{21}$+ $e_{22}$+ ··· + $e_{p_2-1}$) mod (t − 1)]. If every element 0, 1, … , t − 2 appears exactly once in the new set of shifts S*, then the design from the set of shifts will be CNB in periods of sizes k₁, k₂ and k₃. The sum of any 2, 3, … , (k − 3) successive elements of the set of shifts S_i + 1 should not be 0 mod (t − 1). If so rearrange the elements.

Example

Sets of shifts S₁= [3, 4, 9, 5, 8], S₂= [1, 2, 11, 10] and S₃= [6]t give the following CNBs for t = 13 in k₁= 6, k₂= 5 and k₃ = 3 with ES = 98%.

Table

Subjects
1	2	3	4	5	6	7	8	9	10	11	12
05	16	27	38	49	510	611	70	81	92	103	114
30	41	52	63	74	85	96	107	118	09	110	211
73	84	95	106	117	08	19	210	311	40	51	62
47	58	69	710	811	90	101	112	03	14	25	36
94	105	116	07	18	29	310	411	50	61	72	83
59	610	711	80	91	102	113	04	15	26	37	48

Table

Subjects
13	14	15	16	17	18	19	20	21	22	23	24
00	11	22	33	44	55	66	77	88	99	1010	1111
10	21	32	43	54	65	76	87	98	109	1110	011
31	42	53	64	75	86	97	108	119	010	111	20
23	34	45	56	67	78	89	910	1011	110	01	12
02	13	24	35	46	57	68	79	810	911	100	111

Table

Subjects
25	26	27	28	29	30	31	32	33	34	35	36
012	112	212	312	412	512	612	712	812	912	1012	1112
60	71	82	93	104	115	06	17	28	39	410	511
126	127	128	129	1210	1211	120	121	122	123	124	125

Here, 11₆ is the treatment 11 applied in the present period, while the subscript ‘6’ is the treatment applied in the previous period.

Generator 6.1:

If t = 2zi + 2a + 2b, i is an integer, k₁ = 2a, z ≠ a, z, a, b > 1, k₂ = 2a + 1 and k₃ = 2b, then CNBs can be obtained from the sets given below where j = 0, 1, … , i − 1. $\begin{array}{rl}{S_j}_{+1}& =\left[zj+1,zj+2,\dots ,zj+z,t-1-\left(zj+1\right),t-1-\left(zj+2\right),\dots ,t-1-\left(zj+z-1\right)\right];\\ {S_i}_{+1}& =\left[zi+1,zi+2,\dots ,zi+a,t-1-\left(zi+1\right),t-1-\left(zi+2\right),\dots ,t-1-\left(zi+a\right)\right];\\ {S_i}_{+2}& =[zi+a+1,zi+a+2,\dots ,zi+a+b-1,t-1-(zi+a+1),t-1-(zi+a+2),\\ & \dots ,t-1-\left(zi+a+b-1\right)]t.\end{array}$

Example 6.1:

CNBs for t = 13, k₁=  8, k₂ = 7 and k₃ = 4 are constructed from the sets given below with 98% efficiency. $S_1=\left[1,2,3,4,16,15,14\right],S_2=\left[5,6,7,12,11,10\right]\text{and}{S}_3=\left[8,9\right]t.$

Generator 6.2:

If t = 2zi + 2a + 2b − 1, i is an integer, k₁ = 2z, z ≠ a ≠ b > 1, k₂ = 2a and k₃ = 2b, then CNBs can be constructed from the sets given below where j = 0, 1, … , i − 2. $\begin{array}{rl}{S_j}_{+1}& =\left[zj+1,zj+2,\dots ,zj+z,t-1-\left(zj+1\right),t-1-\left(zj+2\right),\dots ,t-1-\left(zj+z-1\right)\right];\\ {S_i}_{+1}& =\left[zi+1,zi+2,\dots ,zi+a,t-1-\left(zi+1\right),t-1-\left(zi+2\right),\dots ,t-1-\left(zi+a-1\right)\right];\\ {S_i}_{+2}& =[zi+a+1,zi+a+2,\dots ,zi+a+b-1,t-1-(zi+a+1),t-1-(zi+a+2),\\ & \dots ,t-1-\left(zi+a+b-2\right),0]t,\text{for}b>2;\\ {S_i}_{+2}& =\left[\left(t-1\right)/2,0\right]t,\text{for}b=2.\end{array}$

