Repeated Measurement Designs of Five Periods: Estimating the Parameter of Carryover Effects

Abstract

This study investigates the derivation of optimal repeated measurement designs of two treatments, five periods, and n experimental units for carryover effects. The optimal designs are determined for cases where n = 0, 1 (mod 2). The adopted optimality criterion focuses on minimizing the variance of the estimated carryover effect, thereby ensuring maximum precision in parameter estimation and design efficiency. The results presented here extend and complement earlier research of Chalikias and Kounias on optimal two-treatment repeated measurement designs for a smaller number of periods, and are closely related to the more recent findings on optimal designs for direct effects. Overall, the present work contributes to the theoretical framework of optimal design methodology by providing new insights into the structure and efficiency of repeated measurement designs, particularly in the presence of carryover effects, and sets the ground for future extensions incorporating treatment–period interactions.

1. Introduction

The first recorded use of crossover experimental designs can be traced back to the field of agriculture, where they were originally introduced as a means of reducing variability between plots and increasing the reliability of comparisons between treatments. In 1864, Lawes and Gilbert [1] conducted one of the earliest and most influential crossover experiments, in which two different treatments were applied to the same plot of land to examine their relative effects on crop yield. Their landmark study—spanning two decades of continuous wheat cultivation on the same land—remains a reference point for experimental design and is considered a precursor to the formal statistical methodologies developed in the 20th century. By applying multiple treatments to the same experimental unit, they effectively controlled for the inherent heterogeneity of the land, thus allowing for more precise treatment comparisons. This concept became the foundation for the principle that crossover or repeated measurement designs can substantially improve efficiency and precision by using each experimental unit as its own control.

While the origins of crossover experimentation lie in agriculture, the fundamental idea quickly spread to other scientific fields. Medicine and pharmacology, in particular, adopted the crossover structure for clinical trials, where the same patient receives multiple treatments in successive periods separated by washout phases. Such designs have been especially useful in comparing pharmaceutical drugs or therapies, where between-subject variability can obscure treatment differences. In these cases, a crossover design allows researchers to isolate the effects of treatments more effectively, as each subject’s response under different conditions can be directly compared to their own baseline. Similar designs have been successfully implemented in psychology, physiology, and industrial engineering, where repeated measurements under controlled sequences of conditions yield valuable insights into treatment or process effects.

In general terms, a repeated measurement design (or crossover design) involves a sequence of treatments applied to each experimental unit (e.u.) over a series of time periods, with one treatment administered per period. This design structure provides a rich framework for statistical analysis, but it also introduces additional complexity compared to simple or parallel designs. Specifically, responses measured in later periods may be influenced not only by the treatment applied in that period but also by residual effects (carryover effects) from treatments applied in previous periods.

So, researchers are interested in estimating two types of effects: (a) the direct effect, which is the immediate effect of the treatment applied during a specific period, and (b) the carryover effect, which is the residual effect of the treatment applied in the preceding period. In a two-treatment case, with treatments A and B, the carryover effects are denoted ${\delta }_A$ and ${\delta }_B$. In our case, the parameter of principal interest is $\delta ={\delta }_A-{\delta }_B$, which quantifies the comparative efficacy or impact of the two treatments. Estimating δ accurately requires carefully balanced designs to eliminate or control for effects such as period effects and subject effects.

The design of experiments has long focused on finding optimal designs, that is, experiments that allow for the most efficient estimation of model parameters. Most researchers in this field concentrate on constructing designs that minimize the variance of estimators, satisfy some criteria of optimality, or satisfy all the criteria of optimality (universally optimal designs). These principles have been developed extensively in the literature, forming a solid theoretical framework for repeated measurement and crossover designs.

A pivotal step in the formalization of crossover design theory was the introduction of statistical models that explicitly incorporate direct and carryover effects. Hedayat and Afsarinejad [2] presented one of the foundational models for two-treatment repeated measurement designs, providing a clear mathematical basis for analyzing treatment effects across multiple periods. Their model laid the groundwork for subsequent studies on design optimality. Ten years later, Afsarinejad [3] extended this framework by considering the case where subject effects are random—an assumption that aligns more closely with real-world medical and biological experiments. Under such conditions, the efficiency of crossover designs can be enhanced by accounting for variability among subjects while still preserving within-subject control advantages.

Further notable contributions to this field include the work of Stufken [4], who identified families of optimal and efficient repeated measurement designs, and Hedayat and Zheng [5], who developed designs based on Type I orthogonal arrays suitable for models with correlated errors and mixed carryover effects. These developments have significantly broadened the applicability of crossover methodologies, enabling researchers to handle more complex scenarios, including multi-treatment and multi-period studies with realistic error structures.

The model of Hedayat and Afsarinejad [2] has been widely adopted as a benchmark. Using this model as a foundation, Kounias and Chalikias [6] explored optimal designs for estimating direct and carryover effects in two-period experiments, identifying configurations that yield minimum variance estimates of both parameters. Their later work extended these findings to experiments involving three periods and considered more advanced models incorporating treatment–period interactions [7]. Such interactions arise when the effect of a treatment depends not only on its identity but also on the period in which it is applied—an important consideration in many biological and industrial contexts where time-dependent effects can occur. Kounias and Chalikias [8] also addressed the four-period case, identifying optimal designs for direct effects and providing theoretical results on the structure and properties of these optimal arrangements. Collectively, these studies have progressively extended the theory of optimal crossover design to cover experiments of increasing complexity.

Our current focus is on the specific case of two treatments (A and B) administered over five periods. The goal is to construct designs that achieve optimal estimation of carryover effects, specifically, the difference $\delta ={\delta }_A-{\delta }_B$, while maintaining the highest possible efficiency and balance in treatment sequences.

The present study continues this line of research by examining the case of two treatments across five periods, focusing on the estimation of carryover effects, extending the optimization that has been investigated for direct effects [8].

