Optimal Designs for Direct Effects: The Case of Two Treatments and Five Periods

Abstract

Cross-Over Designs or Repeated Measurements Designs are experimental designs in which treatments (e.g., medicines, fertilizers, diets) are applied to experimental units (usually humans) in different time periods. A common problem is to find the distribution of n experimental units in order to find the optimal experimental design for the well-known criteria of optimality (A, D, E optimality, etc.). If there is only one parameter of interest, the criterion is the minimization of the variance of the parameter estimator. In this case, a Repeated Measurements Design with one parameter of interest (the direct effect of the treatment) is examined and the distribution of n which minimizes the variance of that parameter is found. The objective of the research is the estimation of the variance of the Ordinal Least-Squares estimators of the Repeated Measurements Design model for two treatments and five periods. Heydayat and Afsarinejad introduced the basic model which is used. The optimal Repeated Measurements Designs are derived for $n$ experimental units. Optimality criterion is the minimization of the variance of the estimated direct effects.

1. Introduction

Cross Over Designs or Repeated Measurements Designs are experimental designs when treatments (e.g., medicines, fertilizers, diets) are applied to experimental units (usually humans) in different time periods. The used notification is $RMD\left(t,n,m\right),$ where t is the number of treatments, $n$ is the number of experimental units (e.u), and m is the number of periods.

A common problem is to find the distribution of n experimental units in order to find the optimal experimental designs for the well known criteria of optimality (A, D, E optimality, etc.). If there is only one parameter of interest the criterion is the minimization of the variance of the estimator of the parameter. In this case, a Repeated Measusurement Design with one parameter of interest (the direct effect of the treatment) is examined and the distribution of n which minimizes the variance of that parameter is found. The objective of the research is the estimation of variance of the OLS estimators of the Repeated Measurement Design’s model for two treatments and five periods. In the majority of cases, the estimation of the treatment parameter applied in the time period considered (the direct effect of the treatment) or the estimation of the treatment parameter applied in the previous period from that which is considered (the residual or carry-over effect of the treatment) are of interest.

Repeated Measurements Designs have advantages and disadvantages. The main advantages are as follows:

• With n experimental units (e.u), nxm observations are available.
• Optimal estimators are derived if the variability within experimental units is less than the variability between experimental units.
• Repeated Measurements Designs are used in clinical trials when the purpose is to improve a disease rather than cure it.

The main disadvantages are as follows:

• Some experimental units are left out of the experiments if their duration is extended too much.
• When the duration of experiments is extended too much, the values of some variables change.

The first Repeated Measurements Design took place in the 19th century in agriculture experiments [1] in which ammonia and potash fertilizers were compared according to the output of cultivation.

In [2], Repeated Measurement Designs in dietetics were used in order to compare four diets (treatments). In that experiment, the author was the first to use a period in order to eliminate the carry-over effect (washout period).

Later, Cohran et al. [3] defined uniform designs and were the first to use carry -ver effects.

Balanced designs were defined in [4], where Williams showed that when the number of treatments is even and t = m, balanced designs are constructed using a Latin square.

One year later, the same author [5] defined second-order carry-over designs. In [6], complete balanced designs were defined.

The first author to present a theoretical model for two treatments with interactions was Balaam [7]. Moreover, three years [8] later, he proposed designs with two periods when the number of e.us was $t^2$, RMD(t, $t^2$, 2).

In 1974 and 1975, Hedayat and Afsarinejad [9,10] introduced the model of direct and carry-over effects that we have used. Optimal designs of direct and carry-over effects can be found in [9,10,11,12,13]. Many papers focus on universally optimal designs (optimal designs for all optimality criteria). In the case of two treatments, some papers for universally optimal designs are available [14,15,16]. Moreover, estimators when the observations are correlated are also of interest [17,18]. For smaller dimensions and in the cases of two, three, and four periods, optimal designs have been found for various cases [14,19,20]. Specifically, for the case of two treatments and two periods for two parameters, optimal designs with direct and carry-over effects were found [14]. Furthermore, in the same paper, for the two parameters simultaneously (direct and carry-over effect), universally optimal designs were derived. For the case of two treatments and three periods for two parameters simultaneously (direct and carry-over effect), Φ optimal designs were found in [19]. In [20], for the case of four periods and the parameter of carry-over effects, optimal designs were found.

In Section 2, the model is presented; in Section 3, the methodology is presented; in Section 4, the optimal designs are shown (the distribution for all the cases of n); and the final section is a discussion of the results.

2. The Model

There are 32 sequencies of treatments A, Β, and the following notification is used:

A	A	A	A	A	$u_0$	A	A	A	A	B	$u_{16}$
B	A	A	A	A	$u_1$	B	A	A	A	B	$u_{17}$
…
B	B	B	B	A	$u_{15}$	B	B	B	B	B	$u_{31}$

Binary system is used in order to enumerate the sequences, corresponding to 1 for treatment B and 0 for A, and is exposed to j − 1 (for j-th period). For example, for the 21st sequence, ΒAΒAΒ is ($2^0+2^2+2^4=21$).

The notification of $u_i,i=0,1,\dots ,31$ is the number of experimental units that received the i-th sequence of treatments, so $u_0+u_1\cdots +u_{31}=n$, and all e.us are n.