Example 6.2:

CNBs for t = 23, k₁=  10, k₂ = 8 and k₃ = 6 are constructed from the sets given below with 99% efficiency. $S_1=\left[1,2,3,4,5,21,20,19,18\right],S_2=\left[6,7,8,9,16,15,14\right]\text{ and}{S}_3=\left[10,11,12,0\right]t.$

Generator 6.3:

If t = 2zi + 2a + 2b + 1, k₁ = 2z, i is an integer, k₂ = 2a + 1,k₃ = 2b + 1, a ≠ b > 1 and z > 1, then CNBs can be obtained from the sets given below where j = 0, 1, … , i − 1. $\begin{array}{rl}{S_j}_{+1}& =\left[zj+1,zj+2,\dots ,zj+z,t-1-\left(zj+1\right),t-1-\left(zj+2\right),\dots ,t-1-\left(zj+z-1\right)\right];\\ {S_i}_{+1}& =\left[zi+1,zi+2,\dots ,zi+a,t-1-\left(zi+1\right),t-1-\left(zi+2\right),\dots ,t-1-\left(zi+a\right)\right];\\ {S_i}_{+2}& =[zi+a+1,zi+a+2,\dots ,zi+a+b,t-1-(zi+a+1),\\ & t-1-\left(zi+a+2\right),\dots ,t-1-\left(zi+a+b-1\right)]t.\end{array}$

Example 6.3:

CNBs for t = 19, k₁=  8, k₂ = 7 and k₃ = 5 are constructed from the sets given below with 99% efficiency. $S_1=\left[1,2,3,4,17,16,15\right],S_2=\left[5,6,13,7,12,11\right],S_3=\left[8,9,10\right]t.$

7. Concluding remarks

CSBs help in estimating direct and carryover effects independently and providing high efficiency of separability. Despite the importance of RMDs, there is no sufficient literature available on the construction of the CSBs and CNBs with high efficiency of separability for unequal period sizes. In situations where CSBs cannot be constructed, they are preferable as they give efficiencies close to CSBs. Through well-known MOCS Rule II, these designs have been constructed for unequal period sizes with high efficiency of separability which covers the lack of designs for unequal period sizes. The experimenters now have more choices when dealing with different period sizes. All the proposed designs possess an efficiency close to 100% and, therefore, they should be a better alternative to CSBs.

Acknowledgements

The authors are thankful to the reviewers for the valuable suggestions and corrections.

Appendix

Table A1. List of sets of shifts for CNBs in two different sizes of periods and treatments smaller than 30.

