Construction of circular strongly partially-balanced repeated measurements designs

1 Introduction

A repeated measurements design 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). Williams (1949, 1950) initiated repeated measurements designs. Magda (1980) introduced the idea of a circular balance repeated measurements design when proper balance for different effects is considered. Cheng and Wu (1980) explained two different types of repeated measurements designs (RMD), the balanced uniform RMD and the strongly balanced uniform RMD. Afsarinejed (1990) presented some construction methods for repeated measurements design. Afsarinejed (1994) gave an easy method of constructing balanced and strongly balanced minimal repeated measurements designs with unequal period sizes. Iqbal and Jones (1994) constructed (i) efficient repeated measurements designs with equal and unequal period sizes using method of cyclic shifts, (ii) Strongly balanced repeated measurements designs, and (iii) combinatorial balanced designs for two unequal period sizes. Hedayat and Yang (2003) discussed the universal optimality of balanced cross over designs for v ≥ 3 and λ ≤ (v − 1)/2. Sharma et al. (2003) introduced a general strategy of construction of balanced repeated measurements designs for odd number of treatments and their analysis. Iqbal and Tahir (2009) constructed CSBRMD (circular strongly balanced repeated measurements designs) for some classes. Iqbal et al. (2010) constructed some first- and second-order CBRMD (circular balanced repeated measurements designs). They also constructed some CSBRMDs. Their construction is for period sizes less than the number of treatments. Strongly balanced repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. The situations where minimal strongly balanced repeated measurements designs cannot be constructed, strongly partially-balanced repeated measurements designs are preferable. This paper deals with the construction of circular strongly partially-balanced repeated measurements designs through method of cyclic shifts. Chalikias and Kounias (2012) extended the conditions of Cheng and Wu, in order to present a method to construct Φ-optimal designs for the case of p = 3 and n = 0 mod 4. Chalikias (2017) gave the impact review of two treatment repeated measurement designs with uncorrelated observations. Kounias and Chalikias (2015) extracts the universally optimal designs (strongly balanced on the periods and on the treatments) for the model with interactions.

The rest of the paper is organized as follows: In Section 2, we briefly explained the method of cyclic shifts. In Section 3, infinite series to generate CSPBRMDs are developed. In Sections 4 and 5, CSPBRMDs are constructed by method of cyclic shifts, Rule I and Rule II respectively.

2 Method of cyclic shifts

Method of cyclic shifts is explained here briefly. For detail, see Iqbal & Tahir (2009) and Iqbal et al. (2010).

Rule I: Let S = [q₁, q₂, …, q_p−1] be a set of shifts, where 0 ≤ q_i ≤ v − 1. If each element 0, 1, 2, …, v − 1 appears an equal number of times, say λ′ in a new set of shifts S^∗, where S^∗ = [q₁, q₂, …, q_p−1, v − (q₁ + q₂ + … + q_p−1) mod v] then it will be circular strongly balanced repeated measurement design. If S^∗ contains all of 0, 1, 2, …, v − 1 exactly λ_i times then it will be CSPBRMD.

Example 2.1. CSPBRMD is constructed for v = 10 and p = 4 through the following two sets of shifts. S₁ = [2, 3, 8], S₂ = [5, 6, 9]

Rule II: Let S_a = [q_a1, q_a2, …, q_a(p−1)] and S_b = [q_b1, q_b2, …, q_b(p−2)]t be sets of shifts, where 0 ≤ q_i ≤ v − 2. If at least one set of shifts is of type [q₁, q₂, …, q_(p−2)]t then S^∗ will be [q_a1, q_a2, …, q_a(p−1), (v − 1) − (q_a1 + q_a2 + … + q_a(p−1))mod(v − 1), q_b1, q_b2, …, q_b(p−2)]. If each element 0, 1, 2, …, v − 2 appears an equal number of times, say λ′ in a new set of shifts S^∗ then it will be circular strongly balanced repeated measurements design. If S^∗ contains all of 0, 1, 2, …, v − 2 exactly λ_i times then it will be CSPBRMD.

Example 2.2: Sets of shifts S₁ = [1, 3, 8, 6] & S₂ = [2, 4, 7]t provide following CPBRMD for v = 10 with p = 5.

