A Complete Class of Type 1 Optimal Block Designs with Unequal Replications

Abstract

For block designs comparing v treatments in b incomplete blocks of size k, consider settings where bk = vr + 1 and $r(k-1)=\lambda (v-1)$ for integers r and $\lambda$. These settings admit designs that possess the symmetry of balanced incomplete block designs and which, though nonbinary, are candidates for optimality in some standard senses. For k = 3, earlier authors have established a class of one binary and one nonbinary design that is complete with respect to all type 1 optimality criteria. Here a solution for the complete class problem for type 1 optimality is obtained for k = 5. The complete class includes two binary and two nonbinary designs.

Appendix: Results on Eigenvalues

Bounds involving the eigenvalues of a real symmetric matrix are provided here. In what follows, A will denote a real symmetric $n \times n$ matrix with row sum zero. The rows and columns of A are indexed by $N = \{1, \ldots n\}$. For $I = \{i_1, \ldots , i_t\} \subset N$, the principal submatrix of A with rows and columns indexed by I will be denoted by A(I). In design optimality arguments, A will be either $C_d$ or $\Delta _d$, and n will be v.

The first result follows from Ky Fan’s Maximum principle (see page 24 of Bhatia [3], for instance).

Lemma 8.1

Consider $n \times 1$ nonzero vectors $x_i, i = 1, \ldots , p \; (p < n)$ satisfying $x^{\prime }_i 1_n = 0 = x^{\prime }_i x_j,\, i \ne j, \; i,j =1, \ldots , p$. Then the following holds.

$$\begin{aligned} \sum _{j=1}^{i}\mu _j(A) \le \sum _{j=1}^{i}(x_j^{\prime }A x_j)/(x_j^{\prime }x_j) \le \sum _{j=1}^{i}\mu _{n-j} (A), 1 \le i \le p. \end{aligned}$$

Corollaries 8.1 and 8.2 are immediate consequences of Lemma 8.1.

Corollary 8.1

(a) Let $I \subset N, \; |I| = m$ and $A_1 = A(I)$. For $m \times 1$ nonzero vectors $x_i, \, i = 1, \ldots , p$ ($p < m)$ satisfying $x^{\prime }_i 1_m = 0 = x^{\prime }_i x_j,$ $i\ne j =1, \ldots , p$,

$$\begin{aligned} \sum _{j=1}^{i}\mu _j(A) \le \sum _{j=1}^{i}(x_j^{\prime }A_1 x_j)/(x_j^{\prime }x_j) \le \sum _{j=1}^{i}\mu _{n-j} (A), \ 1 \le i \le p. \end{aligned}$$

(b) Let $I_1$ and $I_2$ be two disjoint subsets of N of sizes $n_1$ and $n_2$, respectively, and let $A_t = A(I_t), \ t = 1,2$. Let $x_{i1}, \ldots x_{im_i}$ be $m_i$ $n_i \times 1$ vectors satisfying $x^{\prime }_{ij} 1_{n_i} = 0 = x^{\prime }_{ij} x_{il}, \ l \ne j,\; 1 \le l,j \le m_i, i = 1,2$. Consider integers $p_1,p_2$ such that $1 \le p_i \le m_i$. Let $q = p_1 + p_2$. Then

$$\begin{aligned} \sum _{i=1}^{q}\mu _i(A) \le \sum _{t=1}^2 \sum _{j=1}^{p_i}(x_j^{\prime }A_t x_j)/(x_j^{\prime }x_j) \le \sum _{i=1}^{q} \mu _{n-i} (A), \ 2 \le q \le m_1 + m_2. \end{aligned}$$

Corollary 8.2

If A has a diagonal element p, then

$$\begin{aligned} \mu _1 (A) \le n p/(n-1) \le \mu _{n-1} (A). \end{aligned}$$

Lemma 8.2

If A has any one of the matrices $M_1,\ldots ,M_6$ listed below as a principle submatrix, then $\mu _1(A) \le -3/2$.

$$\begin{aligned} \begin{array}{lrlrlrl} M_1 & = & \left( \begin{array}{ll} 0 & 1^{\prime }_p \\ 1_p & 0_{p \times p} \end{array} \right) ,\,\, p \ge 3 & & M_2 & = & \left( \begin{array}{rrr} 0 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{array} \right) \\ & & & & & & \\ M_3 & = & \left( \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right) & & M_4 & = & \left( \begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{array} \right) \\ & & & & & & \\ M_5 & = & \left( \begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & -1 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array} \right) & & M_6 & = & \left( \begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \end{array} \right). \end{array} \end{aligned}$$

Proof

By Corollary 8.1(a) with $p=1$, for each $q \times q$ matrix $M_i$ it is enough to find a $q \times 1$ vector x such that $x^{\prime }1_q = 0$ and $x^{\prime }M_i x/(x^{\prime }x) \le -3/2$. Here are the x’s and the upper bounds for $\mu _1(M_i)$ they produce:

Submatrix	x-vector	Upper bound for $\mu _1$
$M_1$	$(p, -1^{\prime }_p)$	$-2p/(p+1)$
$M_2$	(2, −1, −1)	−5/3
$M_3$	(1, −1, 1, −1)	−3/2
$M_4$	(1, 1, −1, −1)	−3/2
$M_5$	(1, −1, −1, 1)	−2
$M_6$	(1, 1, −1, −1)	−3/2