Circular strongly partially-balanced repeated measurement designs in periods of two different sizes using method of cyclic shifts (Rule II)

1 Introduction

A repeated measurements design (RMD) is balanced with respect to the first-order residual effects if each treatment is immediately preceded same number of times, say λ′ times by each other treatment (excluding itself). Williams (1949, 1950) first initiated RMDs. Magda (1980) introduced the idea of a circular balanced RMDs. Cheng and Wu (1980) constructed balanced and strongly balanced RMD. RMD is strongly balanced with respect to the first-order residual effects if each treatment is immediately preceded λ′ times by each other treatment (including itself). Afsarinejed (1994) constructed balanced and strongly balanced minimal RMDs with unequal period sizes. RMDs in unequal period sizes are very useful if there is a restriction on the total number of treatments, some experimental units can receive on the total length of time while some experimental units can remain in the trial. Using method of cyclic shifts, Iqbal and Jones (1994) constructed (i) efficient RMDs with equal and unequal period sizes (ii) Strongly balanced RMDs for two unequal period sizes. Iqbal and Tahir (2009) constructed CSBRMD (circular strongly balanced RMDs) for some classes. Iqbal et al. (2010) constructed some first- and second-order CBRMD and CSBRMDs. Rasheed et al. (2018) developed some infinite series to obtain the minimal CSBRMDs in periods of three different sizes. The situations where minimal CSBRMDs cannot be constructed, minimal CSPBRMDs are preferred. RMD is strongly partially balanced if each treatment is not immediately preceded same number of times by each other treatment (including itself). If λ′_i takes only two values as λ′₁ =  λ′₂ + 1 then it is very close to the balanced. Strongly balanced and strongly partially-balanced RMDs are useful for the estimation of direct and residual effects independently. Minimal CSPBRMDs are preferred for the cases where minimal SBRMDs cannot be constructed. Jabeen et al. (2019) constructed minimal CSPBRMDs in equal period sizes almost for every cases. Using method of cyclic shifts (Rule I), Nazeer et al. (2018) constructed these designs in periods of two different sizes only for a few cases of v. For the remaining cases of v, minimal CSPBRMDs in periods of two different sizes could be constructed through method of cyclic shifts (Rule II) which should be constructed. Therefore, in this article, these designs are constructed for the remaining cases through method of cyclic shifts (Rule II).

The rest of the paper is organized as follows: In Section 2, method of cyclic shifts (Rule II) is explained to generate the CSPBRMDs in periods of two different sizes. In Section 3, efficiency of proposed designs is discussed. Using method of cyclic shifts (Rule II), some infinite series are developed in Section 4 to obtain minimal CSPBRMDs. These series are very useful for researchers and experimenters. They can get the required designs just by putting the values needed for the series. Contribution of this research is discussed in Section 5.

2 Method of cyclic shifts

In this article, Method of Cyclic Shifts introduced by Iqbal (1991) is used as a methodology for construction of the proposed designs. This method is preferred because it provides an easy construction of several types of cyclic designs such as (i) balanced incomplete block designs, (ii) polygonal designs, (iii) neighbor balanced designs, (iv) strongly balanced neighbor designs, (v) balanced RMDs, (vi) strongly balanced RMDs, (vii) weakly balanced RMDs, and (viii) strongly partially-balanced RMDs. All these designs can be constructed through this method in linear and circular periods/blocks of equal and unequal sizes. Furthermore, this method has edge over the existing methods because without studying the complete design, one can check the standard property of treatment balance and other balance properties such as for neighbor effects and residual effects, etc. Method of cyclic shifts (Rule II) is explained here briefly only for the construction of CSPBRMDs. For detail, see Iqbal & Tahir (2009) and Iqbal et al. (2010).

Rule II: Let S₁ = [ $q_{11}$ , $q_{12}$ , …, $q_{{1(p}_1-1)}$ ] and S₂ = [ $q_{21}$ , $q_{22}$ , …, $q_{{2(p}_2-2)}$ ]t be sets of shifts, where ${0\le q}_{\mathit{ij}}$  ≤ v − 2. If each element 0, 1, 2, …, v − 2 appears an equal number of times, say λ′ in a new set of shifts S*, where S* = [ $q_{11}$ , $q_{12}$ , …, $q_{{1(p}_1-1)}$ , $q_{21}$ , $q_{22}$ , …, $q_{2(p_2-2)}$ , v − 1-( $q_{11}$ + $q_{12}$ + …+ $q_{1(p_1-1)}$ ) mod v − 1] then it will be CSBRMD in periods of sizes p₁ & p₂, otherwise CSPBRMD.