3 Infinite series to generate CSPBRMDs

Series 3.1: CSPBRMDs can be constructed for v = 2mi + 2, i integer and p = 2 m, m > 1 through the following i sets of shifts.

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

• S_i = [v/2, (v + 2)/2, v−1] for p = 4.

• S_i = [(v − (p − 2))/2, …, (v − 6)/2, (v − 4)/2, v/2, (v + 2)/2, (v + 4)/2, (v + 6)/2, …, (v + p − 2)/2, v − 1] for p > 4.

Among v² ordered pairs, here 2v ordered pairs {(0, 1), (1, 2), …, (v − 2, v − 1), (v − 1, 0), (0, (v − 2)/2), (1, v/2), …, (v/2, v − 1), ((v + 2)/2, 0), ((v + 4)/2, 1), …, (v − 1, (v − 4)/2)} do not appear together while all other v(v − 2) appear once.

Example 3.1(a). CSPBRMD can be constructed for v = 4i + 2 and p = 4 through the following i sets of shifts.

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

• S_i = [v/2, (v + 2)/2, v − 1]

Example 3.1(b). CSPBRMD can be constructed for v = 6i + 2 and p = 6 through the following i sets of shifts.

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

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

Example 3.1(c). CSPBRMD can be constructed for v = 8i + 2 and p = 8 through the following i sets of shifts.

• S_j+1 = [4j + 2, 4j + 3, 4j + 4, 4j + 5, v − 2 − 4j, v − 3 − 4j, v − 4 − 4j]; j = 0, 1, …, i − 2.

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

Series 3.2: CSPBRMDs can be constructed for v = 2 m, m > 1 integer and p = 2 m + 1 through the following set of shifts.

• S_j+1 = [0, 1, 2, …, v − 1];

If sum of any two, three, …, v − 1 consecutive elements is 0 mod v, rearrange the elements to get the binary designs. Among v² ordered pairs, here v ordered pairs {(0, v/2), (1, (v + 2)/2), …, ((v − 2)/2, v − 1), (v/2, 0), ((v + 2)/2, 1), …, (v − 1, (v − 2)/2)} appear twice together while all other v(v − 1) appear once.

Example 3.2(a). CSPBRMD is constructed for v = 6 and p = 7 through the following set of shifts.

• [0, 1, 4, 3, 2, 5]

Example 3.2(b). CSPBRMD is constructed for v = 8 and p = 9 through the following set of shifts.

• [0, 1, 2, 3, 4, 5, 6, 7]

4 Construction of CSPBRMDs by Rule I

Construction 4.1: CSPBRMDs can be constructed for v = ri + 1, i odd and p = r > 1 odd through the following i sets of shifts.

• S_j = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …, i.

• 0 ≤ q_j1, q_j2, …, q_j(r−1) ≤ v − 1 but ≠ v/2,

• S^∗ contains each of 0, 1, 2, …, v − 1 exactly once except v/2 which does not appear.

• S^∗ = [q_j1, q_j2, …_, q_j(r−1), v−(q_j1 + q_j2 + … + q_j(r−1))]

Among v² ordered pairs, here v ordered pairs {(0, v/2), (1, (v + 2)/2), …, ((v − 2)/2, v − 1), (v/2, 0), ((v + 2)/2, 1), …, (v − 1, (v − 2)/2)} do not appear together while all other v(v − 1) appear once. Designs constructed from this method for p = 3, p = 5, p = 7 and p = 9 for v ≤ 100 are presented in Tables A1–A4 respectively given in Appendix A.

Construction 4.2: CSPBRMDs can be constructed for v = ri − 1, i odd and p = r through the following i sets of shifts.

• S_j = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …, i.

• 0 ≤ q_j1, q_j2, …, q_j(r−1) ≤ v − 1,

• S^∗ contains each of 0, 1, 2, …, v − 1 exactly once except v/2 which appears twice.

• S^∗ = [q_j1, q_j2, …_, q_j22, v − (q_j1 + q_j2 + … + q_j22)]

Among v² ordered pairs, here v ordered pairs {(0, v/2), (1, (v + 2)/2), …, ((v − 2)/2, v − 1), (v/2, 0), ((v + 2)/2, 1), …, (v − 1, (v − 2)/2)} appear twice together while all other v(v − 1) appear once. Designs constructed from this method for p = 3, p = 5, p = 7 and p = 9 for v ≤ 100 are presented in Tables A5–A8 respectively given in Appendix A.

