Optimal Circular Balanced Repeated Measurements Designs for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 37.8125 13.4804" width="37.8125pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g185-40" d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"/></g><g transform="matrix(.017,0,0,-0.017,12.533,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.017,0,0,-0.017,27.401,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg> through the Method of Cyclic Shift (MCS) Rule II

Shehzad, Farrukh; Rajab, Muhammad; Daniyal, Muhammad; Ahmed, Rashid; Tawiah, Kassim

doi:https://doi.org/10.1155/2022/2875183

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 2875183 | https://doi.org/10.1155/2022/2875183

Optimal Circular Balanced Repeated Measurements Designs for through the Method of Cyclic Shift (MCS) Rule II

Farrukh Shehzad,¹Muhammad Rajab,¹Muhammad Daniyal,¹Rashid Ahmed,¹and Kassim Tawiah^2,3

Academic Editor: Gianpaolo Di Bona

Received23 Jun 2022

Revised10 Aug 2022

Accepted10 Sept 2022

Published06 Oct 2022

Abstract

Background. One of the common characteristics of preclinical genetic experimentation is the result of repeated measurements, and for this purpose, repeated measurements designs (RMDs) have gained much more significance. In the class of RMDs, balanced repeated measurements designs (BRMDs) are preferred as they balance out the residual effects since the experimentation is repeated over time on different subjects. This study provides the theoretical framework of universal optimal criteria proposed by Kiefer (1975) for the newly proposed circular balanced repeated measurements designs (CBRMDs) by Rajab et al. (2018). These universal optimality criteria were proved for the special class of designs where the number of treatments is equal to the number of periods. Universal optimality has been discussed considering all the possible effects in the models, i.e., units, subjects, treatments, and periods. Methodology. This study characterized CBRMDs, where several treatments and periods are equal in the contest of three separate models with their matrices of information in simplified form. We used these simplified matrices of information to ascertain the criteria for universally optimal CBRMDs under different conditions. These new CBRMDs have been constructed using the well-known method of cyclic shifts (MCS) rule II. Results. Universal optimality of the new proposed classes of designs has been discussed theoretically. Universally optimal CBRMDs were constructed for using the MCS rule II along with the confirmation of the universal optimality criteria proposed in the existing theory. Conclusions. The proposed class of new CBRMDs has been proven to have theoretically universally optimal designs, which have been constructed by the method of cyclic shifts rule II when the number of treatments is equal to the number of periods.

1. Introduction

Repeated measurements designs (RMDs) have wide applications to ensure the validity and reproducibility of research in plant breeding and genetics research. Such designs have been extensively applied in areas where the treatment could be a plant variety, fertilizer, irrigation, or pesticide [1–3]. In crop experiments, the neighbor effects have been recognized by Haines and Benzian [4] as well as Jenkyn and Dyke [5] due to competition for light, poaching of nutrients by plant roots, and the spread of disease pathogens. Particularly, the competition effect has been discussed in other experiments as associated with a height difference, tillering ability, canopy size, date of maturity, and yield of roots, respectively, on insects, kale, disease screening, forests, trees, and sunflowers [6–8].

The problem of universal optimality of balanced repeated measurements designs (BRMDs) has been widely studied in the literature. Hedayat and Afsarinejad [9, 10] provided the foundation for the optimality of RMDs, whereas Hedayat and Afsarinejad [10] and Cheng and Wu [11] established the universally optimal estimation of treatment as well as residual effects.

With a maximum of twice the number of subjects as the number of treatments, Kunert [12, 13] demonstrated that uniform balanced designs can be considered optimally universal in estimating treatment effects for numerous design classes. Kunert [12, 13] also established that for sufficiently large numbers of experimental subjects, a balanced uniform design may no longer be optimal. Such designs are no longer universally optimal for the estimation of residual effects when there exist special cases of designs. Stufken [14] constructed some universally optimal designs using orthogonal arrays of type I.

Kempton et al. [15] proposed a model where carry-over effects are proportional to direct treatment effects; see Bailey and Kunert [16] for optimal designs related to this model. Bailey and Druilhet [17] considered total effects for models without interactions and showed that binary designs are efficient for such effects. Unfortunately, total effects under the Afsarinejad and Hedayat [18] model are not estimable for designs with no treatment preceded by themselves, and binary designs cannot be used in that context. Druilhet [19] considered optimal cross-over designs for total effects under the model proposed by Afsarinejad and Hedayat [18]. Rajab et al. [20] proposed the optimality criteria for CBRMDs developed by the MCS rule I, but that criteria did not cover the designs proposed by the MCS rule II, which provides the designs for .