S₁ = [1,3,2,8,7], S₂ = [4,0]t

3 Statistical model and efficiency of the design

In order to consider the efficiency of the constructed designs, the model for circular RMDs proposed by Davis and Hall (1969) is used.

Using the identities, D′D = R′D = bpI_v, D′R = L, D′U = N, D′P = bE_v,p, R′U = N, U′U = pI_n, U′U = E_k,q, P′P = nI_n. The reduced normal equations for $\widehat{\mathit{\delta }}$ and $\widehat{\mathit{\rho }}$ will be: $$\mathit{C}\left[\begin{array}{c}\widehat{\mathit{\delta }}\\ \widehat{\mathit{\rho }}\end{array}\right]=\left[\begin{array}{cc}\mathit{\theta }& \mathit{\pi }\\ {\mathit{\pi }}^{\mathbf{\prime }}& \mathbf{\Theta }\end{array}\right]\left[\begin{array}{c}\widehat{\mathit{\delta }}\\ \widehat{\mathit{\rho }}\end{array}\right]=\left[\begin{array}{c}\mathit{T}\\ \mathit{S}\end{array}\right]$$ $$\left[\begin{array}{cc}\mathit{bp}{\mathit{I}}_{\mathit{v}}-p^{-1}\mathit{N}{\mathit{N}}^{\mathbf{\prime }}& {\mathit{L}}^{\mathbf{\prime }}-p^{-1}\mathit{N}{\mathit{N}}^{\mathbf{\prime }}\\ {\mathit{L}}^{\mathbf{\prime }}-p^{-1}\mathit{N}{\mathit{N}}^{\mathbf{\prime }}& \mathit{bp}{\mathit{I}}_{\mathit{v}}-p^{-1}\mathit{N}{\mathit{N}}^{\mathbf{\prime }}\end{array}\right]\left[\begin{array}{c}\widehat{\mathit{\delta }}\\ \widehat{\mathit{\rho }}\end{array}\right]=\left[\begin{array}{c}{\mathit{D}}^{\mathbf{\prime }}\mathit{Y}-p^{-1}\mathit{N}{\mathit{U}}^{\mathbf{\prime }}\mathit{Y}\\ {\mathit{R}}^{\mathbf{\prime }}\mathit{Y}-p^{-1}\mathit{N}{\mathit{U}}^{\mathbf{\prime }}\mathit{Y}\end{array}\right]$$

θ =  $\mathit{bp}{\mathit{I}}_{\mathit{v}}-p^{-1}\mathit{N}{\mathit{N}}^{\mathbf{\prime }}$ $,$ Θ =  $\mathit{bp}{\mathit{I}}_{\mathit{v}}-p^{-1}\mathit{N}{\mathit{N}}^{\mathbf{\prime }}$ , $\mathit{\pi }$  =  ${\mathit{L}}^{\mathbf{\prime }}-p^{-1}\mathit{N}{\mathit{N}}^{\mathbf{\prime }}$ , $\mathit{T}={\mathit{D}}^{\mathbf{\prime }}\mathit{Y}-p^{-1}\mathit{N}{\mathit{U}}^{\mathbf{\prime }}\mathit{Y}$ , $\mathit{S}=$ ${\mathit{R}}^{\mathbf{\prime }}\mathit{Y}-p^{-1}\mathit{N}{\mathit{U}}^{\mathbf{\prime }}\mathit{Y}$ ,