The model is analogous to the model of Hedayat and Afsarinejad [9,10]:

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

(1)

μ is the mean of the model;

j corresponds to the j-th period, j = 1, 2, 3, 4, 5;

i refers to the i-th sequence, $i=0, 1,\dots 31$;

k refers to the unit k = 1, 2, … n;

h = A, B ${\tau }_A,{\tau }_{{\rm B}}$ are direct effects of treatments A and Β;

${\pi }_j$: is the effect of the j-th period;

${\delta }_{i,j-1}$i refers to the treatment of the i-th sequence, $i=0, 1,\dots 31, \mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h} \mathrm{i}\mathrm{s} \mathrm{a}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} ($j − 1)th period, j = 1, 2, 3, 4, 5 (the values are either ${\delta }_A,{\delta }_B$) (the residual effects of A and B);

${\gamma }_i$: is the effect of the i-th sequence;

$e_{ijk}$: independent of the errors normally distributed.

The errors $e_{ijk}$ are assumed to be independent between sequences and within sequences.

$\mathrm{A}\mathrm{s} y_{ijk}$ and the errors $e_{ijk}$ of the model are continuous variables which follow the normal distribution, all the parameters of the model, the mean, direct effect of the treatment, the period effect, and the residual and sequence effect $(\mu ,{\tau }_h,{\pi }_j,{\delta }_{i,j-1} \mathrm{a}\mathrm{n}\mathrm{d} {\gamma }_i$) are real numbers.

The above model (in overparameterized form) is written as

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

(2)

and $\mathit{{\rm Y}},\mathit{\mu },{\mathit{\tau }}_A,{\mathit{\tau }}_{{\rm B}},{\mathit{\delta }}_A,{\mathit{\delta }}_{{\rm B}},{\mathit{\pi }}_1,\cdots {\mathit{\pi }}_5,{\mathit{\gamma }}_0,\cdots {\mathit{\gamma }}_{31},\mathit{e}$ are $5n\times 1$ vectors; the direct effect vector is 1 if the treatment is A, and zero if it is B. For example, for the sequence ABB…, ${\tau }_A=\left[\begin{array}{c}1\\ 0\\ 0\\ ⋮\end{array}\right]$ analogous ${\tau }_B=\left[\begin{array}{c}0\\ 1\\ 1\\ ⋮\end{array}\right]$, ${\delta }_A=\left[\begin{array}{c}0\\ 1\\ 0\\ \begin{array}{c}⋮\end{array}\end{array}\right]$ and, in the same way, ${\mathit{\delta }}_{\mathit{{\rm B}}},{\mathit{\pi }}_i, \mathrm{and}{\mathsf{\gamma }}_i$ are defined so that ${\mathit{\tau }}_{\mathit{A}}+{\mathit{\tau }}_{\mathit{{\rm B}}}=1_{5\mathit{n}}$, ${\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{{\rm B}}}+{\mathit{\pi }}_1=1_{5\mathit{n}}$, and ${\mathit{\pi }}_1+{\mathit{\pi }}_2+{\mathit{\pi }}_3+{\mathit{\pi }}_4+{\mathit{\pi }}_5=1_{5\mathit{n}}$. Also, 1 is used when the ith unit is employed, and 0 is used elsewhere, so ${\mathit{\gamma }}_0+{\mathit{\gamma }}_1+{\mathit{\gamma }}_2+\dots +{\mathit{\gamma }}_{31}=1_{5\mathit{n}}$. So, in Equation (2) there are linearly dependent vectors.

• In this model, in order to eliminate the second-order carry-over effect, if it is necessary, an extension of the time between the periods is applied (the washout period).
• An interesting idea was proposed by Freedman: in order to have a carry-over effect from the first period, a measurement before the first period is defined. This measurement is called the baseline measurement [21].
• In order to face the same problem (to have a carry-over effect from the first period), cyclic experimental designs are used where the first period comes after the last [22].
• Some models follow the idea of Fleis [23], who proposed that for the case of two treatments in the sequences AA, AΒ, the carry-over effect of A is not the same for both sequences.

In a vector form:

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

(3)

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

In our case, r = 1 and it is referred to as the parameter of interest for the difference of the direct effects, $\tau ={\tau }_A-{\tau }_{{\rm B}}$, which is an estimable parameter as shown in Proposition 2.

Proposition 1. 

If $\mathit{\mu },{\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3,{\mathit{\pi }}_4,{\mathit{\pi }}_5,{\mathit{\delta }}_{\mathit{A}},{\mathit{\delta }}_{\mathit{{\rm B}}},{\mathit{\gamma }}_0,{\mathit{\gamma }}_1,\dots ,{\mathit{\gamma }}_{31}$ are the columns of ${\mathit{{\rm X}}}_2$ which correspond to the parameters $\mathit{\mu },{\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3,{\mathit{\pi }}_4,{\mathit{\pi }}_5,{\delta }_{\mathit{A}},{\delta }_{\mathit{{\rm B}}},{\mathit{\gamma }}_0,{\mathit{\gamma }}_1,\dots ,{\mathit{\gamma }}_{31}$, the vectors, ${\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3,{\mathit{\pi }}_4,\mathit{\delta }={\mathit{\delta }}_{\mathit{A}}-{\mathit{\delta }}_{\mathit{{\rm B}}},{\mathit{\gamma }}_0,{\mathit{\gamma }}_1,\dots ,{\mathit{\gamma }}_{31}$, are basis of the linear space produced from the columns of ${\mathit{{\rm X}}}_2$ and notified as $R\left({\mathit{X}}_2\right)$.

Proof. 

If we replace ${\mathsf{\tau }}_{\mathsf{A}},{\mathsf{\tau }}_{\mathsf{{\rm B}}}$ with ${\mathsf{\tau }}_{\mathsf{A}}+{\mathsf{\tau }}_{\mathsf{{\rm B}}},{\mathsf{\tau }}_{\mathbf{A}}-{\mathsf{\tau }}_{\mathbf{B}}$, the linear space $\mathbf{R}\left({\mathbf{X}}_2\right)$ does not change. Moreover, $1_{5n}=\mathit{\mu }={\mathit{\tau }}_{\mathit{A}}+{\mathit{\tau }}_{\mathit{{\rm B}}}$, and ${\mathit{\gamma }}_0+{\mathit{\gamma }}_1+\cdots +{\mathit{\gamma }}_{31}=1_{5\mathit{n}}$, so μ can be omitted and ${\mathit{\delta }}_{\mathit{A}}+{\mathit{\delta }}_{\mathit{{\rm B}}}$ can be replaced with ${\mathit{\gamma }}_{31}$. □

Corollary 1. 