In this article, newly constructed CBRMDs for through the MCS rule II by Rajab et al. [21] will be proved universally optimal based on Kiefer’s [22] criteria. These designs have neither been constructed before nor their universal optimality has been proved theoretically. To make the designs useful for practical purposes, their universal optimality should be proved theoretically. All the possible models, including different effects, have been used to prove the universally optimal criteria that have not been practiced earlier. The expressions for the information matrices proposed by Rajab et al. [20] will be used for proving the optimality criteria. Kiefer [22] defined universally optimal notions. He proposed the following set of conditions adequate for universally optimal designs, as follows:

Suppose is the information matrix of a design for estimating particular effects is such that (i) is completely symmetric. (ii) , then is universally optimal in .

Thus, the CBRMDs established by Rajab et al. [21] and the expressions proposed by Rejab et al. [20] will be utilized to ascertain the universally optimal designs under the conditions of Kiefer [22].

In the next sections of the paper, we discuss the models for CBRMDs and universal optimality. The conclusion ends the paper.

2. Models for CBRMDs with Identities

Let be a class of BRMDs with treatments juxtaposed with units of experiment in the period, . In total, the following four types of effects in the designs were studied; effects of the period, effects of the unit, effects of treatment, and effects of residuals. The potential linear models with distinct effects are as follow:where is np observations vector, is the overall mean, treatment effects vector residual effect vector is n × 1 unit vector, period effects vector and column vector with zero and fixed variance . treatment effect incidence matrix. residual effect incidence matrix with the order. unit effect incidence matrix. period effect incidence matrix.

Following are the identities:where matrix of 1's, and identity matrix.

Matrices of information for approximating treatment and the effects of residuals were derived by incorporating model 1. Equivalently, model (1) may be formulated as follows:

The design matrix can be partitioned as; with and . Subsequently, matrices of information for effects of treatment and residual under model (1) may be derived by beginning with the matrices of information, , of the model (4) is as follows:

With i.e., belonging to , the space column. So, in its reduced form can be written as follows when we substitute the following matrices of incidence:

Considering the design , assume , and , to be respectively, the effects of treatment, effects of residual, and jointly treatment and residual effects information matrices. Rajab et al. [20] derived the following matrix of information for evaluating the effects of treatment and residual:

3. Universal Optimality

Here, we discuss how our stipulated CBRMDs satisfy the constraints of Kiefer’s [22] optimality criteria.

3.1. Universally Optimal Designs: Second Class (C2)

Suppose be circular RMDs connected in the same class with treatments , units of subjects/experimental , and periods . Suppose is a design satisfying the conditions.(i)Each treatment appears at most once in each unit (binary).(ii)Each treatment is preceded by every other treatment exactly once, .

Theorem 1. A design is universally optimal, under model (1), for estimating treatment and effects of residual effects of the incidence matrix . Recall the expressions for the joint information matrix, and the information matrices for the estimation of treatment and residual effects under model (1) are as follows:For the design , Rajab et al. [20] proposed the following expression:To ascertain the universally optimal for estimating the effects of treatment, we illustrate that the matrices of information have a maximum trace and are unconditionally symmetric. This maximation of the trace shows that has minimum given by . Thus, has a minimum trace provided the matrix of incidence is of the form , where is some known scaler. Subsequently, the matrix of information minimizes to the form with and being scalars, indicating the matrices are unconditionally symmetric. Therefore, by Kiefer’s [22] conditions of sufficiency, the design is optimally universal for estimating the effects of treatment.

Theorem 2. A design is optimally universal, in the context of the model (2), in estimating treatment together effects of residual within the matrix of incidence .

Proof. We know from previous discussions that the joint matrix of information and the matrices of information for estimating the effects of residuals and treatment, and in the context of the model (2), are as follows:This joint matrix of information can be partitioned asThe matrices of information for the estimation of effects of treatment effects and residual effects in the context model (2) are; . For the design , Rajab et al. [20] proposed the following expression:The matrix of information subsequently reduces form to indicating that it is completely symmetric. Thus, by sufficient conditions of Kiefer [22], the design is also universally optimal for the estimation of treatment effects.

Theorem 3. The design is optimally universal, in the context of the model (3), for estimating of effects of treatment and effects of residual if the matrix of incidence is .

Proof. Recall the expressions for the joint information matrix and the information matrices for estimation of treatment and residual effects under model (3) are as follows:This joint matrix of information can be partitioned as follows:The information matrices for estimating treatment effects and residual effects under model (3) are and . For the design , Rajab et al. [20] proposed the following expression:Subsequently, the matrix of information reduces form to showing it is completely symmetric. Therefore, the design is also universally optimal for the estimation of treatment effects.

3.2. Universally Optimal RMDs Uniform on Subjects

Kunert [12] showed that circular balanced designs that are uniform across subjects are universally optimal for the estimation of treatment as well as residual effects. He deals with the situation where . He showed that, for the estimation of treatment effects, balanced uniform designs are universally optimal over if or . If n is sufficiently large, they are no longer optimal. For the estimation of residual effects, can never be universally optimal over and cannot be optimally universal over provided special other designs exist.