Construction 4.3: CSPBRMDs can be constructed for v = ri + 2, i odd, r > 1(odd) and p = r through the following i sets of shifts.

• S_j = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …,i.

where,

• 0 ≤ q_j1, q_j2, …, q_j(r−1) ≤ v−1 but ≠ (v − 1)/2, (v + 1)/2,

• S^∗ contains each of 0, 1, …, v − 1 exactly once except (v − 1)/2,(v + 1)/2 which does not appear.

• S^∗ = [q_j1, q_j2, …_, q_j(r−1), v − (q_j1 + q_j2 + … + q_j(r−1)]

Among v² ordered pairs, here 2v ordered pairs {(0, (v − 1)/2), (1, (v + 1)/2), …, ((v − 1)/2, v − 1), ((v + 1)/2, 0), ((v + 3)/2, 1), …, (v − 1, (v − 3)/2), (0, (v + 1)/2), (1, (v + 3)/2), …, ((v − 3)/2, v − 1), ((v − 1)/2, 0), ((v + 1)/2, 1), …, (v − 1, (v − 1)/2),} do not appear together while all other v(v − 2) appear once. Designs constructed from this method for p = 5, p = 7 and p = 9 for v ≤ 100 are presented in Tables A9–A11 respectively given in Appendix A.

Construction 4.4: CSPBRMDs can be constructed for v = ri + 2, i even, r > 1(odd) and p = r through the following i sets of shifts.

• S_j = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …,i.

• 0 ≤ q_j1, q_j2, …, q_j(r−1) ≤ v − 1 but ≠ 2, (v − 4)/2,

• S^∗ contains each of 0, 1, …, v − 1 exactly once except 2, (v − 4)/2 which do not appear.

• S^∗ = [q_j1, q_j2, …_, q_j(r−1), v − (q_j1 + q_j2 + … + q_j(r−1)]

Among v² ordered pairs, 2v ordered pairs {(0, 2), (1, 3), …, (v − 3, v − 1), (v − 2, 0), (v − 1, 1), (0, (v − 4)/2), (1, (v − 2)/2), (2, v/2), …, ((v + 2)/2, v − 1), ((v + 4)/2, 0), ((v + 6)/2, 1), …, (v − 1, (v − 6)/2)} do not appear together while all other v(v − 2) appear once. Designs constructed from this method for p = 5, p = 7 and p = 9 for v ≤ 100 are presented in Tables A12–A14 respectively given in Appendix A.

5 Construction of PSBRMDs by Rule II

Construction 5.1: CSPBRMDs can be constructed for v = ri, i even and p = r > 3(odd) through the following i sets of shifts.

• S_j+1 = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …, i−1.

• S_i = [q_i1, q_i2, …, q_i(r−2)]t

• 0 ≤ q_j1, q_j2, …, q_j(r−1), q_i1, q_i2, …, q_i(r−2) ≤ v − 2 but ≠ v/2.

• S^∗ contains each of 0, 1, …, v − 2 exactly once except v/2 which does not appear.

• S^∗ = [q_j1, q_j2, …_, q_j(r−1), v − 1 − (q_j1 + q_j2 + … + q_j(r−1)), q_i1, q_i2, …, q_i(r−2)]

Here ordered pairs {(0, (v − 2)/2), (1, v/2), …, ((v − 2)/2, v − 1), (v/2, 0), ((v + 2)/2, 1), …, (v − 1, (v − 4)/2), (v − 1, v − 1)} does not appear together while all other appear once. Designs constructed from this method for p = 5, p = 7 and p = 9 for v ≤ 60 are presented in Tables B1–B3 respectively given in Appendix B.

Construction 5.2: CSPBRMDs can be constructed for v = ri, i > 1(odd) and p = r > 3(odd) through the following i sets of shifts.

• S_j+1 = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …, i−1.

• S_i = [q_i1, q_i2, …, q_i(r−2)]t

• 0 ≤ q_j1, q_j2, …, q_j(r−1), q_i1, q_i2, …, q_i(r−2) ≤ v − 2 but ≠ (v − 1)/2,

• S^∗ contains each of 0, 1, …, v − 2 exactly once except (v − 1)/2 which does not appear.