For the period of two different sizes information matrix can be presented as: $${\mathit{C}}^{\ast }=\left[\begin{array}{cc}\mathit{bp}{\mathit{I}}_v-p_1^{-1}{\mathit{N}}_1{{\mathit{N}}^{\mathbf{\prime }}}_1-p_2^{-1}{\mathit{N}}_2{{\mathit{N}}^{\mathbf{\prime }}}_2& {\mathit{L}}^{\mathbf{\prime }}-p_1^{-1}{\mathit{N}}_1{{\mathit{N}}^{\mathbf{\prime }}}_1-p_2^{-1}{\mathit{N}}_2{{\mathit{N}}^{\mathbf{\prime }}}_2\\ {\mathit{L}}^{\mathbf{\prime }}-p_1^{-1}{\mathit{N}}_1{{\mathit{N}}^{\mathbf{\prime }}}_1-p_2^{-1}{\mathit{N}}_2{{\mathit{N}}^{\mathbf{\prime }}}_2& \mathit{bp}{\mathit{I}}_v-p_1^{-1}{\mathit{N}}_1{{\mathit{N}}^{\mathbf{\prime }}}_1-p_2^{-1}{\mathit{N}}_2{{\mathit{N}}^{\mathbf{\prime }}}_2\end{array}\right]$$

The information matrix for direct and residual effects denoted by $\theta$ and Θ respectively can be specified by their initial rows: $$\theta =\left[{\mathrm{\theta }}_0,{\mathrm{\theta }}_1,\dots {,\theta }_{\mathrm{t}-1}\right]\mathrm{and}\mathrm{\Theta }=\left[{\mathrm{\Theta }}_0,{\mathrm{\Theta }}_1,\dots ,{\mathrm{\Theta }}_{\mathrm{t}-1}\right]$$

According to the duality presented in the model (1), both direct and residual effects share the same information matrix. The non-zero Eigen values of information matrix C* are called the canonical efficiency factors, see James and Wilkinson (1971) and Pearce et al. (1974). The canonical efficiency factor is calculated by working out harmonic mean of non-zero Eigen values of their respective information matrix relative to that of an orthogonal with the same number of treatments having same number of replications. It is further assume that σ² is the same for the proposed design and the orthogonal design to which it is compared. The high value of E_r shows that design is suitable for the estimation of residual effects. Our proposed designs have high value of E_r, therefore, these designs are suitable for the estimation of residual effects while using periods of two different sizes.

4 Infinite Series to generate CSPBRMDs in periods of two different sizes

In this section, some infinite series are developed by method of cyclic shifts (Rule II) to generate minimal CSPBRMDs in periods of two different sizes. In the all following series, S* contains all values from 0, 1, 2, …, v − 1 either 0 or 1 time, therefore, all series provide minimal CSPBRMDs. Here ordered pairs {(0, v/2), (1, (v + 2)/2), …, ((v − 4)/2, v − 2), ((v − 2)/2, 0), (v/2, 1), …, (v − 2, (v − 2)/2), (v − 1, v − 1)} do not appear together while all other appear once in Series 3.1–3.9 while ordered pairs {(0, (v + 1)/2), (1, (v + 3)/2), …, ((v − 3)/2, v − 2), ((v − 1)/2, 0), ((v + 1)/2, 1), …, (v − 2, (v − 3)/2), (v − 1, v − 1)} do not appear together while all other appear once in Series 3.10 to 3.20. In these series sum of any two, three, …, (p − 2) consecutive elements should not be 0 (mod v). If so, reorder the elements.

Series 4.1: CSPBRMDs can be constructed for v = 2mi + 4, m > 2, i integer, p₁ = 2 m and p₂ = 4 through the following (i + 1) sets of shifts.

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 2)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 4)/2, (v − 2)/2, (v + 2)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 6)/2, (v − 4)/2, (v − 2)/2, (v + 2)/2, (v + 4)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 6)/2, (v − 4)/2, (v − 2)/2, (v + 2)/2, (v + 4)/2]t