A design with the following properties:(i)the information matrix of for the estimation of treatment (resp. residual) effects is completely symmetric (i.e., all diagonal elements are the same and all off-diagonal elements are identical);(ii)this information matrix has maximal trace over and is universally optimal over for the estimation of treatment (resp. residual) effects. The criterion of “universal optimality” includes the commonly applied criteria of , and optimality [22]. Over the class . Druilhet [23] and Baily and Druilhet [17] extended this to designs where the number of periods is any multiple of .

Theorem 4. Universally optimal minimal CBRMDs can be constructed for from the set of shifts, using the MCS rule II. , Where the sum of any two, three, …, consecutive shifts (elements) should not be 0 mod . If so, rearrange the shifts.

Proof. Each element appears once in S; therefore, by the MCS rule II, it is minimal CBRMD. The proposed CBRMDs uniform on subjects is optimal based on the following conditions:
the row sum of the information matrix is zero.
The information matrix of each design, which is uniform on subjects, is symmetric. i.e., all the diagonal elements are the same, and all the off-diagonal elements are the same.
The trace of the information matrix of every design that is uniform on subjects is maximum. Table 1 shows the list of optimal designs through the MCS rule II for .
Considering the optimal design for , obtained from the set of shifts ; the following are the matrix , matrix and N matrix of the CBRMD (Table 2).Kiefer [22] showed that the design is considered optimal if the trace of the information matrix C_D is maximum and it is symmetric as well. It was also proposed by Kiefer [22] that balance designs are optimal when . Considering the above-proposed design of CBRMD, the following is the information matrix for treatment and residual effects with .From the above matrices, it can be observed that both share the same information matrices and are symmetric as well. Hinkelmann and Kempthorne [24] suggested the method of relative efficiency. This method also allows for comparing an incomplete block design (IBD) with a completely randomized design (CRD) or randomized complete block design (RCBD) and two competing IBDs with each other. In this method, the residual variances are assumed to be the same for both designs to be compared. Relative efficiency (RE) is the harmonic mean of nonzero eigenvalues of matrix C. The efficiency of both treatment and residual effects is 100% and is an optimal design in the class of . It is uniform in subjects, as , and its concurrence matrix is as follows:Here is the treatment concurrence matrix, whose diagonal elements are repetitions of each treatment and off-diagonal elements are the number of times a pair of treatments appear together in the same blocks. is the incidence matrix of treatments versus neighbors (left and right). Diagonal elements of the matrix for a design in which no treatment appears as a neighbor to itself are zero, and the off-diagonal matrix is the number of times a pair of treatments appear as a neighbor to each other in the same blocks. For further detail, see Iqbal and Tahir [25] and Iqbal et al. [26].
Taking into consideration another example for , obtained from a set of shift , reperesented in Table 3 with matrix, matrix, and matrix of the CBRMD are as follows:following are the information matrices of treatment and residual effects; also .The efficiency of both treatment and residual effect is 100%, and this is an optimal design in the class of Ω (9, 9, 9). It is uniform in subjects as p = with a concurrence matrix as follows:

4. Conclusion

This study provided the theoretical framework of universal optimal criteria proposed by Kiefer [22] for the newly proposed CBRMDs by Rajab et al. [20, 21]. These universal optimality criteria were proved for the special class of designs where the number of treatments is equal to the number of periods. This special class of design has been constructed, and its universal optimality has been discussed considering all the possible effects in models, i.e., unit, subject, treatment, and period. These designs satisfied the conditions of universal optimality criteria proposed by Kiefer [22]. The proposed optimal designs have relative importance in scientific experiments, which may involve clinical trials of human beings and animal behavior, preclinical genetic experimentation, and responses for comparing several curative and noncurative treatments for their effectiveness. Consider the application in which the goal of the analysis is to compare four competing treatments. BRMDs could involve five animals, each measured at four-period sizes, once under each experimental condition. Such designs involve a smaller number of animals, which is both efficient and ethically appealing. The application of such designs has also been discussed in family studies by Ball et al. [27], in environmental impact and monitoring studies by Green [28], in clinical trials by Rana et al. [29], and by Kong et al. [30], where both approximate and exact designs for estimating the total effects under one crossover and two interference models were discussed. However, the proposed theoretical framework of the universal optimality criteria does not apply to the other class of designs because they have their own structure of construction and limitations. These criteria did not include the correlation between the effects of the model. It is worth noting that new theoretical results can be developed to cover these limitations.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

MR, FS, MD, RA, and KT conceived the idea, suggested the methodology, planned the study, and drafted the manuscript. All authors read and approved the final manuscript.

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Copyright © 2022 Farrukh Shehzad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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