• S^∗ = [q_j1, q_j2, …_, q_j(r−1), v − 1 − (q_j1 + q_j2 + … + q_j(r−1)), q_i1, q_i2, …, q_i(r−2)]

Here ordered pairs {(0, (v − 1)/2), (1, (v + 1)/2), …, ((v − 1)/2, v − 1), ((v + 1)/2, 0), ((v + 3)/2, 1), …, (v − 1, (v − 3)/2), (v − 1, v − 1)} does not appear together while all other appear once. Designs constructed from this method for p = 5, p = 7 and p = 9 for v ≤ 60 are presented in Tables B4–B6 respectively given in Appendix B.

Construction 5.3: CSPBRMDs can be constructed for v = ri + 1, i even and p = r > 3(odd) through the following i sets of shifts.

• S_j+1 = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …, i−1.

• S_i = [q_i1, q_i2, …, q_i(r−2)]t

• 0 ≤ q_j1, q_j2, …, q_j(r−1), q_i1, q_i2, …, q_i(r−2) ≤ v − 2 but ≠ 2 and (v − 5)/2.

• S^∗ contains each of 0,1,…, v − 2 exactly once except 2 & (v − 5)/2 which do not appear.

• S^∗ = [q_j1, q_j2, …_, q_j(r−1), v − 1 − (q_j1 + q_j2 + … + q_j(r−1)), q_i1, q_i2, …, q_i(r−2)]

Here ordered pairs {(0, 2), (1, 3), …, (v − 4, v − 2), (v − 3, 0), (v − 2, 1), (0, (v − 5)/2), (1, (v − 3)/2),…, (v − 5, v − 2), (v − 4, 0), (v − 3, 1), (v − 2, 2), (v − 1, v − 1)} do not appear together while all other v(v−2) appear once. Designs constructed from this method for p = 5, p = 7 and p = 9 for v ≤ 60 are presented in Tables B7–B9 respectively given in Appendix B.

Construction 5.4: CSPBRMDs can be constructed for v = ri + 1, i > 1(odd) and p = r > 3(odd) through the following i sets of shifts.

• S_j+1 = [q_j1, q_j2, …, q_j(r−1)]; j = 1, 2, …, i − 1.

• S_i = [q_i1, q_i2, …, q_i(r−2)]t

• 0 ≤ q_j1, q_j2, …, q_j(r−1), q_i1, q_i2, …, q_i(r−2) ≤ v − 3 but ≠ 1.

• S^∗ contains each of 0, 1, …, v − 2 exactly once except 1 & (v − 2) which do not appear.

• S^∗ = [q_j1, q_j2, …_, q_j(r−1), v − 1 − (q_j1 + q_j2 + … + q_j(r−1)), q_i1, q_i2, …, q_i(r−2)]

Here ordered pairs {(0, 1), (1, 2), …, (v−2, v−1), (v−1, 0), (0, (v−2)/2), (1, v/2), …, (v/2, v−1), ((v + 2)/2, 0), ((v + 4)/2, 1), …, (v−1, (v−4)/2), (v−1, (v−1)} do not appear together while all other v(v−2) appear once. Designs constructed from this method for p = 5, p = 7 and p = 9 for v ≤ 60 are presented in Tables B10–B12 respectively given in Appendix B.

Conclusions and Ideas for Future Constructions: Series 3.1 works as a generator which generates the CSPBRMDs for every period size p (even). It provides CSPBRMDs for every v which is two plus multiple of p. Series 3.2 gives a design for a single value of v for each value of p, where v = 2 m, m > 1 integer and p = 2 m + 1. Constructions in Sections 4 and 5 are for p (odd). Generators to generate these designs are not developed yet for p (odd), therefore, these should be developed to generate (i) CSBRMDs in equal period sizes, (ii) CSBRMDs in unequal period sizes, (iii) CSPBRMDs in equal period sizes, (iv) CSPBRMDs in unequal period sizes, (v) CBRMDs in equal period sizes, (vi) CBRMDs in unequal period sizes, (vii) CWBRMDs (Circular weakly balanced repeated measurements designs) in equal period sizes, (viii) CWBRMDs in unequal period sizes.