## Annals of Statistics

### Resolvable designs with large blocks

#### Abstract

Resolvable designs with two blocks per replicate are studied from an optimality perspective. Because in practice the number of replicates is typically less than the number of treatments, arguments can be based on the dual of the information matrix and consequently given in terms of block concurrences. Equalizing block concurrences for given block sizes is often, but not always, the best strategy. Sufficient conditions are established for various strong optimalities and a detailed study of E-optimality is offered, including a characterization of the E-optimal class. Optimal designs are found to correspond to balanced arrays and an affine-like generalization.

#### Article information

Source
Ann. Statist., Volume 35, Number 2 (2007), 747-771.

Dates
First available in Project Euclid: 5 July 2007

https://projecteuclid.org/euclid.aos/1183667292

Digital Object Identifier
doi:10.1214/009053606000001253

Mathematical Reviews number (MathSciNet)
MR2336867

Zentralblatt MATH identifier
1117.62075

Subjects
Primary: 62K05: Optimal designs

#### Citation

Morgan, J. P.; Reck, Brian H. Resolvable designs with large blocks. Ann. Statist. 35 (2007), no. 2, 747--771. doi:10.1214/009053606000001253. https://projecteuclid.org/euclid.aos/1183667292

#### References

• Bailey, R. A., Monod, H. and Morgan, J. P. (1995). Construction and optimality of affine-resolvable designs. Biometrika 82 187--200.
• Bailey, R. A. and Speed, T. P. (1986). Rectangular lattice designs: Efficiency factors and analysis. Ann. Statist. 14 874--895.
• Bhatia, R. (1997). Matrix Analysis. Springer, New York.
• Bose, R. C. (1942). A note on the resolvability of balanced incomplete block designs. Sankhyā 6 105--110.
• Chakravarti, I. M. (1956). Fractional replication in asymmetrical factorial designs and partially balanced arrays. Sankhy$\bara$ 17 143--164.
• Chakravarti, I. M. (1961). On some methods of construction of partially balanced arrays. Ann. Math. Statist. 32 1181--1185.
• Cheng, C. S. (1978). Optimality of certain asymmetrical experimental designs. Ann. Statist. 6 1239--1261.
• Craigen, R. and Kharaghani, H. (2007). Hadamard matrices and Hadamard designs. In Handbook of Combinatorial Designs, 2nd ed. (C. J. Colbourn and J. H. Dinitz, eds.) 273--280. Chapman and Hall/CRC, Boca Raton, FL.
• Fuji-Hara, R. and Miyamoto, N. (2000). Balanced arrays from quadratic functions. J. Statist. Plann. Inference 84 285--293.
• Ghosh, S. and Teschmacher, L. (2002). Comparisons of search designs using search probabilities. J. Statist. Plann. Inference 104 439--458.
• Harshbarger, B. (1946). Preliminary report on the rectangular lattices. Biometrics 2 115--119.
• Harshbarger, B. (1949). Triple rectangular lattices. Biometrics 5 1--13.
• John, J. A., Russell, K. G., Williams, E. R. and Whitaker, D. (1999). Resolvable designs with unequal block sizes. Aust. N. Z. J. Stat. 41 111--116.
• Kirkman, T. P. (1851). Query VI on Fifteen young ladies....'' Lady's and Gentleman's Diary No. 148 48.
• Kunert, J. (1985). Optimal repeated measurements designs for correlated observations and analysis by weighted least squares. Biometrika 72 375--389.
• Kuriki, S. (1993). On existence and construction of balanced arrays. Discrete Math. 116 137--155.
• Morgan, J. P. (1996). Nested designs. In Design and Analysis of Experiments (S. Ghosh and C. R. Rao, eds.) 939--976. North-Holland, Amsterdam.
• Morgan, J. P. (2007). Optimal incomplete block designs. J. Amer. Statist. Assoc. 102 655--663.
• Patterson, H. D. and Hunter, E. A. (1983). The efficiency of incomplete block designs in National List and Recommended List cereal variety trials. J. Agricultural Sci. 101 427--433.
• Patterson, H. D. and Silvey, V. (1980). Statutory and recommended list trials of crop varieties in the United Kingdom. J. Roy. Statist. Soc. Ser. A 143 219--252.
• Patterson, H. D. and Williams, E. R. (1976). A new class of resolvable incomplete block designs. Biometrika 63 83--92.
• Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York.
• Reck, B. H. (2002). Nearly balanced and resolvable block designs. Ph.D. dissertation, Old Dominion Univ.
• Shah, K. R. and Sinha, B. K. (1989). Theory of Optimal Designs. Lecture Notes in Statist. 54. Springer, New York.
• Sinha, K., Dhar, V. and Kageyama, S. (2002). Balanced arrays of strength two from block designs. J. Combin. Des. 10 303--312.
• Uddin, N. and Morgan, J. P. (1992). Optimal block designs with maximum blocksize and minimum replication constraints. Comm. Statist. Theory Methods 21 179--195.
• Williams, E. R. (1975). A new class of resolvable designs. Ph.D. dissertation, Univ. Edinburgh.
• Williams, E. R., Patterson, H. D. and John, J. A. (1976). Resolvable designs with two replications. J. Roy. Statist. Soc. Ser. B 38 296--301.
• Yates, F. (1936). A new method or arranging variety trials involving a large number of varieties. J. Agricultural Sci. 26 424--455.
• Yates, F. (1940). Lattice squares. J. Agricultural Sci. 30 672--687.