t	k1	k2	Sets of shifts	ES
9	6	4	[1,3,2,7,6] + [4,0]t	96
10	6	5	[1,3,2,8,7] + [4,5,0]t	97
11	8	4	[1,3,2,4,9,8,7] + [5,0]t	97
13	6	4	[1,3,2,11,10] + [4,5,8] + [6,0]t	98
13	8	6	[1,3,2,4,11,10,9] + [5,6,7,0]t	98
13	10	4	[1,3,2,4,5,11,10,9,8] + [6,0]t	98
14	8	7	[1,3,2,4,12,11,10] + [5,6,7,8,0]t	98
15	6	4	[1,3,2,13,12] + [4,5,6,10,9] + [7,0]t	98
15	6	5	[1,2,3,13,12] + [4,5,10,9] + [6,7,8]t	98
15	8	4	[1,3,2,4,13,12,11] + [5,6,9] + [7,0]t	98
16	6	5	[1,3,2,14,13] + [4,5,6,11,10] + [7,8,0]t	98
17	10	4	[1,3,2,4,5,15,14,13] + [6,7,10] + [8,0]t	99
17	10	8	[1,3,2,4,5,15,14,13,12] + [6,7,8,10,9,0]t	99
18	10	9	[1,3,2,4,5,16,15,14,13] + [6,7,8,9,10,11,0]t	99
19	6	4	[1,3,2,17,16] + [4,5,6,14,13] + [7,8,11] + [9,0]t	99
19	8	4	[1,3,2,4,17,16,15] + [5,6,7,8,13,12,11] + [9,0]t	99
21	6	4	[1,3,2,19,18] + [4,5,6,16,15] + [7,8,9,13,12] + [10,0]t	99
21	6	5	[1,2,3,19,18] + [4,5,6,16,15] + [7,8,13,12] + [9,10,11]t	99
21	8	6	[1,3,2,4,19,18,17] + [5,6,7,8,15,14,13] + [9,11,10,0]t	99
21	8	7	[1,2,3,4,19,18,17] + [5,6,7,15,14,13] + [8,9,10,12,11]t	99
22	6	5	[1,3,2,20,19] + [4,5,6,17,16] + [7,8,9,14,13] + [10,11,0]t	99
22	8	7	[1,3,2,4,20,19,18] + [5,6,7,8,16,15,14] + [9,10,11,12,0]t	99
23	8	4	[1,3,2,4,21,20,19] + [5,6,7,8,17,16,15] + [9,10,13] + [11,0]t	99
23	10	4	[1,3,2,4,5,21,20,19,18] + [6,7,8,9,10,16,15,14,13] + [11,0]t	99
25	6	4	[1,3,2,23,22] + [4,5,6,20,19] + [7,8,9,17,16] + [10,11,14] + [12,0]t	99
27	6	4	[1,3,2,25,24] + [4,5,6,22,21] + [7,8,9,19,18] + [10,11,12,16,15] + [13,0]t	99
27	6	5	[1,2,3,25,24] + [4,5,6,22,21] + [7,8,9,19,18] + [10,11,16,15] + [12,13,14]t	99
27	8	4	[1,3,2,4,25,24,23] + [5,6,7,8,21,20,19] + [9,10,11,12,17,16,15] + [13,0]t	99
27	10	4	[1,3,2,4,5,25,24,23,22] + [6,7,8,9,10,20,19,18,17] + [11,12,15] + [13,0]t	99
27	10	8	[1,3,2,4,5,25,24,23,22] + [6,7,8,9,10,20,19,18,17] + [11,12,13,15,14,0]t	99
27	10	9	[1,2,3,4,5,25,24,23,22] + [6,7,8,9,20,19,18,17] + [10,11,12,16,15,14,13]t	99
28	6	5	[1,3,2,26,25] + [4,5,6,23,22] + [7,8,9,20,19] + [10,11,12,17,16] + [13,14,0]t	99
28	10	9	[1,3,2,4,5,26,25,24,23] + [6,7,8,9,10,21,20,19,18] + [11,12,13,14,15,16,0]t	99
29	8	6	[1,3,2,4,27,26,25] + [5,6,7,8,23,22,21] + [9,10,11,12,19,18,17] + [13,15,14,0]t	99
29	8	7	[1,2,3,4,27,26,25] + [5,6,7,8,23,22,21] + [9,10,11,19,18,17] + [12,13,14,16,15]t	99

Table A2. List of sets of shifts for CNBs in three different sizes of periods and treatments smaller than 30.

t	k1	k2	k3	Sets of shifts	ES
14	6	5	4	[1,2,3,12,11] + [4,5,9,8] + [6,7]t	98
17	8	6	4	[1,2,3,4,15,14,13] + [5,6,7,11,10] + [8,0]t	99
19	8	7	5	[1,2,3,4,17,16,15] + [5,6,7,13,12,11] + [8,9,10]t	99
20	6	5	4	[1,2,3,18,17] + [4,5,6,15,14] + [7,8,12,11] + [9,10]t	99
20	8	7	6	[1,2,3,4,18,17,16] + [5,6,7,14,13,12] + [8,9,11,10]t	99
23	10	8	6	[1,2,3,4,5,21,20,19,18] + [6,7,8,9,16,15,14] + [10,11,12,0]t	99
25	8	6	4	[1,2,3,4,23,22,21] + [5,6,7,8,19,18,17] + [9,10,11,15,14] + [12,0]t	99
25	10	9	7	[1,2,3,4,5,23,22,21,20] + [6,7,8,9,18,17,16,15] + [10,11,12,13,14]t	99
26	6	5	4	[1,2,3,24,23] + [4,5,6,21,20,19] + [7,8,9,18,17,16] + [10,11,15,14] + [12,13]t	99
26	10	9	8	[1,2,3,4,5,24,23,22,21] + [6,7,8,9,19,18,17,16] + [10,11,12,15,14,13]t	99
27	8	7	5	[1,2,3,4,25,24,23] + [5,6,7,8,21,20,19] + [9,10,11,17,16,15] + [12,13,14]t	99
28	8	7	6	[1,2,3,4,26,25,24] + [5,6,7,8,22,21,20] + [9,10,11,18,17,16] + [12,13,15,14]t	99