## 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.

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## References

Bagchi S (2006) On the optimality of a class of designs with three concurrences. Linear Algebra Appl 417(1):8–30

Bagchi B, Bagchi S (2001) Optimality of partial geometric designs. Ann Stat 29(2):577–594

Bhatia R (1997) Matrix analysis, volume 169 of Graduate texts in mathematics. Springer, New York

Cheng CS (1978) Optimality of certain asymmetrical experimental designs. Ann Stat 6(6):1239–1261

Cheng C-S, Bailey RA (1991) Optimality of some two-associate-class partially balanced incomplete-block designs. Ann Stat 19(3):1667–1671

Hanani H (1972) On balanced incomplete block designs with blocks having five elements. J Combin Theory A 12(2):184–281

Jacroux M (1985) Some sufficient conditions for the type 1 optimality of block designs. J Stat Plan Inference 11(3):385–398

Jacroux M (1989) Some sufficient conditions for type 1 optimality with applications to regular graph designs. J Stat Plan Inference 23(2):193–215

Kiefer J (1975) Construction and optimality of generalized Youden designs. In: Srivastava JN (ed) A survey of statistical design and linear models. North-Holland, Amsterdam, pp 333–353

Marshall AW, Olkin I, Arnold BC (2011) Inequalities: theory of majorization and its applications. Springer series in statistics, 2nd edn. Springer, New York

Morgan JP (2015) Blocking with independent responses. In: Dean A, Morris M, Stufken J, Bingham D (eds) Handbook of design and analysis of experiments. CRC Press, Boca Raton, FL, pp 99–157

Morgan JP, Reck B (2007) E-optimal design in irregular BIBD settings. J Stat Plan Inference 137(5):1658–1668

Morgan JP, Srivastav SK (2000) On the type-1 optimality of nearly balanced incomplete block designs with small concurrence range. Stat Sin 10(4):1091–1116

Morgan JP, Uddin N (1995) Optimal, non-binary, variance balanced designs. Stat Sin 5(2):535–546

Reck B, Morgan JP (2005) Optimal design in irregular BIBD settings. J Stat Plan Inference 129(1–2):59–84

Roy BK, Shah KR (1984) On the optimality of a class of minimal covering designs. J Stat Plan Inference 10(2):189–194

Shah KR, Sinha BK (1989) Theory of optimal designs. Lecture notes in statistics, vol 54. Springer, Berlin

Tomić M (1949) Théorème de Gauss relatif au centre de gravité et son application. Bull Soc Math Phys Serbie 1:31–40

## Acknowledgements

This work began when the first author visited the Department of Statistics, Virginia Tech in May, 2016. She expresses her gratitude to everyone in the department for their hospitality during the visit. She also thanks Professor B.V. Rajarama Bhat, professor-in-charge, Stat-Math division, Indian Statistical Institute and Professor Abhyuday Mandal, Department of Statistics, University of Georgia for making this visit possible.

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Part of special issue guest edited by Pritam Ranjan and Min Yang—Algorithms, Analysis and Advanced Methodologies in the Design of Experiments.

## Appendix: Results on Eigenvalues

### 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.*

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* \(x^{\prime }_i 1_m = 0 = x^{\prime }_i x_j,\) \(i\ne j =1, \ldots , p\),

*(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*

### Corollary 8.2

*If* *A* *has a diagonal element* *p*, *then*

### Lemma 8.2

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

###
*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 |

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Bagchi, S., Morgan, J.P. A Complete Class of Type 1 Optimal Block Designs with Unequal Replications.
*J Stat Theory Pract* **13**, 59 (2019). https://doi.org/10.1007/s42519-019-0057-4

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DOI: https://doi.org/10.1007/s42519-019-0057-4

### Keywords

- Block designs
- Optimality
- Type 1 criteria
- Complete class