Nearest neighbor balanced block designs for autoregressive errors

Abstract

In this paper we study the problem of finding neighbor optimal designs for a general correlation structure. We give universal optimality conditions for nearest-neighbor (NN) balanced block designs when observations on the same block are modeled by an autoregressive AR(m) process with arbitrary order m. The cases $m=1,2$ have been studied by Grondona and Cressie (Sankhyā Indian J Stat Ser A 55(2):267–284, 1993) for AR(2) and by Gill and Shukla (Biometrika 72(3):539–544, 1985a, Commun Stat Theory Methods 14(9):2181–2197, 1985b) and Kunert (Biometrika 74(4):717–724, 1987) for AR(1); we extend these results to the cases $m \ge 3$.

Appendix: Proofs

This appendix begins by the proof of formula (23) on $c=\mathbf{1 }_{k}'{\mathbb {M}}\mathbf{1 }_{k}$ in Proposition 3. To establish this formula we need the (essential) technical Lemma 1 on the sums of entries of a row of matrix ${\mathbb {M}}$ which is in Sect. 6.1 too. It is probably the difficulty in establishing this lemma that has long prevented the generalization to any m of the optimal conditions for the AR(m) process. The next four sections are devoted to the respective proofs of Propositions 3, Theorems 1 and 2 and Proposition 5. We will end in Sect. 6.6 with the proofs of Identities (3), (4), (19) and (20).

1.1 Sum c of entries of matrix ${\mathbb {M}}$, Identity (23)

We want establish the formula (23) of Proposition 3. This formula on c, the sum of entries of ${\mathbb {M}}$, is essentially based on Lemma 1 which gives the sum $p_\ell$ of entries of row $\ell \in {\llbracket 1,k\rrbracket }$; it will be also used to establish Identity (44) in the proof of Proposition 3. We first prove Identity (23) using Lemma 1 that will come after.

By definition, $c =\mathbf{1 }_{k}'{\mathbb {M}}\mathbf{1 }_{k}= \sum _{\ell =1}^{k}\sum _{\ell '=1}^{k }\gamma _{\ell ,\ell '}=2\sum _{\ell =1}^{m}p_{\ell } + \sum _{\ell =m+1}^{k-m}p_{\ell }$. After that, from Formula $p_{\ell }=a_0(a_0-a_\ell )$ of Lemma 1, we find:

$$\begin{aligned} c= & {} 2 a_{0}\sum _{\ell =1}^{m}(a_{0}-a_{\ell }) + \sum _{\ell =m+1}^{k-m}a_0^2 = 2 a_{0}\sum _{\ell =1}^{m}(\theta _{0}+\cdots +\theta _{\ell -1}) + (k-2m)a_0^2 \quad \\= & {} 2a_0 \sum _{\ell =0}^{m-1}(m-\ell )\theta _\ell \; + \; (k-2m)a_0^2 \quad \end{aligned}$$

because in the sum $\sum _{\ell =1}^{m}(\theta _{0}+\cdots +\theta _{\ell -1})$ we count m times $\theta _0$, $m-1$ times $\theta _1$, and so on until only once $\theta _{m-1}$. Thus Formula (23) is proved $\square$

Lemma 1

Assume $k >2m \ge 2$. Let $p_{\ell }=\displaystyle \sum _{\ell '=1}^k\gamma _{\ell ,\ell '}$ be the sum of the entries of row $\ell \in {\llbracket 1,k\rrbracket }$ of matrix ${\mathbb {M}}$, $a_\ell =\displaystyle \sum _{u=\ell }^m\theta _u\;$ for $\ell \le m$ and $a_{\ell }=0\;$ for $\ell >m$. Then:

$$\begin{aligned} p_\ell =a_0(a_0-a_\ell )=(1-\theta _1-\cdots -\theta _m)(1-\theta _1-\cdots -\theta _{\ell -1}) \quad \end{aligned}$$

(28)

for $\ell \in {\llbracket 1,k-m\rrbracket }$ and, as ${\mathbb {M}}$ is symmetric with respect to its second diagonal, $p_\ell =p_{k-\ell +1}\;$ for $\ell \in {\llbracket k-m +1,k\rrbracket }$.

In particular, $p_\ell =p_{m+1}=a_0^{2}\;$ for $\ell \in {\llbracket m+1,k-m\rrbracket }$.

Proof

We consider the matrix ${\mathbb {M}}=(\gamma _{\ell ,\ell ^{\prime }})_{1\le \ell ,\ell ^{\prime }\le k}$ and we would like to express the sum $p_\ell =\sum _{\ell '=1}^k\gamma _{\ell ,\ell '}$ of the entries of row $\ell$ in the form given in Lemma 1. By symmetry of ${\mathbb {M}}$ we can suppose that $\ell \in {\llbracket 1,k-m\rrbracket }$. We write $p_\ell =\alpha _\ell + \beta _\ell$ where $\alpha _\ell = \sum _{\ell '=\ell }^k\gamma _{\ell ,\ell '}$ and $\beta _\ell =\sum _{\ell '=1}^{\ell -1}\gamma _{\ell ,\ell '}$.

First we compute $\alpha _\ell = \sum _{\ell '=\ell }^k\gamma _{\ell ,\ell '}$. From Identity (8) of Proposition 1, we have the following expression of each $\gamma _{\ell ,\ell '}$ for $\ell ' \in {\llbracket \ell ,k\rrbracket }$:

$$\begin{aligned} \gamma _{\ell ,\ell + s} = \sum _{u=0}^{\ell -1} \theta _u\theta _{u+s} \quad \text{ for } s \in {\llbracket 0,k-\ell \rrbracket } \quad . \end{aligned}$$

(29)

Then, as $\alpha _\ell =\sum _{s=0}^{k-\ell } \sum _{u=0}^{\ell -1} \theta _u\theta _{u+s}$, we obtain:

$$\begin{aligned} \alpha _\ell = \sum _{u=0}^{\ell -1} \left( \theta _u \sum _{s=0}^{k-\ell } \theta _{u+s} \right) = \sum _{u=0}^{\ell -1} \left( \theta _u \sum _{b=u}^{k + u-\ell } \theta _{b} \right) = \sum _{u=0}^{\ell -1} \left( \theta _u \sum _{b=u}^{m} \theta _{b} \right) \quad \end{aligned}$$

(30)

because for each $b>m$ we have $\theta _b=0$ and for each $u \in {\llbracket 0,\ell -1\rrbracket }$ we have $k+u-\ell \ge k - \ell \ge k - (k-m)=m$.

Now consider $\beta _\ell =\sum _{\ell '=1}^{\ell -1}\gamma _{\ell ,\ell '} = \sum _{\ell '=1}^{\ell -1}\sum _{u=0}^{\ell ' -1}\theta _u\theta _{u+(\ell -\ell ')}$ and search to establish this formula:

$$\begin{aligned} \beta _\ell =\sum _{a=1}^{\ell -1}\theta _a \sum _{b=0}^{a-1}\theta _b \quad . \end{aligned}$$

(31)

The expression $\beta _\ell = \sum _{\ell '=1}^{\ell -1}\sum _{u=0}^{\ell ' -1}\theta _u\theta _{u+(\ell -\ell ')}$ is a double sum and $\ell$ is fixed. Let us consider the square matrix $B=(b_{u,\ell '})$ of size $\ell -1$ indexed by $\ell ' \in {\llbracket 1,\ell -1\rrbracket }$ for the columns and by $u \in {\llbracket 0,\ell -2\rrbracket }$ for the rows. We define $b_{u,\ell '}$ as follows: $b_{u,\ell '}= \theta _u\theta _{u+(\ell -\ell ')}$ for $u\le \ell '$, otherwise $b_{u,\ell '} = 0$ (B is upper triangular). Note that $\sum _{u=0}^{\ell ' -1}\theta _u\theta _{u+(\ell -\ell ')}$ is both the inner sum of the double sum $\beta _{\ell }$ and the sum of the entries of column $\ell '$; thus the sum of all the entries of B is $\beta _{\ell }$.