Despite the extensive literature on optimal repeated measurement and crossover designs, the problem of optimal estimation of carryover effects for experiments involving five periods has not yet been addressed. Existing results are primarily confined to designs with two, three, or four periods, where the structure of optimal sequences and balance conditions is considerably simpler. In particular, while optimal designs for carryover effects have been established for three-period and four-period experiments, these results do not extend in a straightforward manner to the five-period case due to the increased complexity of treatment sequences and carryover patterns.

The five-period setting is of practical relevance in many applications, especially in longitudinal clinical trials and pharmacological studies involving chronic treatments, where a larger number of periods is required to adequately capture treatment dynamics. From a theoretical perspective, increasing the number of periods introduces additional challenges in achieving balance between direct, period, and carryover effects, rendering the characterization of optimal designs substantially more complex [9,10].

The present study aims to fill this gap by systematically investigating optimal repeated measurement designs for two treatments over five periods, with a particular focus on the estimation of the carryover effect contrast ${\delta =\delta }_A-{\delta }_B$. Building on the modeling framework of Hedayat and Afsarinejad and extending recent results obtained for direct effects in the five-period case, we derive conditions for optimality and identify classes of designs that achieve efficient estimation of carryover effects. To the best of our knowledge, this is the first work to provide a complete characterization of optimal designs for carryover effect estimation in two-treatment, five-period repeated measurement experiments.

The remainder of the paper is organized as follows. In Section 2, we present the statistical model and underlying assumptions, together with the methodological framework used in the analysis. Section 3 describes the resulting optimal designs and the corresponding distributions of experimental units. Finally, Section 4 provides a discussion of the results and concluding remarks.

2. Methodology

2.1. The Model

The 32 sequences of treatments A and B are presented in the following

Table 1. The 32 sequences of treatments A, B.

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

A

A

B

B

B

B

A

A

A

A

B

B

B

B

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

$u_0$

$u_1$

$u_2$

$u_3$

$u_4$

$u_5$

$u_6$

$u_7$

$u_8$

$u_9$

$u_{10}$

$u_{11}$

$u_{12}$

$u_{13}$

$u_{14}$

$u_{15}$

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

A

A

B

B

B

B

A

A

A

A

B

B

B

B

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

$u_{16}$

$u_{17}$

$u_{18}$

$u_{19}$

$u_{20}$

$u_{21}$

$u_{22}$

$u_{23}$

$u_{24}$

$u_{25}$

$u_{26}$

$u_{27}$

$u_{28}$

$u_{29}$

$u_{30}$

$u_{31}$

, where $u_i (i=\mathrm{0,1},\cdots ,31)$ is the number of experimental units that received the i-th sequence of treatments, $u_0+\cdots +u_{31}=n$, and all experimental units are n. Specifically, a dual system is used to enumerate the sequences. We assign 2 for treatment B and 0 for A, raised to the power of j-1 (for period j). For example, the 21st sequence is defined by ($2^0+2^2+2^4=21$).

As it is referred in the introduction, the model was first used by Hedayat and Afsarinejad [2,8]:

$$y_{ijk}=\mu +{\tau }_h+{\pi }_j+{\delta }_{i,j-1}+{\gamma }_i+e_{ijk}$$

(1)

where y_ijk denotes the response observed for the unit k that receives the sequence i during period j.

• As it concerns the indices
• The index j corresponds to the j-th period, j = 1, 2, 3, 5;
• The index i refers to the i-th sequence, $\mathrm{i}=0,1,...31$;
• The index k refers to the unit k = 1, 2, … n.
• As it concerns the model components
• τ_h: is the direct effect of treatment A or B $({\tau }_A,{\tau }_B)$;
• π_j: is the period effect of the j-th period; the period effect accounts for systematic differences among the measurement periods.
• δ_A, δ_B: the residual (carryover) effect of the treatment administered in the previous period of A and B;
• γ_i: the sequence effect (the effect of the i-th sequence), capturing differences attributable to the specific order in which treatments are administered;
• e_ijk: the random error term, representing unexplained variability of the response.
• The error terms $e_{ijk}$ are assumed to be

• Mutually independent,
• Normally distributed with mean zero,
• Having constant variance,
• Independent both between different sequences and within the same sequence.

The above model (in overparameterized form) is written as follows:

$$\mathit{{\rm Y}}={\tau }_A{\mathit{\tau }}_A+{\tau }_B{\mathit{\tau }}_B+{\delta }_A{\mathit{\delta }}_A+{\delta }_B{\mathit{\delta }}_B+{\pi }_1{\mathit{\pi }}_1+\cdots +{\pi }_5{\mathit{\pi }}_5++{\gamma }_0{\mathit{\gamma }}_0+\cdots +{\gamma }_{31}{\mathit{\gamma }}_{31}+\mathit{e}$$

(2)

and $\mathit{{\rm Y}},{\mathit{\tau }}_A,{\mathit{\tau }}_B,{\mathit{\delta }}_A,{\mathit{\delta }}_B,{\mathit{\pi }}_1,\cdots {\mathit{\pi }}_5,{\mathit{\gamma }}_0,\cdots {\mathit{\gamma }}_{31},\mathit{e}$ are vectors of dimension $1\times 5n$;

The construction of the vectors is defined in [8], using 1 if the treatment is A, and zero if it is B. Apply that to ${\mathit{\tau }}_{\mathit{A}} \mathrm{f}\mathrm{o}\mathrm{r}$ ABB…, ${\tau }_A=\left[\begin{array}{c}1\\ 0\\ 0\\ ⋮\end{array}\right]$ and ${\tau }_B=\left[\begin{array}{c}0\\ 1\\ 1\\ ⋮\end{array}\right]$, ${\delta }_A=\left[\begin{array}{c}0\\ 1\\ 0\\ \begin{array}{c}⋮\end{array}\end{array}\right]$ and, in the same way, ${\mathit{\delta }}_{\mathit{B}},{\mathit{\pi }}_i \mathrm{and} {\mathsf{\gamma }}_i$ are defined so that ${\mathit{\tau }}_{\mathit{A}}+{\mathit{\tau }}_{\mathit{B}}=1_{5\mathit{n}}$, ${\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{B}}+{\mathit{\pi }}_1=1_{5\mathit{n}}$, and ${\mathit{\pi }}_1+{\mathit{\pi }}_2+{\mathit{\pi }}_3+{\mathit{\pi }}_4+{\mathit{\pi }}_5=1_{5\mathit{n}}$. Also, it takes the value 1 when the i-th unit is employed and 0 otherwise, so that ${\mathit{\gamma }}_0+{\mathit{\gamma }}_1+{\mathit{\gamma }}_2+\cdots +{\mathit{\gamma }}_{31}=1_{5\mathit{n}}$.