If with ${\tilde{\mathit{X}}}_2$ we note the space of ${\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3,{\mathit{\pi }}_4,\mathit{\delta }={\mathit{\delta }}_{\mathit{A}}-{\mathit{\delta }}_{\mathit{{\rm B}}},{\mathit{\gamma }}_0,{\mathit{\gamma }}_1,\dots ,{\mathit{\gamma }}_{31}$, then $R\left({\mathit{X}}_2\right)=R\left({\tilde{\mathit{X}}}_2\right)$.

$${\tilde{\mathit{X}}}_1=\left[\begin{array}{c}\begin{array}{c}{\mathit{X}}_{10}\}{u}_0\\ {\mathit{X}}_{11}\}{u}_1\end{array}\\ ⋮\\ {\mathit{X}}_{131}\}{u}_{31}\end{array}\right]$$

$${\tilde{\mathit{X}}}_2=\left[\begin{array}{cccccc}{\mathit{X}}_{20}& 1_5& 0& \cdots & 0& 0\\ {\mathit{X}}_{21}& 0& 1_5& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ {\mathit{X}}_{230}& 0& 0& \cdots & 0& 1_5\\ {\mathit{X}}_{231}& 0& 0& 0& 0& 0\end{array}\right]$$

$X_{1i}$,$X_{2i}$ are columns of the design matrix X and their elements are indicators of 1 when the effect exists and 0 when it does not exist.

Matrices ${\tilde{\mathit{X}}}_{1\mathit{i}},{\tilde{\mathit{X}}}_{2\mathit{i}},i=0,1,2\dots 31$

With ${\tilde{\mathit{X}}}_{1\mathit{i}}$, i corresponds to the i-th sequence, so for i = 0 for the sequence AAAAA, ${\tilde{\mathit{X}}}_{10}$ is defined as ${\tilde{X}}_{10}=\left[\begin{array}{l}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]$ in the same way:

$${\tilde{X}}_{11}=\left[\begin{array}{l}0\\ 1\\ 1\\ 1\\ 1\end{array}\right], {\tilde{X}}_{12}=\left[\begin{array}{l}1\\ 0\\ 1\\ 1\\ 1\end{array}\right], {\tilde{X}}_{13}=\left[\begin{array}{l}0\\ 0\\ 1\\ 1\\ 1\end{array}\right], {\tilde{X}}_{14}=\left[\begin{array}{l}1\\ 1\\ 0\\ 1\\ 1\end{array}\right], {\tilde{X}}_{15}=\left[\begin{array}{l}0\\ 1\\ 0\\ 1\\ 1\end{array}\right], {\tilde{X}}_{16}=\left[\begin{array}{l}1\\ 0\\ 0\\ 1\\ 1\end{array}\right], {\tilde{X}}_{17}=\left[\begin{array}{l}0\\ 0\\ 0\\ 1\\ 1\end{array}\right],$$

$${\tilde{X}}_{1,31-i}=1_5-{\tilde{X}}_{1i},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{X}}_{1,16+i}={\tilde{X}}_{1i}-\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 1\end{array}\right]\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{\tilde{X}}_{1,15-i}=\left[\begin{array}{l}1\\ 1\\ 1\\ 1\\ 2\end{array}\right]-{\tilde{X}}_{1i}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=0,\hspace{0.17em}1,\dots ,7$$

and ${\tilde{\mathit{X}}}_{2\mathit{i}}$ are the vectors

$${\tilde{X}}_{2i}={\tilde{X}}_{2,16+i}=\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\\ x_4\\ 0\end{array}\right]+x_5\cdot {\tilde{X}}_{22i},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{X}}_{2,15-i}={\tilde{X}}_{2,31-i}=\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\\ x_4\\ 0\end{array}\right]-x_5\cdot {\tilde{X}}_{22i},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=0,\hspace{0.17em}1,\dots ,7$$

$${\tilde{\mathit{X}}}_{220}=\left[\begin{array}{c}0\\ \begin{array}{c}1\\ 1\\ \begin{array}{c}1\\ 1\end{array}\end{array}\end{array}\right],{\tilde{\mathit{X}}}_{221}=\left[\begin{array}{c}0\\ \begin{array}{c}-1\\ 1\\ \begin{array}{c}1\\ 1\end{array}\end{array}\end{array}\right],{\tilde{\mathit{X}}}_{222}=\left[\begin{array}{c}0\\ \begin{array}{c}1\\ -1\\ \begin{array}{c}1\\ 1\end{array}\end{array}\end{array}\right],{\tilde{\mathit{X}}}_{222}=\left[\begin{array}{c}0\\ \begin{array}{c}-1\\ -1\\ \begin{array}{c}1\\ 1\end{array}\end{array}\end{array}\right]$$

$${\tilde{\mathit{X}}}_{224}=\left[\begin{array}{c}0\\ \begin{array}{c}1\\ 1\\ \begin{array}{c}-1\\ 1\end{array}\end{array}\end{array}\right],{\tilde{\mathit{X}}}_{225}=\left[\begin{array}{c}0\\ \begin{array}{c}-1\\ 1\\ \begin{array}{c}-1\\ 1\end{array}\end{array}\end{array}\right],{\tilde{\mathit{X}}}_{226}=\left[\begin{array}{c}0\\ \begin{array}{c}1\\ -1\\ \begin{array}{c}-1\\ 1\end{array}\end{array}\end{array}\right],{\tilde{\mathit{X}}}_{227}=\left[\begin{array}{c}0\\ \begin{array}{c}-1\\ -1\\ \begin{array}{c}-1\\ 1\end{array}\end{array}\end{array}\right].$$