To obtain the right member of (31), we will sum the entries for each diagonal of B. As B is upper triangular, each of the sums of the diagonals below the main diagonal is zero; for the $\ell -1$ upper diagonals, let a be in ${\llbracket 1,\ell -1\rrbracket }$; the sum of the entries of the diagonal at distance $\ell -1 -a$ from the main diagonal is $\theta _a \sum _{b=0}^{a-1}\theta _b$. For example, for the main diagonal ($a=\ell -1$ and the distance is 0), the sum of the entries equals $\theta _{\ell -1}(\theta _0+\theta _1+\cdots +\theta _{\ell -2})$; for the diagonal just above the main diagonal ($a=\ell -2$ and the distance is 1), the sum of entries is $\theta _{\ell -2}(\theta _0+\theta _1+\cdots +\theta _{\ell -3})$; the last diagonal is reduced to the only one element $\theta _1\theta _0$ ($a=1$ and the distance is $\ell -2$). Then (31) is proved.

From (30) and (31), we deduce Formula (28) of Lemma 1:

$$\begin{aligned} p_\ell = \alpha _\ell + \beta _\ell = \sum _{u=0}^{\ell -1} \theta _u \sum _{b=0}^{m} \theta _{b} = a_0(a_0 - a_\ell )\quad \end{aligned}$$

with $a_\ell =\sum _{b=\ell }^m\theta _b$ for $\ell \in {\llbracket 1,m\rrbracket }$ and $a_\ell =0$ for $\ell >m$. In particular, for $\ell \in {\llbracket m+1,k-m\rrbracket }$, the formula becomes $p_{\ell }=p_{m+1}=a_0^2=(1-\theta _1-\cdots -\theta _m)^2$. Consequently, Lemma 1 is proved $\square$

1.2 Proof of Proposition 3 on entries of the information matrix

As the design d is fixed in ${\varOmega }_{v, b, k}$, it will be omitted in the indices. In Sect. 6.1, we have already established Identity (23) on c. We still have to establish Identities (21) and (22) about the entries $\sigma ^{2}\mathbf{C }_{j,j}$ and $\sigma ^{2}\mathbf{C }_{j,j'}$ ($j\ne j'$) of the matrix $\sigma ^{2}\mathbf{C }_{d}$. The information matrix is given by Identity (15) rewritten below:

$$\begin{aligned} \sigma ^{2}\mathbf{C }_{d}= \displaystyle \sum _{i=1}^{b} T_{i}'\, {\mathbb {M}} \, T_{i} - c^{-1} \sum _{i=1}^{b} T_{i}'\,{\mathbb {M}} \,\mathbf{1 }_k \mathbf{1 }_k'\, {\mathbb {M}} \, T_{i} = {{\mathcal {A}}} - c^{-1}{{\mathcal {B}}}\quad \end{aligned}$$

(32)

where $T_{i}=(\mathbf{t }_1(i), \ldots , \mathbf{t }_v(i))$, ${{\mathcal {A}}}=\sum _{i=1}^{b} T_{i}'\, {\mathbb {M}} \, T_{i}$ and ${{\mathcal {B}}}=\sum _{i=1}^{b} T_{i}'\,{\mathbb {M}} \, \mathbf{1 }_k \mathbf{1 }_k'\, {\mathbb {M}} \, T_{i}$. The entries $\gamma _{\ell ,\ell '}$ of the matrix ${\mathbb {M}}=\sigma ^{2}V^{-1}$ are described in Proposition 1. We will find:

$$\begin{aligned} \mathbf{C }_{j,j} = \tau - \omega _{j,j} \quad \text { and } \quad \mathbf{C }_{j,j'} = \mu - \omega _{j,j'} \end{aligned}$$

where $\tau$ and $\mu$ come from ${{\mathcal {A}}}$ and $\omega _{j,j}$ and $\omega _{j,j'}$ come from $c^{-1}{{\mathcal {B}}}$. In the following, we will look for formulas on $\tau$, $\mu$, $\omega _{j,j}$ and $\omega _{j,j'}$ by first considering the matrix ${{\mathcal {A}}}$ and then the matrix $c^{-1}{{\mathcal {B}}}$. Before that, we introduce some necessary tools.

Preliminary notations and remarks

For $r\in {\llbracket 1,k\rrbracket }$, ${\varvec{e}}_r=({\varvec{e}}_{r,s})_{1\le s \le k}$ denotes the r-th canonical vector of ${\mathbb {R}}^k$, i.e. ${\varvec{e}}_{r,s}=\delta _{r,s}$ (where $\delta$ is the Kronecker symbol). Note that each entry of a $k\times k$-matrix A is expressed in the form $A_{r,s}={\varvec{e}}_r' \, A \, {\varvec{e}}_{s}$.

For each treatment $j\in {\llbracket 1,v\rrbracket }$, the jth column vector $\mathbf{t }_j(i)$ of the matrix $T_i$ defined in (11) can be expressed as follows: for each $i \in {\llbracket 1,b\rrbracket }$, we set

$$\begin{aligned} \mathbf{t }_j(i) =\left\{ \begin{array}{l} {\varvec{e}}_\ell \quad \text { if } j \text { is applied to } i\text { -th patient at period } \ell \in {\llbracket 1,k\rrbracket } \\ \mathbf{0 }_{k} \quad \text{ otherwise. } \end{array} \right. \end{aligned}$$

(33)

Hence, following notations of Sect. 3.3, for each period $\ell \in {\llbracket 1,k-m\rrbracket }$, we find

$$\begin{aligned} \phi _{j}^\ell =\left\{ \begin{array}{ll} \# \{i : \mathbf{t }_j(i)\in \{{\varvec{e}}_\ell ,{\varvec{e}}_{k-\ell +1} \} \} &{} \quad \text{ if } \ell \in {\llbracket 1,m\rrbracket } \\ \\ \# \{i : \mathbf{t }_j(i)={\varvec{e}}_\ell \} &{} \quad \text{ if } \ell \in {\llbracket m+1,k-m\rrbracket } \quad . \end{array} \right. \end{aligned}$$

(34)

Remark 4

For each treatment j, exactly r vectors $\mathbf{t }_j(i)$ are non-zero because exactly r patients receive the treatment j.

Remark 5

As the designs we consider in this paper are binary, each patient i receives at most one time the same treatment j; consequently, for each $\ell \in {\llbracket 1,m\rrbracket }$ and because $\ell \ne k-\ell +1$ since $k>2m$, we have:

$$\begin{aligned} \{i : \mathbf{t }_j(i)={\varvec{e}}_\ell \}\cap \{i : \mathbf{t }_j(i)= {\varvec{e}}_{k-\ell +1} \}=\emptyset . \end{aligned}$$

Now we fix two distinct treatments $j,j'$ in ${\llbracket 1,v\rrbracket }$. To establish Identities (21) and (22) about $\sigma ^{2}\mathbf{C }_{j,j}$ and $\sigma ^{2}\mathbf{C }_{j,j'}$ of the matrix $\sigma ^{2}\mathbf{C }_{d}$, we will examine separately the contributions of each of the two sums of the right member of (32), namely ${{\mathcal {A}}}$ (for $\tau$ and $\mu$) then ${{\mathcal {B}}}$ (for $\omega _{j,j}$ and $\omega _{j,j'}$). Then we will achieve the proof.