For example, for the sequence ABBAA,

$${\mathit{\tau }}_{\mathit{A}}+,{\mathit{\tau }}_{\mathit{B}}=\left[\begin{array}{c}1\\ 0\\ \begin{array}{c}0\\ 1\\ 1\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 1\\ \begin{array}{c}1\\ 0\\ 0\end{array}\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ \begin{array}{c}1\\ 1\\ 1\end{array}\end{array}\right]=1_{5\mathit{n}}$$

$${\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{B}}+{\mathit{\pi }}_1=\left[\begin{array}{c}0\\ 1\\ \begin{array}{c}0\\ 0\\ 1\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \begin{array}{c}1\\ 1\\ 0\end{array}\end{array}\right]+\left[\begin{array}{c}1\\ 0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ \begin{array}{c}1\\ 1\\ 1\end{array}\end{array}\right]=1_{5\mathit{n}}$$

$${\mathit{\pi }}_1+{\mathit{\pi }}_2+{\mathit{\pi }}_3+{\mathit{\pi }}_4+{\mathit{\pi }}_5=\left[\begin{array}{c}1\\ 0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 1\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \begin{array}{c}1\\ 0\\ 0\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 1\\ 0\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 0\\ 1\end{array}\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ \begin{array}{c}1\\ 1\\ 1\end{array}\end{array}\right]{=1}_{5\mathit{n}}$$

The model in a vector form

$$\mathit{{\rm Y}}=\mathit{X}\mathit{b}+\mathit{e}\iff \mathit{Y}=\left(\begin{array}{cc}{\mathit{X}}_1& {\mathit{X}}_2\end{array}\right)\left(\begin{array}{c}{\mathit{b}}_1\\ {\mathit{b}}_2\end{array}\right)$$

(3)

where Y is (5n) × 1, b is s × 1, X is (5n) × 1, e is (5n) × 1, $\mathit{b}=\left(\begin{array}{cc}{\mathit{b}}_1& {\mathit{b}}_2\end{array}\right)$, where ${\mathit{b}}_1$ is the (vector of) carryover effect, and ${\mathit{b}}_2$ is the remaining parameters.

Proposition 1. 

In the model (1), for every number of periods, the residual effects  ${\delta }_A,{\delta }_B,{\delta }_A+{\delta }_B are not estimable on the contrary estimable is the difference$ ${\delta }_A-{\delta }_B$ with $\mathit{var}({\widehat{\mathit{\delta }}}_{\mathit{A}}-{\widehat{\mathit{\delta }}}_{\mathit{B}})={\sigma }^2{\mathit{G}}^{-1},\mathit{G}={\mathit{\delta }}_{\mathit{A}}^{\mathit{T}}(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{A}}={\mathit{\delta }}_{\mathit{B}}^{\mathit{T}}(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{B}}=-{\mathit{\delta }}_{\mathit{A}}^{\mathit{T}}(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{B}}$.

Proof. 

As it was previously referred to, ${\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{B}}+{\mathit{\pi }}_1=1_{5\mathit{n}}$, then ${\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{B}}$ is a linear combination of the vectors of ${\mathit{X}}_2$ so $({\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{B}})\in R\left({\mathit{X}}_2\right)$ and $(\mathit{I}-\mathit{P})({\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{B}})=0$, then ${\mathit{\delta }}_A^T(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{A}}=-{\mathit{\delta }}_A^{\tau }(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{B}})={\mathit{\delta }}_B^{\tau }(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{B}}=\mathit{G}$.

On the contrary, ${\mathit{\delta }}_{\mathit{A}}-{\mathit{\delta }}_{\mathit{B}}$ does not arise as a linear combination of the remaining columns of the matrix ${\mathit{X}}_2$.

From (3), we have

$$\left[\begin{array}{cc}{\mathit{\delta }}_{\mathit{A}}^{\mathit{T}}(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{A}}& {\mathit{\delta }}_{\mathit{A}}^{\mathit{T}}(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{B}}\\ {\mathit{\delta }}_{\mathit{B}}^{\mathit{T}}(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{A}}& {\mathit{\delta }}_{\mathit{B}}^{\mathit{T}}(\mathit{I}-\mathit{P}){\mathit{\delta }}_{\mathit{B}}\end{array}\right]\left[\begin{array}{c}{\widehat{\mathit{\delta }}}_{\mathit{A}}\\ {\widehat{\mathit{\delta }}}_{\mathit{B}}\end{array}\right]=\left[\begin{array}{c}\mathit{G}({\widehat{\mathit{\delta }}}_{\mathit{A}}-{\widehat{\mathit{\delta }}}_{\mathit{B}})\\ -\mathit{G}({\widehat{\mathit{\delta }}}_{\mathit{A}}-{\widehat{\mathit{\delta }}}_{\mathit{B}})\end{array}\right]=\left[\begin{array}{c}{\mathit{\delta }}_{\mathit{A}}^{\mathit{T}}(\mathit{I}-\mathit{P})\mathit{Y}\\ {\mathit{\delta }}_B^{\mathit{T}}(\mathit{I}-\mathit{P})\mathit{Y}\end{array}\right]$$

and $\mathit{var}({\widehat{\mathit{\delta }}}_{\mathit{A}}-{\widehat{\mathit{\delta }}}_{\mathit{B}})={\sigma }^2{\mathit{G}}^{-1}$. So, the parameters ${\delta }_A,{\delta }_B,{\delta }_A+{\delta }_B$ cannot be estimated; nevertheless, the ${\delta }_A-{\delta }_B$ can be estimated. □

The columns of ${\mathit{X}}_2$ are not linearly independent, as it is shown in the following proposition.