3. Methodology

3.1. Estimation of the Direct Effects

As referred to in the previous paragraph, our purpose is the estimation of variance of the unknown parameter (direct effect). In order to find the variance, the formula of the OLS estimators was used. The variance is a product of the matrixes ${\mathit{X}}_1,{\mathit{X}}_2$ and the projection matrix P of $R\left({\mathit{X}}_2\right)$. It is proposed to find variance as a distance of the vector spaces $R\left({\mathit{X}}_1\right)$ and $R\left({\mathit{X}}_2\right)$. This method has been applied in the previous work [14] in the case of four periods and carry-over effects.

Using the Ordinary Least-Squares Estimators which are also the Best Linear Unbiased Estimators (BLUE) of τ = ${\mathit{\tau }}_{\mathit{A}}$ [13], the model is:

$$({\mathit{{\rm X}}}_1^{\mathit{{\rm T}}}{\mathit{{\rm X}}}_1-{\mathit{{\rm X}}}_1^{\mathit{{\rm T}}}\mathit{{\rm P}}{\mathit{{\rm X}}}_1)\left({\mathit{\tau }}_{\mathit{A}}\right)={\mathit{X}}_1^{\mathit{T}}(\mathit{I}-\mathit{P})\mathit{Y}$$

(4)

where ${\mathit{X}}_1=\left[\begin{array}{c}{\mathit{\tau }}_{\mathit{A}}\end{array}\right]$, ${\mathit{X}}_2=\left[{\mathit{\pi }}_1,{\mathit{\pi }}_2,{\mathit{\pi }}_3{,\mathit{\pi }}_4,{\mathit{\delta }}_A-{\mathit{\delta }}_{{\rm B}},{\mathit{\gamma }}_1,{\mathit{\gamma }}_2,\dots ,{\mathit{\gamma }}_{31}\right]$ 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 $R\left({\mathit{X}}_2\right)$.

$$\mathit{var}({\mathit{\tau }}_{\mathit{A}})={\sigma }^2{Q}^{-1},Q={\mathit{\tau }}_{\mathit{A}}^{\mathit{{\rm T}}}({\mathit{I}}_{5\mathit{n}}-\mathit{P}){\mathit{\tau }}_{\mathit{A}}={\mathit{X}}_1^{\mathit{T}}({\mathit{I}}_{5\mathit{n}}-\mathit{P}){\mathit{X}}_1$$

So, the goal is the estimation of ${\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\mathit{P}\right){\mathit{X}}_1$.

Proposition 2. 

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

$\mathit{P}{\mathit{X}}_1$ is the orthogonal projection of ${\mathit{X}}_1$ to the linear space of $\mathit{R}\left({\mathit{X}}_2\right)$. Nevertheless, $R\left({\mathit{X}}_2\right)=R\left({\tilde{\mathit{X}}}_2\right)$, so $\mathit{P}{\mathit{X}}_1=\tilde{\mathit{P}}{\mathit{X}}_1$ and ${\mathit{X}}_1^{\mathit{T}}({\mathit{I}}_{5\mathit{n}}-\mathit{P}){\mathit{X}}_1={\mathit{X}}_1^{\mathit{T}}({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}){\mathit{X}}_1$. □

3.2. Calculation of $X_1^T(I_5-\tilde{P})X_1$

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

As $\tilde{\mathit{P}}$ is the projection matrix of $R\left({\tilde{\mathit{X}}}_2\right)$, then ${\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}\right){\mathit{X}}_1$ is the distance of the two spaces $R\left({\mathit{X}}_1\right)$ and ${R(\mathit{X}}_2)$. That distance is estimated in Proposition 3.

Proposition 3. 

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

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

Proof. 

$Q={\mathit{X}}_1^{\mathit{T}}\left({\mathit{I}}_{5\mathit{n}}-\tilde{\mathit{P}}\right){\mathit{X}}_1 \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e} R\left({\mathit{X}}_2\right)(as\mathit{P}={\mathit{X}}_2({\mathit{X}}_2^{\mathit{T}}{\mathit{X}}_2{)}^{-1}{\mathit{X}}_2^{\mathit{T}}$ is the $\left(5n\right)\times \left(5n\right)$ projection matrix in $R\left({\mathit{X}}_2\right)) \mathrm{from} {\mathit{X}}_1$. For a point Τ of $\mathit{R}\left({\tilde{\mathit{X}}}_2\right)$ $T={\tilde{\mathit{X}}}_2\mathit{w}$, 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 $R\left({\mathit{X}}_1\right)$, 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

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

where

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

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

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

In order to find the minimum of $F(x,z)$, minimization for x and z is carried out separately.