• S_i+1 = [0, (v − 8)/2, (v − 6)/2, (v − 4)/2, (v − 2)/2, (v + 2)/2, (v + 4)/2, (v + 6)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 10)/2, (v − 8)/2, (v − 6)/2, (v − 4)/2, (v − 2)/2, (v + 2)/2, (v + 4)/2, (v + 6)/2, (v + 8)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S₂ = [(v − 2)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S₂ = [(v − 4)/2, (v − 2)/2, (v + 2)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S₂ = [(v − 6)/2, (v − 4)/2, (v − 2)/2, (v + 2)/2, (v + 4)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S₂ = [(v − 8)/2, (v − 6)/2, (v − 4)/2, (v − 2)/2, (v + 2)/2, (v + 4)/2, (v + 6)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 3)/2, (v − 1)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 5)/2, (v − 3)/2, (v − 1)/2, (v + 3)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 7)/2, (v − 5)/2, (v − 3)/2, (v − 1)/2, (v + 3)/2, (v + 5)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 9)/2, (v − 7)/2, (v − 5)/2, (v − 3)/2, (v − 1)/2, (v + 3)/2, (v + 5)/2, (v + 7)/2]t

• S_j+1 = [mj + 1, mj + 2, …, mj + m, v − 2-mj, v − 3-mj, …, v − m-mj]; j = 0, 1, …, i − 1.

• S_i+1 = [0, (v − 11)/2, (v − 9)/2, …, (v − 1)/2, (v + 3)/2, (v + 5)/2, (v + 7)/2, (v + 9)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S_j+2 = [2j + m + 1, 2j + m + 2, v − 2-m-2j]; j = 0, 1, …, i − 2.

• S_i+1 = [(v − 3)/2, (v − 1)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S_j+2 = [3j + m + 1, 3j + m + 2, 3j + m + 3, v − 2-m-3j, v − 3-m-3j]; j = 0, 1, …, i − 2.

• S_i+1 = [(v − 5)/2, (v − 3)/2, (v − 1)/2, (v + 3)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S_j+2 = [4j + m + 1, 4j + m + 2, 4j + m + 3, 4j + m + 4, v − 2-m-4j, v − 3 m-4j, v − 4-m-4j]; j = 0,1, …, i − 2.

• S_i+1 = [(v − 7)/2, (v − 5)/2, (v − 3)/2, (v − 1)/2, (v + 5)/2, (v + 3)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S_j+2 = [5j + m + 1, 5j + m + 2, …, 5j + m + 5, v − 2-m-5j, v − 3-m-5j, …, v − 5-m-5j]; j = 0, 1, …, i − 2.

• S_i+1 = [(v − 9)/2, (v − 7)/2, (v − 5)/2, (v − 3)/2, (v − 1)/2, (v + 7)/2, (v + 3)/2, (v + 5)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S_j+2 = [6j + m + 1, 6j + m + 2, …, 6j + m + 6, v − 2-m-6j, v − 3-m-6j, …, v − 6-m-6j]; j = 0, 1, …, i − 2.

• S_i+1 = [(v − 11)/2, (v − 9)/2, …, (v − 1)/2, (v + 9)/2, (v + 7)/2 , (v + 5)/2, (v + 3)/2]t

• S₁ = [1, 2, …, m, v − 2, v − 3, …, v − 1-m];

• S_j+2 = [7j + m + 1, 7j + m + 2, …, 7j + m + 7, v − 2-m-7j, v − 3-m-7j, …, v − 7-m-7j]; j = 0, 1, …, i − 2.

• S_i+1 = [ (v − 13)/2, (v − 11)/2, …, (v + 11)/2, (v + 9)/2 , (v + 7)/2, (v + 5)/2,(v + 3)/2]t

5 Contribution of this research

Strongly balanced and strongly partially-balanced RMDs are useful for the estimation of direct and residual effects independently. The situations where minimal CSBRMDs cannot be constructed, minimal CSPBRMDs are preferred. In this article, some series are developed which are new one and provide the minimal CSPBRMDs in periods of two different sizes with high efficiency to balance out the residual effects. These designs can be used in the experiments to investigate (i) whether the perceived velocity of a moving point on a computer screen is affected by relative cues such as the presence of either vertical or horizontal lines and the amount of spacing between them, (ii) chronic conditions in clinical trials, (iii) the effect of chlorhexidine gluconate in dental plaque re-growth, (iv) different methods of preoxygenation, and (iv) cellulose membranes. These designs are also useful in asthma trial to compare doses of Budesonide and Fluticasone. Other areas where proposed designs may be applied are agricultural animal feeding trials, bioassay, bioequivalence studies, biomedical or physiological measurements, consumer trials, questionnaires, taste testing experiments, etc.