Proposition 2. 

If $\mathit{\mu },{\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3,{\mathit{\pi }}_4,{\mathit{\pi }}_5,{\mathit{\tau }}_{\mathit{A}},{\mathit{\gamma }}_0,{\mathit{\gamma }}_1,\cdots ,{\mathit{\gamma }}_{31}$ corresponds to the parameters $\mu ,{\pi }_1,{\pi }_2,{\pi }_3,{\pi }_4,{\tau }_A,{\gamma }_0,{\gamma }_1,\cdots ,{\gamma }_{31}$, the vectors ${\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3,{\mathit{\pi }}_4,{\mathit{\tau }}_{\mathit{A}},{\mathit{\gamma }}_0,{\mathit{\gamma }}_1,\cdots ,{\mathit{\gamma }}_{31}$ form a basis of the linear space $R\left({\mathit{X}}_2\right)$.

Proof. 

${\mathit{\tau }}_{\mathit{A}} \mathrm{and} {\mathit{\tau }}_{\mathit{B}}$ can be replaced with ${\mathit{\tau }}_{\mathit{A}}+{\mathit{\tau }}_{\mathit{B}},{\mathit{\tau }}_A-{\mathit{\tau }}_B$ without a change of $R\left({\mathit{X}}_2\right)$. Moreover, we observe that $1_{5n}=\mathit{\mu }={\mathit{\tau }}_{\mathit{A}}+{\mathit{\tau }}_{\mathit{B}}$, and ${\mathit{\gamma }}_0+{\mathit{\gamma }}_1+\cdots +{\mathit{\gamma }}_{31}=1_{5\mathit{n}}$, ${\mathit{\pi }}_1+{\mathit{\pi }}_2+{\mathit{\pi }}_3+{\mathit{\pi }}_4+{\mathit{\pi }}_5=1_{5n},$ μ so and ${\mathit{\pi }}_5$ can be omitted. □

Definition 1. 

${\tilde{\mathit{X}}}_2$ is defined as the space of ${\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3,{\mathit{\pi }}_4,{\mathit{\tau }}_{\mathit{A}},{{\mathit{\tau }}_{\mathit{B}},\mathit{\gamma }}_0,{\mathit{\gamma }}_1,\cdots ,{\mathit{\gamma }}_{31}$.

Corollary 1. 

$R\left({\mathit{X}}_2\right)=R\left({\tilde{\mathit{X}}}_2\right)$.

The matrixes ${\mathit{X}}_1\mathsf{\kappa }\mathsf{\alpha }\mathsf{\iota }{\tilde{\mathit{X}}}_2 \mathrm{a}\mathrm{r}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}$:

$${\mathit{X}}_1=\left[\begin{array}{c}{\mathit{X}}_{10}\}{u}_0\\ {\mathit{X}}_{11}\}{u}_1\\ \begin{array}{c}⋮\\ ⋮\\ {\mathit{X}}_{131}\}{u}_{31}\end{array}\end{array}\right], {\tilde{\mathit{X}}}_2=\left[\begin{array}{cccccc}{\tilde{\mathit{X}}}_{20}& 1_5& 0_5& \cdots & 0_5& 0_5\\ {\tilde{\mathit{X}}}_{21}& 0_5& 1_5& \cdots & 0_5& 0_5\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ {\tilde{\mathit{X}}}_{230}& 0_5& 0_5& \cdots & 0_5& 1_5\\ {\tilde{\mathit{X}}}_{231}& 0_5& 0_5& 0_5& 0_5& 0_5\end{array}\right]$$

where the matrixes of ${\mathit{X}}_{1\mathit{i}}$, ${\tilde{\mathit{X}}}_{2\mathit{i}}$ are presented in Appendix A.

2.2. Estimation of the Residual Effects

In order to find the Best Linear Unbiased Estimators for $\delta ={\delta }_A-{\delta }_B$ we have the following:

$$({\mathit{X}}_1^{\mathit{T}}{\mathit{X}}_1-{\mathit{X}}_1^{\mathit{T}}\mathit{P}{\mathit{X}}_1)\left(\delta ={\delta }_A-{\delta }_B\right)={\mathit{X}}_1^{\mathit{T}}(\mathit{I}-\mathit{P})\mathit{Y}$$

(4)

where ${\mathit{X}}_1=\left[\begin{array}{c}\delta ={\delta }_A-{\delta }_B\end{array}\right]$, ${\mathit{X}}_2=\left[\mathit{\mu },{\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3{,\mathit{\pi }}_4,{\mathit{\pi }}_5,{\mathit{\tau }=\mathit{\tau }}_A-{\mathit{\tau }}_B,{\mathit{\gamma }}_1,{\mathit{\gamma }}_2,\cdots ,{\mathit{\gamma }}_{31}\right]$ and $\mathit{P}={\mathit{X}}_2({\mathit{X}}_2^{\mathit{T}}{\mathit{X}}_2{)}^{-1}{\mathit{X}}_2^{\mathit{T}}$ is the $\left(5n\right)\times \left(5n\right)$ projection matrix in $R\left({\mathit{X}}_2\right)$.

Proposition 3. 

$Q={\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\mathit{P}\right){\mathit{X}}_1={\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}\right){\mathit{X}}_1,$ where $\tilde{\mathit{P}}={\tilde{\mathit{X}}}_2({\tilde{\mathit{X}}}_2^{\mathit{T}}{\tilde{\mathit{X}}}_2{)}^{-1}{\tilde{\mathit{X}}}_2^{\mathit{T}}$.

Proof. 