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

So after partial differentiation for z (for $z_0,z_1,...,z_{31})$ and then for x

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

$F\left(\mathit{x}\right)$ is a function of $u_i$:

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

F(x) is of the form $F\left(\mathit{x}\right)=\frac{1}{5}[R-2{\tilde{\mathit{q}}}^{\mathit{T}}\mathit{x}+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{M}}\mathit{x}$], where

$$\begin{gathered} {\tilde{\mathit{q}}}^T\mathit{x}=u_0[4{\mathit{X}}_{10}^{\mathit{T}}{\tilde{\mathit{X}}}_{20}-({\mathit{X}}_{10}^{\mathit{T}}{1}_5\left)\right({\tilde{\mathit{X}}}_{20}^{\mathit{T}}{1}_5\left)\right] \\ +u_1[4{\mathit{X}}_{11}^{\mathit{T}}{\tilde{\mathit{X}}}_{21}-({\mathit{X}}_{11}^{\mathit{T}}{1}_5\left)\right({\tilde{\mathit{X}}}_{21}^{\mathit{T}}{1}_5\left)\right]+\cdots \\ +u_{15}[4{\mathit{X}}_{1,31}^{\mathit{T}}{\tilde{\mathit{X}}}_{2,31}-({\mathit{X}}_{1,31}^{\mathit{T}}{1}_5\left)\right({\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{1}_5\left)\right] \end{gathered}$$

$$\begin{gathered} {\mathit{x}}^{\mathit{T}}\tilde{\mathit{M}}\mathit{x}=u_0[4{\tilde{\mathit{X}}}_{20}^{\mathit{T}}{\tilde{\mathit{X}}}_{20}-({\tilde{\mathit{X}}}_{2,0}^{\mathit{T}}{1}_5{)}^2]+u_1[4{\tilde{\mathit{X}}}_{21}^{\mathit{T}}{\tilde{\mathit{X}}}_{21}-\left({\tilde{\mathit{X}}}_{2,1}^{\mathit{T}}{1}_5{)}^2\right]+\cdots \\ +u_{31}[4{\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{\tilde{\mathit{X}}}_{2,31}-({\tilde{\mathit{X}}}_{2,31}^{\mathit{T}}{1}_5{)}^2] \end{gathered}$$

The equations are a quadratic form of x and the minimization is

$$Q=\underset{x}{min}F\left(\mathit{x}\right)=\frac{1}{5}[R-{\tilde{\mathit{q}}}^{\mathit{T}}{\tilde{\mathit{M}}}^{-1}\tilde{\mathit{q}}]$$

$$Q=\frac{1}{5}(R-{\tilde{\mathit{q}}}^{\mathit{T}}{\tilde{\mathit{M}}}^{-1}\tilde{\mathit{q}})$$

where

$$\begin{array}{l}R=4(u_0+u_{30})+4(u_2+u_{29})+4(u_4+u_{27})+4(u_8+u_{23})+4(u_{16}+u_{15})+6(u_3+u_{28})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+6(u_5+u_{26})+6(u_6+u_{25})+\hspace{0.17em}6(u_7+u_{24})+6(u_9+u_{22})++6(u_{10}+u_{21})+6(u_{11}+u_{20})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+6(u_{12}+u_{19})+6(u_{13}+u_{18})+6(u_{14}+u_{17})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}6n-6\left(u_0+u_{31}\right)-2[(u_1+u_{30})+(u_2+u_{29})+(u_4+u_{27})+(u_8+u_{23})+(u_{16}+u_{15})]\end{array},$$

$$\tilde{\mathit{q}}=({\tilde{q}}_1,{\tilde{q}}_2,{\tilde{q}}_3,{\tilde{q}}_4,{\tilde{q}}_5{)}^T$$

$$\begin{gathered} {\tilde{q}}_1=-4(u_1-u_{30})+(u_2-u_{29})-3(u_3-u_{28})+(u_4-u_{27})-3(u_5-u_{26})+2(u_6-u_{25}) \\ -2(u_7-u_{24})+(u_8-u_{23})-3(u_9-u_{22})+2(u_{10}-u_{21})-2(u_{11}-u_{20})+2(u_{12}-u_{19}) \\ -2(u_{13}-u_{18})+3(u_{14}-u_{17})-(u_{15}-u_{16}) \end{gathered}$$

$$\begin{gathered} {\tilde{q}}_2=(u_1-u_{30})-4(u_2-u_{29})-3(u_3-u_{28})+(u_4-u_{27})+2(u_5-u_{26})-3(u_6-u_{25}) \\ -2(u_7-u_{24})+(u_8-u_{23})+2(u_9-u_{22})-3(u_{10}-u_{21})-2(u_{11}-u_{20})+2(u_{12}-u_{19}) \\ +3(u_{13}-u_{18})-2(u_{14}-u_{17})-(u_{15}-u_{16}) \end{gathered}$$

$$\begin{gathered} {\tilde{q}}_3=(u_1-u_{30})+(u_2-u_{29})+2(u_3-u_{28})-4(u_4-u_{27})-3(u_5-u_{26})-3(u_6-u_{25}) \\ -2(u_7-u_{24})+(u_8-u_{23})+2(u_9-u_{22})+2(u_{10}-u_{21})+3(u_{11}-u_{20})-3(u_{12}-u_{19}) \\ -2(u_{13}-u_{18})-2(u_{14}-u_{17})-(u_{15}-u_{16}) \end{gathered}$$

$$\begin{gathered} {\tilde{q}}_4=(u_1-u_{30})+(u_2-u_{29})+2(u_3-u_{28})+(u_4-u_{27})+2(u_5-u_{26})+2(u_6-u_{25}) \\ +3(u_7-u_{24})-4(u_8-u_{23})-3(u_9-u_{22})-3(u_{10}-u_{21})-2(u_{11}-u_{20})-3(u_{12}-u_{19}) \\ -2(u_{13}-u_{18})-2(u_{14}-u_{17})-(u_{15}-u_{16}) \end{gathered}$$

$$\begin{gathered} {\tilde{q}}_5=2(u_1+u_{30})-3(u_2+u_{29})+5(u_3+u_{28})-3(u_4+u_{27})-5(u_5+u_{26}) \\ +4(u_7+u_{24})-3(u_8+u_{23})-5(u_9+u_{22})-10(u_{10}+u_{21})-6(u_{11}+u_{20}) \\ -6(u_{13}+u_{18})-(u_{14}+u_{17})-(u_{15}+u_{16}) \end{gathered}$$