Diagonal entry $\sigma ^{2}\mathbf{C }_{j,j}$ : determination of $\tau$

The contribution of ${{\mathcal {A}}}$ to the diagonal entry $\sigma ^{2}\mathbf{C }_{j,j}$ is the value $\tau =\sum _{i=1}^{b} \tau _{i}$ where $\tau _{i}=\mathbf{t }_j'(i) \, {\mathbb {M}} \, \mathbf{t }_j(i)$. From Definition (33) of vectors $\mathbf{t }_j(i)$, we have for each patient i:

$$\begin{aligned} \tau _{i}= \displaystyle \sum _{\ell =1}^k\sum _{\{i ~:~ \mathbf{t }_j(i)={\varvec{e}}_\ell \}}{\varvec{e}}_\ell ' \, {\mathbb {M}} \, {\varvec{e}}_\ell = \sum _{\ell =1}^m\phi _j^\ell \, {\varvec{e}}_\ell ' \, {\mathbb {M}} \, {\varvec{e}}_\ell + \sum _{\ell =m+1}^{k-m}\phi _j^\ell \, {\varvec{e}}_\ell ' \, {\mathbb {M}} \, {\varvec{e}}_\ell . \end{aligned}$$

(35)

Combining the above identity (35) and Lemma 1 applied to the diagonal entries $\gamma _{\ell ,\ell }={\varvec{e}}_\ell ' \, {\mathbb {M}} \, {\varvec{e}}_\ell$ of matrix ${\mathbb {M}}$, we obtain (recall that $\theta _0=-1$):

$$\begin{aligned} \tau= & {} \sum _{\ell =1}^m \phi _j^\ell \, (\theta _0^2+\theta _1^2\cdots +\theta _{\ell -1}^2)+ \sum _{\ell =m+1}^{k-m}\phi _j^\ell \, (\theta _0^2+\theta _1^2\cdots +\theta _{m}^2)\nonumber \\= & {} \phi _j^1 \, \theta _0^{2}+\phi _j^2 \, (\theta _0^{2}+\theta _1^2)+\cdots + \phi _j^{m} \, (\theta _0^{2}+\theta _1^2+\cdots + \theta _{m-1}^2) \nonumber \\&+ \sum _{\ell =m+1}^{k-m}\phi _j^\ell (\theta _0^{2}+\theta _1^2+\cdots +\theta _{m}^2) \nonumber \\= & {} \theta _0^{2}\sum _{\ell =1}^{k-m}\phi _j^\ell +\theta _1^2\sum _{\ell =2}^{k-m}\phi _j^\ell +\theta _2^2\sum _{\ell =3}^{k-m}\phi _j^\ell + \cdots + \theta _{m}^2\sum _{\ell =m+1}^{k-m}\phi _j^\ell \quad . \end{aligned}$$

(36)

As $\sum _{\ell =1}^{k-m}\phi _{j}^\ell =r$ (see Identity (17)), we get:

$$\begin{aligned} \tau= & {} \theta _0^{2} r + \theta _1^2(r-\phi _j^1) + \theta _2^2(r-(\phi _j^1 + \phi _j^2)) +\cdots + \theta _{m}^2(r-(\phi _j^1+\cdots +\phi _j^{m})) \\= & {} r\sum _{u=0}^m\theta _u^2- \phi _j^1\sum _{u=1}^m\theta _u^2 - \phi _j^2\sum _{u=2}^m\theta _u^2 -\cdots -\phi _j^{\ell }\sum _{u=\ell }^m\theta _u^2- \cdots - \phi _j^{m}\theta _{m}^2 . \end{aligned}$$

Finally, the contribution $\tau$ of the term ${{\mathcal {A}}}$ to the diagonal entry $\sigma ^{2}\mathbf{C }_{j,j}$ is:

$$\begin{aligned} \tau= & {} r b_{0} - \phi _j^1 b_{1} - \phi _j^2 b_{2} - \cdots - \phi _j^{m}b_{m} \end{aligned}$$

(37)

with $b_\ell =\displaystyle \sum _{u=\ell }^m\theta _u^{2}$ for $\ell \in {\llbracket 1,m\rrbracket }$, as defined in Proposition 3.

Extra-diagonal entry $\sigma ^{2}\mathbf{C }_{j,j'}$ : determination of $\mu$

Similarly, let us now focus on the contribution $\mu$ of the sum ${{\mathcal {A}}}=\sum _{i=1}^{b} T_{i}'\, {\mathbb {M}}\, T_{i}$ to the extra-diagonal entry $\sigma ^{2}\mathbf{C }_{j,j'}$, where $\mu =\sum _{i=1}^{b} \mu _{i}$ with $\mu _{i}=\mathbf{t }_j'(i){\mathbb {M}}\mathbf{t }_{j'}(i)$. For this purpose, we need to introduce the following notation: for $\ell ,\ell ' \in {\llbracket 1,k\rrbracket }$, we denote by $\phi _{j,j'}^{\ell ,\ell '}$ the number of patients who receive the distinct treatments j and $j'$ at periods $\ell ,\ell '$:

$$\begin{aligned} \phi _{j,j'}^{\ell ,\ell '}=\# \{i \in {\llbracket 1,b\rrbracket }~:~ \mathbf{t }_j(i)+\mathbf{t }_{j'}(i)={\varvec{e}}_\ell + {\varvec{e}}_{\ell '} \}. \end{aligned}$$

(38)

Note that if $\ell '=\ell$ then $\mathbf{t }_j(i)+\mathbf{t }_{j'}(i)\ne {\varvec{e}}_\ell + {\varvec{e}}_{\ell '}$ for each patient i because the distinct treatments j and $j'$ cannot be applied simultaneously to the same patient i at the same period $\ell$. Hence, for any $s \in {\llbracket 1,k-1\rrbracket }$, we can write:

$$\begin{aligned} N_{j,j'}^s=\sum \phi _{j,j'}^{\ell ,\ell '} \end{aligned}$$

(39)

where the sum involves all the distinct periods $\ell ,\ell '$ in ${\llbracket 1,k\rrbracket }$ and $s=|\ell -\ell '| \ne 0$.

From Definition of vectors $\mathbf{t }_j(i)$, as $j\ne j'$, the only non-zero $\mu _{i}=\mathbf{t }_j'(i){\mathbb {M}}\mathbf{t }_{j'}(i)$ are such that $\mathbf{t }_j(i)+\mathbf{t }_{j'}(i)={\varvec{e}}_\ell + {\varvec{e}}_{\ell '}$ for some periods $\ell$ and $\ell '$ which are necessarily distinct. Moreover, as the matrix ${\mathbb {M}}$ is symmetric, when the identity $\mathbf{t }_j(i)+\mathbf{t }_{j'}(i)={\varvec{e}}_\ell + {\varvec{e}}_{\ell '}$ holds, we can suppose that $\mathbf{t }_j(i)= {\varvec{e}}_\ell$ and $\mathbf{t }_{j'}(i)={\varvec{e}}_{\ell '}$ with $\ell <\ell '$.