Let $\mathit{P}{\mathit{X}}_1$ denote the orthogonal projection of ${\mathit{X}}_1$ onto the linear space $R\left({\mathit{X}}_2\right)$. Since the column spaces of ${\mathit{X}}_2$ and ${\tilde{\mathit{X}}}_2$ coincide, i.e., $R\left({\mathit{X}}_2\right)=R\left({\tilde{\mathit{X}}}_2\right)$, the corresponding projection matrices are identical $\mathit{P}=\tilde{\mathit{P}}$ when acting on any vector in the space. Consequently, $\mathit{P}{\mathit{X}}_1=\tilde{\mathit{P}}{\mathit{X}}_1$, and it follows that

$${\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\mathit{P}\right){\mathit{X}}_1={\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}\right){\mathit{X}}_1$$

This equality confirms that the residual component of ${\mathit{X}}_1$ relative to the projection on $R\left({\mathit{X}}_2\right)$, remains unchanged under the transformation from ${\mathit{X}}_2$ to ${\tilde{\mathit{X}}}_2$. □

2.3. Calculation of ${\mathit{X}}_1^{\mathit{T}}({\mathit{I}}_5-\tilde{\mathit{P}}){\mathit{X}}_1$

So, we need to calculate the minimum of the ${\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}\right){\mathit{X}}_1.$

Prepositions 4. 

If $\mathit{w}=(x_1,x_2,x_3,x_4,{x_5,z}_0,z_1, \cdots ,z_{31}{)}^T$, then

$${\mathit{X}}_1^{\mathit{T}}({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}){\mathit{X}}_1=\underset{\mathit{w}}{min}({\mathit{X}}_1-{\tilde{\mathit{X}}}_2\mathit{w}{)}^T({\mathit{X}}_1-{\tilde{\mathit{X}}}_2\mathit{w})$$

(5)

Proof. 

For a point T of $\mathit{R}\left({\tilde{\mathit{X}}}_2\right)$ με $T={\tilde{\mathit{X}}}_2\mathit{w}$, the $({\mathit{X}}_1-{\tilde{\mathit{X}}}_2\mathit{w}{)}^{\mathit{T}}({\mathit{X}}_1-{\tilde{\mathit{X}}}_2\mathit{w})$ is the square of the distance of T from the space ${\mathit{X}}_1$, so

$${\mathit{X}}_1^{\mathit{T}}({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}){\mathit{X}}_1=\underset{w}{min}({\mathit{X}}_1-{\mathit{X}}_2\mathit{w}{)}^{\mathit{T}}({\mathit{X}}_1-{\mathit{X}}_2\mathit{w}),$$

□

If $F(\mathit{x},\mathit{z})=({\mathit{X}}_1-{\tilde{\mathit{X}}}_2\mathit{w}{)}^{\mathit{T}}({\mathit{X}}_1-{\tilde{\mathit{X}}}_2\mathit{w})$, then

$$F(\mathit{x},\mathit{z})={\mathit{X}}_1^{\mathit{T}}{\mathit{X}}_1-2{\mathit{X}}_1^{\mathit{T}}{\tilde{\mathit{X}}}_2\mathit{w}+{\mathit{w}}^{\mathit{T}}{\tilde{\mathit{X}}}_2^{\mathit{T}}{\tilde{\mathit{X}}}_2\mathit{w}$$

(6)

where

$${\mathit{X}}_1^{\mathit{T}}{\mathit{X}}_1=u_0{\mathit{X}}_{10}^{\mathit{T}}{\mathit{X}}_{10}+u_1{\mathit{X}}_{11}^{\mathit{T}}{\mathit{X}}_{11}+\cdots +u_{15}{\mathit{X}}_{1,31}^{\mathit{T}}{\mathit{X}}_{1,31}$$

$$\begin{gathered} {\mathit{X}}_1^{\mathit{T}}{\tilde{\mathit{X}}}_2\mathit{w}=u_0{\mathit{X}}_{10}^{\mathit{T}}({\tilde{\mathit{X}}}_{20}+z_0{1}_5)+u_1{\mathit{X}}_{11}^{\mathit{T}}({\tilde{\mathit{X}}}_{21}+z_1{1}_5)+\cdots \\ +u_{31}{\mathit{X}}_{1,31}^{\mathit{T}}({\tilde{\mathit{X}}}_{2,31}+z_{31}{1}_5) \end{gathered}$$

$$\begin{gathered} {\mathit{w}}^{\mathit{T}}{\tilde{\mathit{X}}}_2^{\mathit{T}}{\tilde{\mathit{X}}}_2\mathit{w}={\mathit{u}}_0[{\tilde{\mathit{X}}}_{20}^{\mathit{T}}{\tilde{\mathit{X}}}_{20}+2z_0{\tilde{\mathit{X}}}_{20}^{\mathit{T}}{1}_5+z_0^2{1}_5^{\mathit{T}}{1}_5] \\ +{\mathit{u}}_1[{\tilde{\mathit{X}}}_{21}^{\mathit{T}}{\tilde{\mathit{X}}}_{21}+2z_1{\tilde{\mathit{X}}}_{21}{1}_5+z_1^2{1}_5^{\mathit{T}}{1}_5]+\cdots \\ +{\mathit{u}}_{31}[{\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{\tilde{\mathit{X}}}_{2,31}+2z_{31}{\tilde{\mathit{X}}}_{2,31}{1}_5+z_{31}^2{1}_5^{\mathit{T}}{1}_5] \end{gathered}$$

The verticals are

F_(x,z), is a vector function of x and z and we minimize separately for x and z,

$$\mathit{min}F(\mathit{x},\mathit{z})=\underset{\mathit{x}}{min}(\underset{\mathit{z}}{min}F(\mathit{x},\mathit{z}))$$

(7)

So, we differ partially for z (for $z_0,z_1,\dots ,z_{31})$

$$\begin{array}{ccc}{\displaystyle \frac{\partial F(\mathit{x},\mathit{z})}{\partial z_0}}& =& 2u_0[-{\mathit{X}}_{10}^{\mathit{T}}{1}_5+{\tilde{\mathit{X}}}_{20}^{\mathit{T}}{1}_5+4z_0]=0\\ ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial F(\mathit{x},\mathit{z})}{\partial z_{31}}}& =& 2u_{31}[-{\mathit{X}}_{1,15}^{\mathit{T}}{1}_4+{\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{1}_5+4z_{31}]=0\end{array}$$