$$\tilde{M}=\left[\begin{array}{ccccc}4n& -n& -n& -n& {\tilde{m}}_{15}\\ -n& 4n& -n& -n& {\tilde{m}}_{25}\\ -n& -n& 4n& -n& {\tilde{m}}_{35}\\ -n& -n& -n& 4n& {\tilde{m}}_{45}\\ {\tilde{m}}_{15}& {\tilde{m}}_{25}& {\tilde{m}}_{35}& {\tilde{m}}_{45}& {\tilde{m}}_{55}\end{array}\right]$$

$${\tilde{m}}_{15}=2(u_7-u_{24})+2(u_{11}-u_{20})+2(u_{13}-u_{18})+2(u_{14}-u_{17})$$

$$\begin{gathered} {\tilde{m}}_{25}=-5(u_3-u_{28})-5(u_5-u_{26})+5(u_6-u_{25})-3(u_7-u_{24})-5(u_9-u_{22})+5(u_{10}-u_{21}) \\ -3(u_{11}-u_{20})+5(u_{12}-u_{19})-3(u_{13}-u_{18})+7(u_{14}-u_{17}) \end{gathered}$$

$$\begin{gathered} {\tilde{m}}_{35}=-5(u_3-u_{28})+5(u_5-u_{26})-5(u_6-u_{25})-3(u_7-u_{24})+5(u_9-u_{22})-5(u_{10}-u_{21}) \\ -3(u_{11}-u_{20})+5(u_{12}-u_{19})+7(u_{13}-u_{18})-3(u_{14}-u_{17}) \end{gathered}$$

$$\begin{gathered} {\tilde{m}}_{45}=5(u_3-u_{28})-5(u_5-u_{26})-5(u_6-u_{25})-3(u_7-u_{24})+5(u_9-u_{22})+5(u_{10}-u_{21}) \\ +7(u_{11}-u_{20})-5(u_{12}-u_{19})-3(u_{13}-u_{18})-3(u_{14}-u_{17}) \end{gathered}$$

$$\begin{gathered} {\tilde{m}}_{55}=20(u_3+u_{28})+20(u_5+u_{26})+20(u_6+u_{25})+16(u_7+u_{24})+20(u_9+u_{22})+20(u_{10}+u_{21}) \\ +16(u_{11}+u_{20})+20(u_{12}+u_{19})+16(u_{13}+u_{18})+16(u_{14}+u_{17}) \end{gathered}$$

4. Optimal Designs for Direct Effects

The minimization of the variance $\mathit{var}({\widehat{\tau }}_A)$, or equivalently the maximization of Q for $u_0,u_1,\dots ,u_{31}$, is of interest. The following proposition will restrict the area of the demand designs:

Proposition 4. 

The maximum value$Q^{*}$ of Q, for $u_0,u_1,\dots ,u_{31}$ with $u_0+u_1+\cdots +u_{31}=n,u_i\ge 0$ integers, has the precaution of

$$u_i=u_{31-i}=0,i=0, 1, 2, 4, 8, 16$$

(5)

Proof. 

Assuming that $u_i=u_{31-i}=0,i=0, 1, 2, 4, 8, 16$, the minimum value of the quadratic form ${\tilde{\mathit{q}}}^{\mathit{T}}{\tilde{\mathit{M}}}^{-1}\tilde{\mathit{q}}$ belongs to (0, 1), so $\underset{u}{Q^{*}=\mathit{max}Q}>\frac{1}{5}(6n-1)$. In another case, if at least one of the variables of (4) is different to zero, then $\underset{u}{Q^{*}=\mathit{max}Q}\le \frac{1}{5}(6n-1)$. Notice that these variables correspond to 5 or 4 of the same treatments (5 or 4 treatments of A, or 5 or 4 treatments of B). □

Using (4) and replacing $\tilde{\mathit{M}}$, Q can be written in the form

$$Q=\frac{1}{5}(6n-G-\frac{F^2}{D})$$

(6)

$$G=\frac{1}{5n}\left[{\tilde{q}}_1^2+{\tilde{q}}_2^2+{\tilde{q}}_3^2+{\tilde{q}}_4^2+({\tilde{q}}_1+{\tilde{q}}_2+{\tilde{q}}_3+{\tilde{q}}_4{)}^2\right]$$

$$\begin{gathered} F=[{\tilde{q}}_5-\frac{1}{5n}({\tilde{q}}_1{\tilde{m}}_{15}+{\tilde{q}}_2{\tilde{m}}_{25}+{\tilde{q}}_3{\tilde{m}}_{35}+{\tilde{q}}_4{\tilde{m}}_{45}+({\tilde{q}}_1+{\tilde{q}}_2+{\tilde{q}}_3+{\tilde{q}}_4)({\tilde{m}}_{15} \\ +{\tilde{m}}_{25}+{\tilde{m}}_{35}+{\tilde{m}}_{45}\left)\right] \end{gathered}$$

$$D={\tilde{m}}_{55}-\frac{1}{5n}[{\tilde{m}}_{15}^2+{\tilde{m}}_{25}^2+{\tilde{m}}_{35}^2+{\tilde{m}}_{45}^2+({\tilde{m}}_{15}+{\tilde{m}}_{25}+{\tilde{m}}_{35}+{\tilde{m}}_{45}{)}^2]$$

Proposition 5. 