Hence, by putting $u_{i,j}=\mathbf{t }_j(i)+\mathbf{t }_{j'}(i)$, $v_{\ell ,\ell '}={\varvec{e}}_\ell + {\varvec{e}}_{\ell '}$ and considering the element $\gamma _{\ell ,\ell '}= {\varvec{e}}_\ell ' \, {\mathbb {M}} \, {\varvec{e}}_{\ell '}$ of the matrix ${\mathbb {M}}$, we obtain:

$$\begin{aligned} \mu = \sum _{1\le \ell<\ell '\le k} \sum _{\{i \, : \, u_{i,j}=v_{\ell ,\ell '}\}}{\varvec{e}}_\ell ' \, {\mathbb {M}} \, {\varvec{e}}_{\ell '} = \sum _{1\le \ell < \ell ' \le k}\gamma _{\ell ,\ell '}\phi _{j,j'}^{\ell ,\ell '}=\sum _{\ell '=2}^{k}\sum _{\ell =1}^{\ell '-1}\gamma _{\ell ,\ell '}\phi _{j,j'}^{\ell ,\ell '} \quad . \end{aligned}$$

For sake of clarity we let $\phi ^{\ell ,\ell '}=\phi _{j,j'}^{\ell ,\ell '}$ for the rest of this proof. We introduce in the expression of $\mu$ the values of the entries $\gamma _{\ell ,\ell '}$ of the matrix ${\mathbb {M}}$ given in Proposition 1. Collecting the factors of each $\theta _\ell$ and $\theta _{\ell }\theta _{\ell '}$, we obtain:

$$\begin{aligned} \mu= & {} -~~ \theta _1\Big (\phi ^{1,2}+\cdots +\phi ^{k-1,k}\Big )-\cdots -\theta _s\Big (\phi ^{\ell ,\ell +s}+\phi ^{\ell +1,\ell +s+1}+\cdots + \phi ^{k-s,k}\Big ) \nonumber \\&-\cdots - \theta _m\Big (\phi ^{1,m+1}+\cdots +\phi ^{k-m,k}\Big ) \nonumber \\&+ \sum _{s=1}^{m-1}\theta _1\theta _{1+s}\Big (\phi ^{2,2+s}+\cdots +\phi ^{k-s-1,k-1}\Big ) \nonumber \\&+~~ \sum _{s=1}^{m-2}\theta _2\theta _{2+s}\Big ( \phi ^{3,3+s}+\cdots +\phi ^{k-s-2,k-2}\Big ) + \cdots \nonumber \\&+~~ \sum _{s=1}^{m-u}\theta _u\theta _{u+s}\Big ( \phi ^{u+1,u+1+s}+ \phi ^{u+2,u+2+s} +\cdots +\phi ^{k-s-u,k-u}\Big ) \nonumber \\&+ \cdots +~~ \sum _{s=1}^{2}\theta _{m-2}\theta _{m-2+s}\Big ( \phi ^{m-1,m-1+s} +\cdots +\phi ^{k-s-(m-2),k-(m-2)}\Big )\nonumber \\&+~~ \theta _{m-1}\theta _{m}\Big ( \phi ^{m,m+1} +\cdots +\phi ^{k-m,k-m+1}\Big ) . \end{aligned}$$

(40)

Recall that Identity (39) says that $N_{j,j'}^s=\phi ^{1,1+s}+\phi ^{2,2+s} +\cdots +\phi ^{k-s-1,k-1}+ \phi ^{k-s,k}$. Putting

$$\begin{aligned} U_{t,s}=\phi ^{t,t+s}+\phi ^{k-t-s+1,k-t+1} \; , \end{aligned}$$

for $s\in {\llbracket 1,m-1\rrbracket }$ and $t\in {\llbracket 1,m-s\rrbracket }$, the expression (40) of $\mu$ becomes:

$$\begin{aligned} \mu= & {} \sum _{s=1}^m\theta _0\theta _sN_{j,j'}^s + \sum _{s=1}^{m-1}\theta _1\theta _{1+s}(N_{j,j'}^s-U_{1,s})\nonumber \\&+ \sum _{s=1}^{m-2}\theta _2\theta _{2+s}(N_{j,j'}^s-(U_{1,s}+U_{2,s}))+ \cdots \nonumber \\&+ \sum _{s=1}^{m-u}\theta _u\theta _{u+s}(N_{j,j'}^s-( U_{1,s}+U_{2,s}+\cdots + U_{u,s})) + \ldots \nonumber \\&+ \sum _{s=1}^{2}\theta _{m-2}\theta _{m-2+s}(N_{j,j'}^s- (U_{1,s}+U_{2,s}+\cdots + U_{m-2,s})) \nonumber \\&+~~\theta _{m-1}\theta _{m}(N_{j,j'}^1-(U_{1,1}+U_{2,1}+\cdots + U_{m-1,1})). \end{aligned}$$

(41)

Collecting the factors of each $N_{j,j'}^s$ and each $U_{t,s}$, we obtain:

$$\begin{aligned} \mu = \sum _{s=1}^{m}N_{j,j'}^s\sum _{u=0}^{m-s}\theta _u\theta _{u+s} -\sum _{s=1}^{m-1}\sum _{t=1}^{m-s}U_{t,s}\sum _{u=t}^{m-s}\theta _u\theta _{u+s} \quad . \end{aligned}$$

Indeed, for each $s\in {\llbracket 1,m-1\rrbracket }$ and $t\in {\llbracket 1,m-s\rrbracket }$, the component $\beta _{t,s}$ of $\mu$ which collects the terms $U_{t,s}\theta _{a,b}$ is the following:

$$\begin{aligned} \beta _{t,s}=-U_{t,s}(\theta _t\theta _{t+s}+\theta _{t+1}\theta _{t+1+s}+\cdots +\theta _{m-s}\theta _m) \; . \end{aligned}$$

In addition, the double sum $\displaystyle \sum _{s=1}^{m-1}\sum _{t=1}^{m-s}\beta _{t,s}$ collects all the terms of the form $U_{t,s}\theta _{a,b}$ of the right-hand side of Identity (41). In order to complete the determination of $\mu$, note that:

Remark 6

We have $\phi _{j,i}^\ell = \delta _{j, d(i,\ell )} + \delta _{j, d(i,k-\ell +1)}$ and $N_{j,j',i}^s \in \{0,1\}$ because d is binary (see Sect. 3.3); then, from Identity (38) about $\phi _{j,j'}^{\ell ,\ell '}$, we find:

$$\begin{aligned} U_{t,s}= & {} \phi _{j,j'}^{t,t+s}+\phi _{j,j'}^{k-t-s+1,k-t+1}\\= & {} \#\Big \{i: \mathbf{t }_j(i)+\mathbf{t }_{j'}(i)\in \{{\varvec{e}}_t+{\varvec{e}}_{t+s}, {\varvec{e}}_{k-t+1}+{\varvec{e}}_{k-(t+s)+1} \} \Big \}\\= & {} \sum _{i=1}^{b}N_{j,j',i}^s(\phi _{j,i}^t\phi _{j',i}^{t+s}+ \phi _{j',i}^t\phi _{j,i}^{t+s}). \end{aligned}$$

Finally, the contribution $\mu$ of the term ${{\mathcal {A}}}=\sum _{i=1}^{b} T_{i}'\, {\mathbb {M}}\, T_{i}$ to the entry $\sigma ^{2}\mathbf{C }_{j,j'}$ is:

$$\begin{aligned} \mu= & {} \sum _{s=1}^{m}N_{j,j'}^s {\varTheta }_{0,s} - \sum _{s=1}^{m-1}\sum _{t=1}^{m-s} {\varTheta }_{t,s} \sum _{i=1}^{b}N_{j,j',i}^s(\phi _{j,i}^t\phi _{j',i}^{t+s}+ \phi _{j',i}^t\phi _{j,i}^{t+s}) \quad \end{aligned}$$

(42)

where ${\varTheta }_{t,s}=\theta _t\theta _{t+s}+\theta _{t+1}\theta _{t+1+s}+\cdots +\theta _{m-s}\theta _m$.