So, we have

$$\begin{gathered} F(\mathit{x})=u_0[{\mathit{X}}_{10}^{\mathit{T}}{\mathit{X}}_{10}-2{\mathit{X}}_{10}^{\mathit{T}}{\tilde{\mathit{X}}}_{20}+{\tilde{\mathit{X}}}_{20}^{\mathit{T}}{\mathit{X}}_{20}-{\displaystyle \frac{({\mathit{X}}_{10}^{\mathit{T}}{1}_5-{\tilde{\mathit{X}}}_{20}^{\mathit{T}}{1}_5{)}^2}{5}}]+\cdots \\ +u_{31}[{\mathit{X}}_{1,31}^{\mathit{T}}{\mathit{X}}_{1,31}-2{\mathit{X}}_{1,31}^{\mathit{T}}{\tilde{\mathit{X}}}_{2,31}+{\tilde{\mathit{X}}}_{2,15}^{\mathit{T}}{\mathit{X}}_{2,15}-{\displaystyle \frac{({\mathit{X}}_{1,31}^{\mathit{T}}{1}_4-{\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{1}_4{)}^2}{5}}] \end{gathered}$$

F(x) is of the form $F(\mathit{x})={\displaystyle \frac{1}{4}}[R-2{\tilde{\mathit{q}}}^{\mathit{T}}\mathit{x}+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{M}}\mathit{x}$], where

$$\begin{gathered} {R22=(u_0+u_1+\cdots +u_{31})\mathit{X}}_{1,i}^{\mathit{T}}{\mathit{X}}_{1,i} \\ {\tilde{\mathit{q}}}^T\mathit{x}=u_0[5{\mathit{X}}_{10}^{\mathit{T}}{\tilde{\mathit{X}}}_{20}-({\mathit{X}}_{10}^{\mathit{T}}{1}_5\left)\right({\tilde{\mathit{X}}}_{20}^{\mathit{T}}{1}_5\left)\right] \\ +u_1[5{\mathit{X}}_{11}^{\mathit{T}}{\tilde{\mathit{X}}}_{21}-({\mathit{X}}_{11}^{\mathit{T}}{1}_5\left)\right({\tilde{\mathit{X}}}_{21}^{\mathit{T}}{1}_5\left)\right]+\cdots \\ +u_{31}[5{\mathit{X}}_{1,31}^{\mathit{T}}{\tilde{\mathit{X}}}_{2,31}-({\mathit{X}}_{1,31}^{\mathit{T}}{1}_4\left)\right({\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{1}_5\left)\right] \\ {\mathit{x}}^{\mathit{T}}\tilde{\mathit{M}}\mathit{x}=u_0[5{\tilde{\mathit{X}}}_{20}^{\mathit{T}}{\tilde{\mathit{X}}}_{20}-({\tilde{\mathit{X}}}_{2,0}^{\mathit{T}}{1}_5{)}^2]+u_1[5{\tilde{\mathit{X}}}_{21}^{\mathit{T}}{\tilde{\mathit{X}}}_{21}-\left({\tilde{\mathit{X}}}_{2,1}^{\mathit{T}}{1}_5{)}^2\right]+\cdots \\ +u_{15}[5{\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{\tilde{\mathit{X}}}_{2,31}-({\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{1}_5{)}^2] \end{gathered}$$

(8)

After substituting ${\mathit{X}}_{1\mathit{i}} \mathrm{a}\mathrm{n}\mathrm{d}$ ${\tilde{\mathit{X}}}_{2\mathit{i}}$, we have that $Q=F(\mathit{x})={\displaystyle \frac{1}{20}}[R-{\tilde{\mathit{q}}}^{\mathit{T}}{\tilde{\mathit{M}}}^{-1}\tilde{\mathit{q}}]$.

So,

$$Q={\displaystyle \frac{1}{20}}(R-{\tilde{\mathit{q}}}^{\mathit{T}}{\tilde{\mathit{M}}}^{-1}\tilde{\mathit{q}})$$

(9)

Using the Relations (8) gives R, the vector q, and the matrix M:

$$\begin{gathered} R=20n-16(u_0+u_{15}+u_{16}+u_{31})-4\left[\right(u_1+u_{30})+(u_2+u_{29})+(u_4+u_{27}) \\ +(u_7+u_{24})+(u_8+u_{23})+(u_{11}+u_{20})+(u_{13}+u_{18})+(u_{14}+u_{17})] \end{gathered}$$

q = (q₁, q₂, q₃, q₄, q₅)^T with

$$\begin{gathered} q_1=0 \\ {q}_2=5[-(u_3-u_{28})-(u_5-u_{26})+(u_6-u_{25})-(u_9-u_{22})+(u_{10}-u_{21})+(u_{12}-u_{19}\left)\right] \\ {q}_3=5[-(u_3-u_{28})+(u_5-u_{26})-(u_6-u_{25})+(u_9-u_{22})-(u_{10}-u_{21})+(u_{12}-u_{19}\left)\right] \\ {q}_4=5\left[\right(u_3-u_{28})-(u_5-u_{26})-(u_6-u_{25})+(u_9-u_{22})+(u_{10}-u_{21})-(u_{12}-u_{19}\left)\right] \\ {q}_5=5\left[\right(u_3+u_{28})-(u_5+u_{26})-(u_9+u_{22})-2(u_{10}+u_{21}\left)\right] \end{gathered}$$

$$M=\left[\begin{array}{ccccc}4n& -n& -n& -n& m_{15}\\ -n& 4n& -n& -n& m_{25}\\ -n& -n& 4n& -n& m_{35}\\ -n& -n& -n& 4n& m_{45}\\ m_{15}& m_{25}& m_{35}& m_{45}& 6n\end{array}\right]$$

$$m_{15}=-3(u_3-u_{28})-3(u_5-u_{26})+2(u_6-u_{25})-3(u_9-u_{22})+2(u_{10}-u_{21})+2(u_{12}-u_{19})$$

$$m_{25}=-3(u_3-u_{28})+2(u_5-u_{26})-3(u_6-u_{25})+2(u_9-u_{22})-3(u_{10}-u_{21})+2(u_{12}-u_{19})$$

$$m_{35}=2(u_3-u_{28})-3(u_5-u_{26})-3(u_6-u_{25})+2(u_9-u_{22})+2(u_{10}-u_{21})-3(u_{12}-u_{19})$$

$$m_{45}=2(u_3-u_{28})+2(u_5-u_{26})+2(u_6-u_{25})-3(u_9-u_{22})-3(u_{10}-u_{21})-(u_{12}-u_{19})$$

3. Results

Proposition 5. 