The maximum value $Q^{*}$ of Q is

(1)

If n mod 2 = 0: $Q^{*}=\frac{6n}{5}$

The optimal designs satisfy the formulas

$$\left\{\begin{array}{l}{u}_i=u_{31-i}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=0,\hspace{0.17em}1,\hspace{0.17em}2,\hspace{0.17em}4,\hspace{0.17em}8,\hspace{0.17em}16\\ {\tilde{q}}_i={\tilde{q}}_2\hspace{0.17em}={\tilde{q}}_3={\tilde{q}}_4=\hspace{0.17em}{\tilde{q}}_5=0\\ (u_3+u_{28})+(u_5+u_{26})+(u_6+u_{25})+(u_7+u_{24})+(u_9+u_{22})\\ +(u_{10}+u_{21})+(u_{11}+u_{20})+(u_{12}+u_{19})+(u_{13}+u_{18})+(u_{14}+u_{17})=n\end{array}\right.$$

(7)

Three groups of solutions of (7) are

(i) $u_i=u_{31-i}=0i=0,1,2,4,8,16$, $u_i=u_{31-i}i=3,5,6,7,9,10,11,12,13,14$

$$5u_3-5u_5+4u_7-5u_9-10u_{10}-6u_{11}-6u_{13}-u_{14}=0$$

$$u_3+u_5+u_6+u_7+u_9+u_{10}+u_{11}+u_{12}+u_{13}+u_{14}=\frac{n}{2}$$

(8)

which means that if we have n experimental units, the sequences AAAAA, BAAAA, ABAAA, BAAAA, AABAA, AAABA, and AAAAB must be applied to zero of them ($u_i=u_{31-i}=0i=0,1,2,4,8,16$), $u_3=u_{28}$ means that in the same number of experimental units the sequences BBAAA AABBB must be applied, and in the same way $u_i=u_{31-i}, i=3, 5, 6, 7, 9, 10, 11, 12, 13, 14$.

In the application for n = 12, a solution is $u_3=u_{28}=u_5=u_{26}=3, and all other{u}_i=0$, and for ${n=14 u}_3=u_{28}=1,u_7=u_{24}=u_9=u_{22}=1 and{u}_{14}=4=u_{17}=4$

(ii) $u_6=u_{25}\in \left\{0,1\dots ,\frac{n}{2}\right\},u_{12}=u_{19}=\frac{n}{2}-u_6,u_i=0,i\ne 6, 12, 19, 25$

(iii) $u_3=u_{28}=u_5=u_{26}=\frac{n}{4},n=0\mathit{mod}4,u_i=0,i\ne 3, 5, 26, 28$

(2)

n mod 2 = 1: $Q^{\mathrm{*}}=\frac{1}{5}\left[6n-\frac{1}{5n-2-\frac{1}{n}}\right]$, $n\ge 5$,

The optimal designs satisfy the Formula (9):

$$\left\{\begin{array}{l}{u}_i=u_{31-i}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=0,\hspace{0.17em}1,\hspace{0.17em}2,\hspace{0.17em}4,\hspace{0.17em}8,\hspace{0.17em}16\\ {\tilde{q}}_i={\tilde{q}}_2={\tilde{q}}_3={\tilde{q}}_4=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tilde{q}}_5=-2\\ {\tilde{m}}_{25}={\tilde{m}}_{35}={\tilde{m}}_{45}=\pm 1,\hspace{0.17em}\hspace{0.17em}{\tilde{m}}_{15}=\mp 4\hspace{0.17em}\\ (u_7+u_{24})+(u_{11}+u_{20})+(u_{13}+u_{18})+(u_{14}+u_{17})=2\hspace{0.17em}\\ (u_3+u_{28})+(u_5+u_{26})+(u_6+u_{25})+(u_7+u_{24})+(u_9+u_{22})\\ +(u_{10}+u_{21})+(u_{11}+u_{20})+(u_{12}+u_{19})+(u_{13}+u_{18})+(u_{14}+u_{17})=n\end{array}\right.$$

(9)

Five groups of solutions of (9) are

(i)

$u_{26}=u_{22}=1,u_{25}=u_7=2,u_{12}-1=u_{19}=\frac{n-7}{2},u_i=0$

$$i\ne 7,12,19,22,25,26,n\ge 7$$

(ii)

$u_5=u_{26}-1=0,u_6=u_{25}-2=b,u_{24}=u_7-2=0,u_{22}-1=u_9=0,u_{12}-1=u_{19}=f$

$$\begin{gathered} b\in \left\{0,1,\dots ,\frac{n-7}{2}\right\},f=\frac{n-7}{2}-b,u_i=0,i\ne 5,6,7,9,12,19,22,24,25,26,n\ge 7, \\ \mathrm{b},\mathrm{f} \text{non-negative} \mathrm{integers}. \end{gathered}$$

(iii)

$u_{28}-1=u_3=a,u_{26}-1=u_5=a+1,u_{25}-1=u_6=c,u_7=2,u_{12}=u_{19}=d,$

$$u_i=0i\ne 3,5,6,7,12,19,25,26,28a\in \{\mathrm{0,1},\dots ,[\frac{n-7}{4}\left]\right\},c+d=\frac{n-7}{2}-2a,n\ge 7$$

(iv)

$u_{28}-1=u_3=a,u_{25}=u_6=b,u_{19}-2=u_{12}=c,u_{13}=u_{14}=1,$

$$u_i=0i\ne 3,6,12,13,14,19,25,28c\in \{\mathrm{0,1},\dots ,[\frac{n-5}{2}\left]\right\},a+b=\frac{n-5}{2}-c,n\ge 5$$

(v)

$u_{26}=u_5=a,u_{25}=u_6-2=b,u_9-1=u_{22}=c,u_{19}=u_{12}=d,u_{17}=u_{24}=1,$

$$u_i=0,i\ne 5,6,9,12,17,24,19,22,25,26b\left\{\mathrm{0,1},\dots ,\left[\frac{n-5}{2}\right]\right\},$$

$$a+c+d=\frac{n-5}{2}-b,n\ge 5$$

Proof. 