Introduce the following notation $\kappa _{j_{1},i}$ for some treatment $j_{1}$ and some patient i:

$$\begin{aligned} \kappa _{j_{1},i} = \mathbf{t }_{j_{1}}'(i) \,{\mathbb {M}} \, \mathbf{1 }_k. \end{aligned}$$

As $\kappa _{j_{1},i}$ is a scalar and ${\mathbb {M}}={\mathbb {M}}'$ (i.e. ${\mathbb {M}}$ is symmetric), we also have:

$$\begin{aligned} \kappa _{j_{1},i} = \mathbf{1 }_k'\,{\mathbb {M}} \, \mathbf{t }_{j_{1}}'(i)\quad . \end{aligned}$$

Then the contribution of $c^{-1}{{\mathcal {B}}}$ to the entry $\sigma ^{2}\mathbf{C }_{j_{1},j_{2}}$ for two treatments $j_{1},j_{2}$, not necessarily distinct, is

$$\begin{aligned} \omega _{j_{1},j_{2}}=c^{-1}\sum _{i=1}^{b} \kappa _{j_{1},i} \kappa _{j_{2},i}. \end{aligned}$$

(43)

In the following, we determine the quantities $\kappa _{j,i}=\mathbf{t }_j'(i)\, {\mathbb {M}} \, \mathbf{1 }_k$ to find $\omega _{j_{1},j_{2}}$.

When the treatment j is not applied to the ith patient, $\kappa _{j,i}=0$ because $\mathbf{t }_j(i)=\mathbf{0 }_{k}$. Otherwise, it is applied only once, at some period $\ell$ and we have

$$\begin{aligned} \kappa _{j,i}=\mathbf{t }_j'(i){\mathbb {M}}\mathbf{1 }_k=\displaystyle \sum _{\ell '=1}^k\gamma _{\ell ,\ell '} \quad . \end{aligned}$$

Recall that the sum of the entries of row $\ell$ in matrix ${\mathbb {M}}$ is given in Lemma 1: for each $\ell \in {\llbracket 1,k-m\rrbracket }$, the value $p_\ell =\displaystyle \sum \nolimits _{\ell '=1}^k\gamma _{\ell ,\ell '}=a_0(a_0-a_\ell )$ (with $a_\ell =\displaystyle \sum \nolimits _{u=\ell }^m\theta _u$ for $\ell \in {\llbracket 1,m\rrbracket }$ and $a_{\ell }=0$ for $\ell >m$) and $p_\ell =p_{k-\ell +1}$ for $\ell \in {\llbracket k-m,k\rrbracket }$. Remark that $p_{\ell }=p_{m+1}=a_0^{2}$ for all $\ell >m$. Thus $\forall$ $\ell \in {\llbracket 1,m\rrbracket }\cup {\llbracket k-m+1,k\rrbracket }$:

$$\begin{aligned} p_{\ell }-p_{m+1}=-a_0a_\ell . \end{aligned}$$

(44)

Now, let’s determine the values of $n_{j,i}$, defined in Sect. 2.1, and $\phi _{j,i}^\ell$ for all $\ell \in {\llbracket 1,m\rrbracket }$. Recall that $\mathbf{t }_j(i)={\varvec{e}}_\ell$ if the treatment j is applied to the ith patient at period $\ell$ and $\mathbf{t }_j(i)=\mathbf{0 }_{k}$ otherwise.

• Case $\mathbf{t }_j(i)=\mathbf{0 }_{k}$: $n_{j,i}=\phi _{j,i}^1=\cdots =\phi _{j,i}^m=0$ because the ith patient does not receive the treatment j.

• Case $\mathbf{t }_j(i)={\varvec{e}}_\ell$ where $~\ell \in {\llbracket 1,m\rrbracket }\cup {\llbracket k-m+1,k\rrbracket }$: $n_{j,i}=\phi _{j,i}^\ell =1$ and $\phi _{j,i}^1=\cdots =\phi _{j,i}^{\ell -1}=\phi _{j,i}^{\ell +1}=\cdots =\phi _{j,i}^m=0$.

• Case $\mathbf{t }_j(i)={\varvec{e}}_\ell$ where $~\ell \in {\llbracket m+1,k-m\rrbracket }$: $n_{j,i}=1$ and $\phi _{j,i}^1=\cdots =\phi _{j,i}^m=0.$

If the treatment j is applied to the ith patient at some period $\ell$ for $\ell \in {\llbracket 1,k\rrbracket }$ then $\kappa _{j,i}=p_\ell$. Otherwise, if the treatment j is not applied to the ith patient then $\kappa _{j,i}=0$. Consequently, we can express the quantity $\kappa _ {j, i}$ in the following form

$$\begin{aligned} \kappa _{j,i}= & {} p_{m+1}n_{j,i}+\phi _{j,i}^1(p_1-p_{m+1})+\phi _{j,i}^2(p_2-p_{m+1})+\cdots \nonumber \\&\cdots + \; \phi _{j,i}^m(p_m-p_{m+1}). \end{aligned}$$

(45)

From Formulas (28) and (44), we deduce that:

$$\begin{aligned} \kappa _{j,i}= & {} a_0 (a_0n_{j,i}-a_1\phi _{j,i}^1-a_2\phi _{j,i}^2-\cdots -a_m\phi _{j,i}^m) \nonumber \\= & {} a_0\left( a_0n_{j,i}-\sum _{\ell =1}^ma_\ell \phi _{j,i}^\ell \right) . \end{aligned}$$

(46)

Diagonal entry $\sigma ^{2}\mathbf{C }_{j,j}$: determination of $\omega _{j,j}$

Recall that the contribution $c^{-1}{{\mathcal {B}}}$ to the entry $\sigma ^2\mathbf{C }_{j,j}$ is the quantity $\omega _{j,j}=c^{-1}\sum _{i} \kappa _{j,i}^{2}$ (see Identity (43)). From Identity (46), we have:

$$\begin{aligned} \kappa _{j,i}^2=a_0^2\left\{ a_0^2n_{j,i}^2+ \sum _{\ell =1}^m a_\ell ^2\phi _{j,i}^\ell -2a_0 \sum _{\ell =1}^m a_\ell n_{j,i}\phi _{j,i}^\ell \right\} \end{aligned}$$

because ${(\phi _{j,i}^{\ell })}^2=\phi _{j,i}^\ell$ $\forall$ $\ell \in {\llbracket 1,m\rrbracket }$, and when $\ell \ne \ell '$, $\phi _{j,i}^{\ell }\phi _{j,i}^{\ell '}=0$. From

$$\begin{aligned} \phi _j^\ell =\displaystyle \sum _{i=1}^b\phi _{j,i}^\ell =\sum _{i=1}^b n_{j,i}\phi _{j,i}^\ell \quad \text { and } \quad r=\displaystyle \sum _{i=1}^bn_{j,i}^2, \end{aligned}$$

we finally obtain:

$$\begin{aligned} c \, w_{j,j}= \displaystyle \sum _{i=1}^b\kappa _{j,i}^2= & {} a_0^2\left\{ a_0^2r+ \displaystyle \sum _{\ell =1}^m a_\ell ^2\phi _j^\ell -2a_0 \sum _{\ell =1}^m a_\ell \phi _j^\ell \right\} \nonumber \\= & {} - a_0^2\left\{ a_0(a_{0} -2a_{0})r - \sum _{\ell =1}^m\phi _j^\ell a_\ell (a_\ell -2a_0)\right\} . \end{aligned}$$

(47)

Extra-diagonal entry $\sigma ^{2}\mathbf{C }_{j,j'}$: determination of $\omega _{j,j'}$