Maximization of Q is when

$${\mathrm{u}}_{\mathrm{i}}=0, \mathrm{i}=\mathrm{0,1,2,4,7,8,11,13,14,15,16,17,18,20,23,24,27,29,30,31}$$

(10)

Proof. 

If the maximum value of $R$ (20n) in formula (10) is attained when all values are equal to zero, ${\mathrm{u}}_{\mathrm{i}}=0$, i = 0, 1, 2, 4, 7, 8, 11, 13, 14, 15, 16, 17, 18, 20, 23, 24, 27, 29, 30, 31 is not valid, then $\underset{{\mathrm{u}}_{\mathrm{i}}}{{\mathrm{Q}}^{\mathrm{*}}=\max \mathrm{Q}}\le {\displaystyle \frac{1}{20}}(20\mathrm{n}-4),$ and the solution is in the next Proposition 6, which gives $Q^{*}>{\displaystyle \frac{1}{20}}(20n-4)$. □

Proposition 6. 

The optimal design for the parameter $({\delta =\delta }_A-{\delta }_B)$ comes up with the maximization of Q, which results as follows:

(α) n = 0 mod 2

Q* = n and all the optimal solutions satisfy the following relations:

$$\{\begin{array}{c}{q}_1=q_2=q_3=q_4=q_5=0\\ (u_3+u_{28})+(u_5+u_{26})+(u_6+u_{25})+(u_9+u_{22})+(u_{10}+u_{21})+(u_{12}+u_{19})=n\end{array}$$

(11)

Three of the solutions are as follows:

$$\mathrm{i}) {\mathrm{u}}_6={\mathrm{u}}_{25}=\mathrm{a},{\mathrm{u}}_{12}={\mathrm{u}}_{19}=\mathrm{b},\mathrm{a}\in \{\mathrm{0,1},\dots ,{\displaystyle \frac{\mathrm{n}}{2}}\},\mathrm{b}={\displaystyle \frac{\mathrm{n}}{2}}-\mathrm{a},{\mathrm{u}}_{\mathrm{i}}=0\mathrm{i}\ne \mathrm{6,12,19,25}$$

$$\begin{gathered} \mathrm{ii}) {\mathrm{u}}_3={\mathrm{u}}_{28}={\displaystyle \frac{\mathrm{n}}{4}},{\mathrm{u}}_5={\mathrm{u}}_{26}=\mathrm{b},{\mathrm{u}}_9={\mathrm{u}}_{22}=\mathrm{c},\mathrm{b}\in \left\{\mathrm{0,1},\cdots ,{\displaystyle \frac{\mathrm{n}}{4}}\right\},\mathrm{c}={\displaystyle \frac{\mathrm{n}}{4}}-\mathrm{b}, \\ {\mathrm{u}}_{\mathrm{i}}=0\mathrm{i}\ne \mathrm{3,5},\mathrm{9,22,26,28},\mathrm{n}=0\mathrm{mod}4 \end{gathered}$$

$$\begin{gathered} \mathrm{i}\mathrm{i}\mathrm{i}) {\mathrm{u}}_3={\mathrm{u}}_{28}={\mathrm{u}}_9={\mathrm{u}}_{22}=\mathrm{a},{\mathrm{u}}_6={\mathrm{u}}_{25}=\mathrm{b},{\mathrm{u}}_{12}={\mathrm{u}}_{19}=\mathrm{c}, \\ \mathrm{a}\in \left\{\mathrm{0,1},\cdots ,\left[{\displaystyle \frac{\mathrm{n}}{4}}\right]\right\},\mathrm{b}+\mathrm{c}={\displaystyle \frac{\mathrm{n}}{2}}-2\mathrm{a},{\mathrm{u}}_{\mathrm{i}}=0\mathrm{i}\ne \mathrm{3,6},\mathrm{9,12,19,22,25,28} \end{gathered}$$

(β) n = 1 mod 2

${\mathrm{Q}}^{\mathrm{*}}=\mathrm{n}-{\displaystyle \frac{1}{\mathrm{n}}}$ and all the optimal solutions satisfy the following relations: ${\mathrm{q}}_1=0$ and

$$\{\begin{array}{c}{\mathrm{q}}_2=\pm 5,{\mathrm{q}}_3=\pm 5,{\mathrm{q}}_4=\mp 5,{\mathrm{q}}_5=0,{\mathrm{m}}_{15}=\pm 2,{\mathrm{m}}_{25}=\pm 2,{\mathrm{m}}_{35}=\mp 3,{\mathrm{m}}_{45}=\mp 3\mathrm{o}\mathrm{r}\\ {\mathrm{q}}_2=\pm 5,{\mathrm{q}}_3=\mp 5,{\mathrm{q}}_4=\mp 5,{\mathrm{q}}_5=0,{\mathrm{m}}_{15}=\pm 2,{\mathrm{m}}_{25}=\mp 3,{\mathrm{m}}_{35}=\mp 3,{\mathrm{m}}_{45}=\pm 2\\ ({\mathrm{u}}_3+{\mathrm{u}}_{28})+({\mathrm{u}}_5+{\mathrm{u}}_{26})+({\mathrm{u}}_6+{\mathrm{u}}_{25})+({\mathrm{u}}_9+{\mathrm{u}}_{22})+({\mathrm{u}}_{10}+{\mathrm{u}}_{21})+({\mathrm{u}}_{12}+{\mathrm{u}}_{19})=\mathrm{n}\end{array}$$

(12)