(a) n mod2 = 0. In this case, assuming that ${\tilde{q}}_1={\tilde{q}}_2={\tilde{q}}_3={\tilde{q}}_4={\tilde{q}}_5=0$ and $u_i=u_{31-i}=0,i=0,1,2,4,8,16,u_0+u_1+\cdots +u_{31}=n$, $Q^{\mathrm{*}}=\frac{6n}{5}$ and there are many solutions which satisfy the Formula (7).

Three groups of solutions which satisfy Equation (7) are the group of Equation (8).

(b) n mod2 = 1. Assuming $u_i=u_{31-i}=0,i=0,1,2,4,8,16$, then

$$(u_3+u_{28})+(u_5+u_{26})+(u_6+u_{25})+(u_7+u_{24})+{\sum }_{i=9}^{14}(u_i+u_{31-i})=n$$

Hence, an odd number of quantities $(u_i+u_{31-i})$ are odds and an odd number of quantities $(u_i-u_{31-i})$ are odds, and therefore ${\tilde{m}}_{25},{\tilde{m}}_{35},{\tilde{m}}_{45}$ are also odds because all the $(u_i-u_{31-i})$ have odd coeficients.

Also, ${\tilde{q}}_1={\tilde{q}}_2={\tilde{q}}_3={\tilde{q}}_4=0$, because $G\ge \frac{10}{5n}=\frac{2}{n}$, and from (6)

$$Q^{*}\le \frac{1}{5}\left[6n-\frac{2}{n}\right]<\frac{1}{5}\left[6n-\frac{1}{5n-2-\frac{1}{n}}\right].$$

The last value of $Q^{*}$ is the solution which is described in (9).

With the usage of the equation,

$$\begin{gathered} \left[{\tilde{q}}_1^2+{\tilde{q}}_2^2+{\tilde{q}}_3^2+{\tilde{q}}_4^2+({\tilde{q}}_1+{\tilde{q}}_2+{\tilde{q}}_3+{\tilde{q}}_4{)}^2\right] \\ =\left(30\right(u_3-u_{28}{)}^2+\cdots +40\left(u_6-u_{25}\right)\left(u_7-u_{24}\right)+\dots +10\left(u_{13}-u_{18}\right)\left(u_{14}-u_{17}\right))\mathit{mod}10=0 \end{gathered}$$

From (5), there is $Q=\frac{1}{5}\left[6n-\frac{F^2}{D}\right]$, the ${\tilde{m}}_{25},{\tilde{m}}_{35},{\tilde{m}}_{45}$ are odds because all the $(u_i-u_{31-i})$ have odd coefficients, and therefore $F=q_5$ and $D\le m_{55}-\frac{1}{5n}\left[16+1+1+1+1\right]=m_{55}-\frac{4}{n}$ because $m_{25},m_{35},m_{45},$ are odds, $m_{15}$ is even, and $(m_{i5}-m_{15})\mathit{mod}5=0,i=2,3,4$, then $m_{25}=m_{35}=m_{45}=\pm 1,m_{15}=\mp 4$.

If $V=(u_7+u_{24})+(u_{11}+u_{20})+(u_{13}+u_{18})+(u_{14}+u_{17})$ then V is even because $m_{15}=\mp 4$. Moreover, $V\ge 2$ because V = 0, then $m_{15}=0$.

Hence, $\frac{F^2}{D}=\frac{q_5^2}{D}\ge \frac{q_5^2}{20n-4V-\frac{4}{n}}\ge \frac{4}{20n-8-\frac{4}{n}}=\frac{1}{5n-2-\frac{1}{n}}$ because $(V+q_5)\mathit{mod}5=0$.

The minimum is when $V=2,q_5=-2$.

There are many solutions which satisfy (9). □

Proposition 6. 

The conjugate of an optimal solution is also an optimal solution.

Proof. 

If conditions are satisfied from a solution they are satisfied from its conjugate solution. □

Observations 1. 

(a) Optimal solutions contain sequences of 3 treatments, A and two B, and vice versa.

(b) For n = 3, the Formula (7) is not valid and the solution found with Mathematica software is $Q^{\mathrm{*}}=\frac{50}{3}<\frac{681}{190}$ with solutions $\left(i\right){ u}_{28}=u_{19}=u_{11}=1,\left(ii\right) u_{28}=u_{19}=u_{13}=1$, $\left(iii\right) u_{25}=u_{26}=u_7=1$, ($iv)u_{25}=u_{14}=u_{19}=1,\left(v\right) u_{22}=u_{25}=u_7=1$ and their conjugates.

5. Discussion

It is obvious that the solutions of this case are much more than the case of four periods or fewer [14,19,20]. Specifically, the number of the optimal designs in the case of five periods is more than ten times larger than the case of four periods [20].

As the problem of finding optimal designs for direct effect in the case of five periods is solved, it is certain that the main goal is the generation of optimal designs for every number of periods m (for models with $m>3$, either m mod 2 = 1 or m mod 2 = 0), but until now there have only been two cases of m being odd ($m=3$ and $m=5$) and two cases of m being even ($m=2$ and $m=4$) [14,20]. A clear conclusion is that optimal designs are constructed with sequences with an equal number of treatments A and B or with sequences in which A and B differ by one (for the case of five periods, $3A$ and $2B$ or $3B$ and $2A$, etc.).

Other issues for the number of periods being $m=5$ include the optimal designs of carry-over effects and the examination of the model with interaction for the estimation of either direct effects or carry-over effects.