Let us determine the contribution $\omega _{j,j'}=c^{-1}\sum _{i=1}^{b}\kappa _{j,i}\kappa _{j',i}$, $j\ne j'$, to the entry $\sigma ^{2}\mathbf{C }_{j,j'}$. From Identity (45), we have:

$$\begin{aligned} \kappa _{j,i}\kappa _{j',i}= & {} a_0^{2}\left( a_0n_{j,i}- \sum _{\ell =1}^m a_\ell \phi _{j,i}^\ell \right) \left( a_0n_{j',i}- \sum _{\ell '=1}^m a_{\ell '}\phi _{j',i}^{\ell '} \right) \\= & {} a_0^2 \left\{ a_0^2n_{j,i}n_{j',i} - a_0 \left( \sum _{\ell =1}^m a_{\ell }n_{j,i}\phi _{j',i}^{\ell } +\sum _{\ell =1}^m a_\ell n_{j',i}\phi _{j,i}^\ell \right) \right. \\&+ \left. \sum _{\ell =1}^m\sum _{{\ell '}=1}^{m} a_{\ell }a_{\ell '} \phi _{j,i}^\ell \phi _{j',i}^{\ell '}\right\} . \end{aligned}$$

From $\lambda _{j,j'}=\sum _{i=1}^bn_{j,i}n_{j',i}$ (see Identity (2)) and

$$\begin{aligned} \phi _{j,j'}^{\ell *}=\displaystyle \sum _{i=1}^b (n_{j',i}\phi _{j,i}^\ell \,+ \,n_{j,i}\phi _{j',i}^{\ell }), \quad \text {for all } \ell \in {\llbracket 1,m\rrbracket }, \end{aligned}$$

(48)

we finally obtain for $c\,\omega _{j,j'}=\sum _{i=1}^b\kappa _{j,i}\kappa _{j',i}$

$$\begin{aligned} c\,\omega _{j,j'} = a_0^4\lambda _{j,j'} -a_0^3\sum _{\ell =1}^m a_\ell \phi _{j,j'}^{\ell *} +a_0^2\sum _{\ell =1}^m\sum _{{\ell '}=1}^{m} \left( a_{\ell }a_{\ell '} \sum _{i=1}^b \phi _{j,i}^\ell \phi _{j',i}^{\ell '} \right) \; . \end{aligned}$$

(49)

End of the proof of Proposition 3. From $\mathbf{C }= {{\mathcal {A}}} - c^{-1}{{\mathcal {B}}}$ (see Identity (32)), we find:

$$\begin{aligned} \mathbf{C }_{j,j} = \tau - \omega _{j,j} \quad \text { and } \quad \mathbf{C }_{j,j'} = \mu - \omega _{j,j'} \end{aligned}$$

where $\tau$ and $\mu$ come from the matrix ${{\mathcal {A}}}$ and $\omega _{j,j}$ and $\omega _{j,j'}$ come from the matrix $c^{-1}{{\mathcal {B}}}$. Using the formulas on $\tau$, $\mu$, $\omega _{j,i}$ and $\omega _{j,j'}$ respectively establish in Identities (37), (42), (47) and (49), we complete the proof of Proposition 3.

1.3 Proof of Theorem 1

Consider $d \in {\varOmega }_{v,b,k}$ a NNm-balanced BIBD$(v,b,r,k,\lambda )$ for the AR(m) model with $k \ge 3$, $m\ge 1$ and $2m< k < v$ (this proof also holds for CBD when $k=v$).

In Remark 1, we have deduced from Proposition 3 that all the competitor designs have the same trace. Hence, from Proposition 4, the universal optimality of the design d is satisfied when the information matrix $\mathbf{C }_d$ of ${\widehat{\gamma }}$ is completely symmetric; which means that its extra-diagonal entries $\mathbf{C }_{d,j,j'}$ are all independent from $j,j'$ $(j\ne j')$ because the sum by row (and by column) of $\mathbf{C }_d$ is null (see Identities (25)). According to the hypothesis of Theorem 1, we will prove that none of the five summation blocks of $\mathbf{C }_{d,j,j'}$ appearing in Identity (22) of Proposition 3 depends on $j,j'$.

As the design d is a NNm-balanced BIBD$(v,b,r,k,\lambda )$, Identities (3) and (4) imply that two of the summation blocks of $\mathbf{C }_{d,j,j'}$ are independent from $j,j'$: those depending on $\lambda =\lambda _{j,j'}$ and $N^s=N_{j,j'}^s$. Therefore, if Identities (i), (ii) and (iii) of Theorem 1 hold then the three others summation blocks of $\mathbf{C }_{d,j,j'}$ are independent from $j,j'$ (see Remark 7 for the case of (iii)).

Remark 7

On the right side of Identity (22), let’s consider the summation block $\sum _{s=1}^{m-1}\sum _{t=1}^{m-s} {\varTheta }_{t,s} {{\overline{\alpha }}}_{s,t}$ of $\mathbf{C }_{d,j,j'}$ where ${{\overline{\alpha }}}_{s,t} = N_{j,j',i}^s(\phi _{j,i}^t\phi _{j',i}^{t+s}+ \phi _{j',i}^t\phi _{j,i}^{t+s})$. Let $\ell \ne \ell '$ in ${\llbracket 1,m\rrbracket }$ and $\alpha _{\ell ,\ell '}=N_{j,j',i}^{|\ell -\ell '|}(\phi _{j,i}^\ell \phi _{j',i}^{\ell '}+ \phi _{j',i}^\ell \phi _{j,i}^{\ell '})$ be the left-hand side of Identity (iii) in Theorem 1. We claim that:

$$\begin{aligned} \{\alpha _{\ell ,\ell '} \mid \ell \ne \ell ' \text { in } {\llbracket 1,m\rrbracket }\} = \{ {{\overline{\alpha }}}_{s,t} \mid s \in {\llbracket 1,m-1\rrbracket } \text { and } t \in {\llbracket 1,m-s\rrbracket } \}. \end{aligned}$$

(50)

For $`` \subset ``$, by symmetry between $\ell ,\ell '$ in $\alpha _{\ell ,\ell '}$, we can suppose that $\ell < \ell '$ and express $\alpha _{\ell ,\ell '}$ as follows: $\alpha _{\ell ,\ell '}= N_{j,j',i}^{|\ell -\ell '|}(\phi _{j,i}^\ell \phi _{j',i}^{\ell +|\ell -\ell '|}+ \phi _{j',i}^\ell \phi _{j,i}^{\ell +|\ell -\ell '|})$. Then $\alpha _{\ell ,\ell '} = {{\overline{\alpha }}}_{s,t}$ with $s=|\ell -\ell '| \in {\llbracket 1,m-1\rrbracket }$ and $\ell =t \in {\llbracket 1,m-s\rrbracket }$ (as expected in the summation in the expression of $\mathbf{C }_{d,j,j'}$). Conversely, let $s \in {\llbracket 1,m-1\rrbracket }$ and $t \in {\llbracket 1,m-s\rrbracket }$. Then we have ${{\overline{\alpha }}}_{s,t} = \alpha _{\ell ,\ell '}$ for the two distinct periods $\ell =t$ and $\ell '=t+s$ in ${\llbracket 1,m\rrbracket }$.

In the following, we will prove Identities (i), (ii) and (iii) of Theorem 1. More precisely, for each identity, we will suppose that the term in the left-hand side is a constant and we prove that it equals to the right-hand side. Recall that $\omega = \frac{2b}{v(v-1)}$.