Some solutions which satisfy (12) are as follows:

$${i) u}_6=u_{25}=a,u_{12}=b+1,u_{19}=b,a\in \{\mathrm{0,1},\dots ,{\displaystyle \frac{n-1}{2}}\},b={\displaystyle \frac{n-1}{2}}-a,u_i=0i\ne \mathrm{6,12,19,25}$$

$${ii) u}_6=u_{25}=a,u_{12}=b-1,u_{19}=b,a\in \{1,\dots ,{\displaystyle \frac{n+1}{2}}\},b={\displaystyle \frac{n+1}{2}}-a,u_i=0i\ne \mathrm{6,12,19,25}$$

$${iii) u}_6=a,u_{25}=a+1,u_{12}=u_{19}=b,a\in \{\mathrm{0,1},\dots ,{\displaystyle \frac{n-1}{2}}\},b={\displaystyle \frac{n-1}{2}}-a,u_i=0i\ne \mathrm{6,12,19,25}$$

$${iv) u}_6=a,u_{25}=a-1,u_{12}=u_{19}=b,a\in \{1,\dots ,{\displaystyle \frac{n+1}{2}}\},b={\displaystyle \frac{n-1}{2}}-a,u_i=0i\ne \mathrm{6,12,19,25}$$

Proof. 

(a) For n = 0mod 2, the maximum value of Q is attained when $q_1=q_2=q_3=q_4=0$, $q_5=0.$ For n observations, we then have Q* = max Q = n. Solutions (i), (ii), and (iii) satisfy the conditions of (11). (b) In the case where n = 1mod 2, if an odd number of ${\mathrm{u}}_{\mathrm{i}}+{\mathrm{u}}_{31-\mathrm{i}}\mathrm{i}=\mathrm{3,5,6,9,10,12}$ are odd numbers, and, as a result, an odd number of ${\mathrm{u}}_{\mathrm{i}}-{\mathrm{u}}_{31-\mathrm{i}}$ is odd because $({\mathrm{u}}_3+{\mathrm{u}}_{28})+({\mathrm{u}}_5+{\mathrm{u}}_{26})+({\mathrm{u}}_6+{\mathrm{u}}_{25})+({\mathrm{u}}_9+{\mathrm{u}}_{22})+({\mathrm{u}}_{10}+{\mathrm{u}}_{21})+({\mathrm{u}}_{12}+{\mathrm{u}}_{19})=\mathrm{n}$.

As a result, ${\mathrm{q}}_1=0$ and the ${\mathrm{q}}_2, {\mathrm{q}}_3, {\mathrm{q}}_4$ are multiples of 5 and odds. In that case, we have $\min [{\mathrm{q}}_1^2+{\mathrm{q}}_2^2+{\mathrm{q}}_3^2+{\mathrm{q}}_4^2+({\mathrm{q}}_1+{\mathrm{q}}_2+{\mathrm{q}}_3+{\mathrm{q}}_4{)}^2]=4\cdot 5^2$.

From (11), we have ${\mathrm{Q}}^{\mathrm{*}}\le {\displaystyle \frac{1}{20}}(20\mathrm{n}-\mathrm{G})={\displaystyle \frac{1}{20}}(20\mathrm{n}-{\displaystyle \frac{20}{\mathrm{n}}})=\mathrm{n}-{\displaystyle \frac{1}{\mathrm{n}}}$.

And the supremum is when the relations (12) are satisfied. □

4. Discussion

In the case of experiments involving five periods and the presence of carryover effects, the optimization process, in all instances, takes into account designs corresponding to $u_i=0$, i = 0, 1, 2, 4, 7, 8, 11, 13, 14, 15, 16, 17, 18, 20, 23, 24, 27, 29, 30. This solution implies that the non-zero values may be $u_3,u_{28},u_5,u_{26},u_6,u_{25},u_9,u_{22},u_{10},u_{21},u_{12},u_{19}$ which correspond to the frequencies of sequences numbered 3, 5, 6, 9, 10, 12, 19, 21, 22, 25, 26, 28. All these sequences include either three treatments of type A and two of type B, or two treatments of type A and three of type B.

The substantive comparison should be made with cases in which the number of periods is less than five, and, in particular, with the corresponding result for an odd number of periods, namely three. As in the three-period case, the optimal designs consist of sequences containing either two treatments of type A and one of type B, or two treatments of type B and one of type A. Analogously, in the present case, the non-zero sequences consist of three treatments of type A and two of type B and vice versa, so that the numbers of treatments A and B differ by one.

Moreover, in the case of three periods, the first two periods contain one treatment A and one treatment B. Analogously, in the case of five periods, all non-zero sequences contain two treatments A and two treatments B in the first four periods, are highlighted in the

Table 2. The 12 non-zero sequences of treatments A, B for the case of 5 periods (are highlighted).

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

A

A

B

B

B

B

A

A

A

A

B

B

B

B

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

$u_0$

$u_1$

$u_2$

$u_3$

$u_4$

$u_5$

$u_6$

$u_7$

$u_8$

$u_9$

$u_{10}$

$u_{11}$

$u_{12}$

$u_{13}$

$u_{14}$

$u_{15}$

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

B

B

A

A

A

A

B

B

B

B

A

A

A

A

B

B

B

B

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

$u_{16}$

$u_{17}$

$u_{18}$

$u_{19}$

$u_{20}$

$u_{21}$

$u_{22}$

$u_{23}$

$u_{24}$

$u_{25}$

$u_{26}$

$u_{27}$

$u_{28}$

$u_{29}$

$u_{30}$

$u_{31}$

.

Similarly, the number of solutions in the three-period case was very limited. In contrast, in the present case, the number of feasible solutions can be determined only through the use of computational methods.

Furthermore, it becomes evident that the number of feasible solutions is considerably higher than in the case of designs with four periods, mainly because the increase in the number of periods leads to a combinatorial expansion of possibilities [8]. These intermediate cases may provide valuable insights toward deriving a general solution applicable to any number of periods p.

Moreover, repeated-measurement designs for models that incorporate interaction effects between periods, including both carryover and direct effects, are of particular interest.