Proof of Identity (i). For each treatment j, we first need to establish the following identity:

$$\begin{aligned} \displaystyle \sum _{j'\ne j}\phi _{j,j'}^{\ell *} = (k-2)\phi _{j}^\ell +2r \quad . \end{aligned}$$

(51)

Proof

Develop $\sum _{j'\ne j}\phi _{j,j'}^{\ell *}$:

$$\begin{aligned} \sum _{j'\ne j}\phi _{j,j'}^{\ell *}= \sum _{j'\ne j} \displaystyle \sum _{i=1}^b (n_{j',i}\phi _{j,i}^\ell + \,n_{j,i}\phi _{j',i}^{\ell }) = \sum _{i=1}^b \phi _{j,i}^\ell \sum _{j'\ne j} n_{j',i} + \sum _{i=1}^b n_{j,i} \sum _{j'\ne j} \phi _{j',i}^{\ell }. \end{aligned}$$

The first term of the right-hand side of the previous identity is

$$\begin{aligned} \alpha =\sum _{i=1}^b \phi _{j,i}^\ell \sum _{j'\ne j} n_{j',i} = \sum _{i=1}^b \phi _{j,i}^\ell \sum _{j'=1}^{v} n_{j',i} - \sum _{i=1}^b \phi _{j,i}^\ell n_{j,i} = k \, \phi _{j}^\ell - \phi _{j}^\ell \end{aligned}$$

by definition of $\phi _{j}^\ell$ and since each patient i receives k treatments. The second term is

$$\begin{aligned} \beta = \sum _{i=1}^b n_{j,i} \sum _{j'\ne j} \phi _{j',i}^{\ell } = \sum _{i=1}^b n_{j,i} \sum _{j'=1}^{v}\phi _{j',i}^{\ell } - \sum _{i=1}^b n_{j,i} \phi _{j,i}^\ell = 2 \, r - \phi _{j}^\ell \end{aligned}$$

because d is equireplicated (i.e. j appears r times in d) and only 2 treatments $j'$ can be applied to a same patient i at periods $\ell$ and $(k-\ell +1)$ (i.e. $\phi _{j',i}^{\ell } =1$ for these two treatments and 0 for the others). Summing $\alpha$ and $\beta$, we obtain Identity (51) $\square$

From Formulas (51) and (16), we obtain finally:

$$\begin{aligned} \sum _{j=1}^{v} \displaystyle \sum _{j'\ne j}\phi _{j,j'}^{\ell *} = 2b(k-2)+2rv= 2b(k-2) +2bk=4b(k-1) \end{aligned}$$

(52)

because $rv=bk$. Suppose that each $\phi _{j,j'}^{\ell *}$ does not depend on $j,j'$. Then we have the equality $\sum _{j=1}^{v} \displaystyle \sum \nolimits _{j'\ne j}\phi _{j,j'}^{\ell *} = v(v-1)\phi _{j,j'}^{\ell *}$. Thus from (52), we obtain Identity

$$\begin{aligned} \begin{array}{cc} (i)&\phi _{j,j'}^{\ell *} = \frac{4b(k-1)}{v(v-1)}=2 \omega (k-1) \quad . \end{array} \end{aligned}$$

Proof of Identity (ii). Consider two distinct periods $\ell$ and $\ell '$ and fix a patient i. Four distinct treatments $j_{1},\ldots ,j_{4}$ are applied to this patient at the respective periods $\ell ,k-\ell +1,\ell ',k-\ell '+1$. Then $\phi _{j_{1},i}^\ell =\phi _{j_{2},i}^\ell =\phi _{j_{3},i}^{\ell '}=\phi _{j_{4},i}^{\ell '}=1$ and the other values $\phi _{j,i}^{\ell }$ and $\phi _{j',i}^{\ell '}$ are zero; consequently:

$$\begin{aligned} \displaystyle \sum _{j=1}^{v}\sum _{j'\ne j}\phi _{j,i}^\ell \phi _{j',i}^{\ell '}= & {} \phi _{j_{1},i}^\ell (\phi _{j_{3},i}^{\ell '} + \phi _{j_{4},i}^{\ell '}) + \phi _{j_{2},i}^\ell (\phi _{j_{3},i}^{\ell '} + \phi _{j_{4},i}^{\ell '})=4 \quad \end{aligned}$$

and

$$\begin{aligned} \displaystyle \sum _{j=1}^{v}\sum _{j'\ne j}\phi _{j,i}^\ell \phi _{j',i}^\ell = \phi _{j_{1},i}^\ell \phi _{j_{2},i}^\ell +\phi _{j_{2},i}^\ell \phi _{j_{1},i}^\ell = 2. \end{aligned}$$

If the quantity $\displaystyle \sum _{i=1}^b\phi _{j,i}^\ell \phi _{j',i}^{\ell '}$ does not depend on $j,j'$ then, by the same reasoning as for (i), we find ($\delta _{\ell ,\ell '}$ is the Kronecker symbol):

$$\begin{aligned} \begin{array}{cc} (ii)&\displaystyle \sum _{i=1}^b\phi _{j,i}^\ell \phi _{j',i}^{\ell '}= \frac{b(2 + 2(1-\delta _{\ell ,\ell '}))}{v(v-1)} = \omega (2-\delta _{\ell ,\ell '}) ~~ \text{ for } \text{ all }~~ \ell ,\ell ' \in {\llbracket 1,m\rrbracket }. \end{array} \end{aligned}$$

Proof of Identity (iii). With reference to Remark 7, prove Identity (iii): $\alpha _{\ell ,\ell '}= 2 \, \omega$ for $\ell \ne \ell '$ in ${\llbracket 1,m\rrbracket }$ is equivalent to prove ${{\overline{\alpha }}}_{s,t} =2 \, \omega$ for $s \in {\llbracket 1,m-1\rrbracket }$ and $t \in {\llbracket 1,m-s\rrbracket }$. Let’s fix $s \in {\llbracket 1,m-1\rrbracket }$ and $t \in {\llbracket 1,m-s\rrbracket }$ and prove that ${{\overline{\alpha }}}_{s,t} =2 \, \omega$. By the same reasoning as above, for a patient i, four distinct treatments $j_{1},\ldots ,j_{4}$ are applied at the respective distinct periods $t,k-t+1,t+s,k-(t+s)+1$. Then

$$\begin{aligned} \beta _{t,s}= & {} \displaystyle \sum _{j=1}^{v}\sum _{j'\ne j}(\phi _{j,i}^t\phi _{j',i}^{t+s}+ \phi _{j',i}^t\phi _{j,i}^{t+s}) \\= & {} \sum _{j=1}^{v}\phi _{j,i}^t\sum _{j'\ne j}\phi _{j',i}^{t+s}+ \sum _{j=1}^{v} \phi _{j,i}^{t+s} \sum _{j'\ne j}\phi _{j',i}^t\\= & {} 2(\phi _{j_{1},i}^t + \phi _{j_{2},i}^t)( \phi _{j_{3},i}^{t+s} +\phi _{j_{4},i}^{t+s}) = 8 \quad . \end{aligned}$$

But, in this sum, there are 4 cases in which two treatments among $j_{1},\ldots ,j_{4}$ are applied at distance s and there are 4 cases in which two treatments among $j_{1},\ldots ,j_{4}$ are applied at distance $\delta$ where $\delta \ge m> s$ because $k>2m$. For the firsts 4 cases, we have $N_{j,j',i}^s=1$ and for the 4 others cases we have $N_{j,j',i}^s=0$. Then

$$\begin{aligned} \sum _{j=1}^{v} \sum _{j'\ne j} N_{j,j',i}^s(\phi _{j,i}^t\phi _{j',i}^{t+s}+ \phi _{j',i}^t\phi _{j,i}^{t+s})= & {} \quad \displaystyle \frac{1}{2} \beta _{t,s} = 4. \end{aligned}$$

Hence, if each quantity ${{\overline{\alpha }}}_{s,t} = \sum _{i=1}^{b} N_{j,j',i}^s(\phi _{j,i}^t\phi _{j',i}^{t+s}+ \phi _{j',i}^t\phi _{j,i}^{t+s})$ does not depend on $j,j'$ ($j\ne j'$), the following identity holds for $\ell \ne \ell '$ in ${\llbracket 1,m\rrbracket }$:

$$\begin{aligned} \begin{array}{cc} (iii)&\alpha _{\ell ,\ell '}={{\overline{\alpha }}}_{s,t} = \frac{4b}{v(v-1)} =2 \, \omega . \end{array} \end{aligned}$$

Then Theorem 1 is proved.

1.4 Proof of Theorem 2

Theorem 2 is a straightforward consequence of the proof of Theorem 1 which also holds for $k=v$ and of Identities (19) and (20) for the NNm-balanced square designs.

1.5 Proof of Proposition 5

Recall that in case of the strength is $t=2$, the index $\omega _2$ is $\omega =\frac{2b}{v(v-1)}$ (see Remark 3). Since an SB(b, k, v, 2) can be interpreted as a BIBD($v,b, r, k, \lambda )$, Identity (27) comes from the identities $v (v-1) \, \omega = 2b$ and $rv = bk$ (see Identity (1)).

Now consider an unordered pair $(j,j')$ of two distinct treatments. For all $m \in {\llbracket 1,k-1\rrbracket }$, the design d is NNm-balanced because $N_{d,j,j'}^s$, the number of times that $(j,j')$ are applied to a same patient at distance $s \in {\llbracket 1,m\rrbracket }$, is a constant $N_{d}^{s}$. More precisely, consider the $k-s$ possible pairs of periods $\ell$ and $\ell +s$ where $\ell$ runs in ${\llbracket 1,k-s\rrbracket }$. Since the strength of d is two, we obtain Identity (4): $N_{d}^{s}=N_{d,j,j'}^s=\omega (k-s)$. To prove the rest of Proposition 5, we use item (a) of Theorem 2 in Martin and Eccleston (1991) which implies that d is universally optimal.

1.6 Proofs of Identities (3), (4), (19) and (20)

Proof of Identity ( 3 )

Let $\beta = \sum _{j=1}^{v}\sum _{j'\ne j} \lambda _{d,j,j'}$. As d is a BIBD, we have $\beta = \sum _{j=1}^{v}\sum _{j'\ne j}\lambda = v(v-1)\lambda$. But we can express $\beta$ differently: $\beta = bk(k-1)$ because there are b patients and exactly $k(k-1)$ distinct pairs of treatments for each of them (recall that $k\le v$). The identification of the two expressions of $\beta$ prove the wanted identity satisfied by $\lambda$:

$$\begin{aligned} \lambda =\lambda _{d,j,j'} =\frac{bk(k-1)}{v(v-1)} = \omega \, \frac{k(k-1)}{2} \quad \quad \forall j,j' \in {\llbracket 1,v\rrbracket }, j\ne j'. \end{aligned}$$

Proof of Identity (4)

Assume that design d is NNm-balanced. Let us fix $s \in {\llbracket 1,m\rrbracket }$ and compute by two ways the sum

$$\begin{aligned} \alpha =\sum _{j=1}^{v}\sum _{j'\ne j} N_{d,j,j'}^s. \end{aligned}$$

Firstly, as the design is NNm-balanced, each $N_{d,j,j'}^s$ equals a constant $N_{d}^{s}$ which does not depend on the choice $j,j'$. So we have:

$$\begin{aligned} \alpha = \sum _{j=1}^{v}\sum _{j'\ne j} N_{d}^s = v(v-1) \, N_{d}^s \quad . \end{aligned}$$

Secondly, suppose that some patient i receives a given treatment j. Recall that j is administered at most once to a same patient. For the ith patient, if j is not applied in the first s or in the last s periods (i.e.when $\sum _{\ell =1}^{s}\phi _{d,j,i}^{\ell } =0$) then there exist $2=\sum _{j'\ne j}N_{d,j,j',i}^s$ treatments at distance s from j. Otherwise, if j is applied in the first s or in the last s periods then $\phi _{d,j,i}^{\ell } =1$ for (only) one period $\ell \in {\llbracket 1,s\rrbracket }$ (i.e. when $\sum _{\ell =1}^{s}\phi _{d,j,i}^{\ell } =1$) and there exists only $1=\sum _{j'\ne j}N_{d,j,j',i}^s$ treatment at distance s from j. Therefore, in both cases, we obtain

$$\begin{aligned} \sum _{j'\ne j}N_{d,j,j',i}^s= 2 -\sum _{\ell =1}^{s}\phi ^\ell _{d,j,i} \quad . \end{aligned}$$

(53)

Moreover, as j appears exactly r times in the design d, by considering all patients i,

$$\begin{aligned} \sum _{j'\ne j} N_{d,j,j'}^s = 2r - \sum _{\ell =1}^{s} \phi _{d,j}^{\ell }. \end{aligned}$$

(54)

Let us sum the above equality for all j and from Identity (16), we obtain this second expression of $\alpha$:

$$\begin{aligned} \alpha =\sum _{j=1}^{v} \sum _{j'\ne j} N_{d,j,j'}^s = \sum _{j=1}^{v}(2r - \sum _{\ell =1}^{s} \phi _{d,j}^{\ell }) = 2rv -2bs \quad . \end{aligned}$$

As $rv=kb$ (see Identity (1)), the identification of the two expressions of $\alpha$ implies the wanted identity (4) satisfied by $N_{d}^{s}$:

$$\begin{aligned} N_{d}^{s}=N_{d,j,j'}^s =\frac{2b(k-s)}{v(v-1)} = \omega \, (k-s) \quad \forall j,j' \in {\llbracket 1,v\rrbracket }, j\ne j'. \end{aligned}$$

Proof of Identities (19) and (20)

Let d be a NNm-balanced design with $k=v$ (i.e. the number of periods equals the number of treatments). We have also $r=b$ because $rv=kb$. We will prove that for each $\ell \in$ ${\llbracket 1,m\rrbracket }$ the quantities $\phi _{d,j}^\ell$ and $\phi _{d,j,j'}^{\ell *}$ do not depend on treatments $j,j'$ ($j\ne j'$); we will express these quantities without j and $j'$.

Let $s\in$ ${\llbracket 1,m\rrbracket }$. Applying Identity (4), as d is a NNm-balanced design, $N_{d,j,j'}^s=N_{d}^{s}=2b(k-s)/v(v-1)=2b(v-s)/v(v-1)$ since $k=v$. Then from (54), we have:

$$\begin{aligned} \sum _{\ell =1}^{s} \phi ^\ell _{d,j} = 2r -(v-1)N_{d}^{s}. \end{aligned}$$

As $r=b$, the previous equality becomes $\sum _{\ell =1}^{s} \phi ^\ell _{d,j} = \frac{2bs}{v}$. Then, for each $s\in {\llbracket 1,m\rrbracket }$, we find:

$$\begin{aligned} \phi ^s_{d,j} = \sum _{\ell =1}^{s} \phi ^\ell _{d,j} - \sum _{\ell =1}^{s-1} \phi ^\ell _{d,j} = \frac{2b}{v} \quad \end{aligned}$$

which is Identity (19). We now prove the second identity. We know that each treatment is administered at most once for each patient; but, as moreover $k=v$, every patient will receive the v distinct treatments once and only once. That means $n_{d,j,i}=1$ for all $j\in {\llbracket 1,v\rrbracket }$. Therefore Identity (48) becomes Identity (20):

$$\begin{aligned} \phi _{d,j,j'}^{\ell *}=\phi _{d,j}^\ell + \phi _{d,j'}^\ell = \frac{4b}{v} \quad ~~ \forall ~~ \ell \in {\llbracket 1,m\rrbracket } \quad \end{aligned}$$

and the two identities on NNm-balanced square designs are proved.