Annals of Statistics

Optimality of neighbor-balanced designs for total effects

R. A. Bailey and P. Druilhet

Full-text: Open access


The purpose of this paper is to study optimality of circular neighbor-balanced block designs when neighbor effects are present in the model. In the literature many optimality results are established for direct effects and neighbor effects separately, but few for total effects, that is, the sum of direct effect of treatment and relevant neighbor effects. We show that circular neighbor-balanced designs are universally optimal for total effects among designs with no self neighbor. Then we give efficiency factors of these designs, and show some situations where a design with self neighbors is preferable to a neighbor-balanced design.

Article information

Ann. Statist., Volume 32, Number 4 (2004), 1650-1661.

First available in Project Euclid: 4 August 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K05: Optimal designs 62K10: Block designs

Neighbor designs, optimal design, universal optimality, total effects.


Bailey, R. A.; Druilhet, P. Optimality of neighbor-balanced designs for total effects. Ann. Statist. 32 (2004), no. 4, 1650--1661. doi:10.1214/009053604000000481.

Export citation


  • Azaïs, J.-M., Bailey, R. A. and Monod, H. (1993). A catalogue of efficient neighbour-designs with border plots. Biometrics 49 1252–1261.
  • Besag, J. and Kempton, R. A. (1986). Statistical analysis of field experiments using neighbouring plots. Biometrics 42 231–251.
  • Cheng, C.-S. and Wu, C. F. (1980). Balanced repeated measurements designs. Ann. Statist. 8 1272–1283. [Correction (1983) 11 349.]
  • Druilhet, P. (1999). Optimality of neighbour balanced designs. J. Statist. Plann. Inference 81 141–152.
  • Hedayat, A. and Afsarinejad, K. (1978). Repeated measurements designs. II. Ann. Statist. 6 619–628.
  • Kempton, R. A. (1982). Adjustment for competition between varieties in plant breeding trials. J. Agricultural Sci. 98 599–611.
  • Kempton, R. A. (1985). Spatial methods in field experiments. Biometric Bulletin 2 (3) 4–5.
  • Kempton, R. A. (1991). Interference in agricultural experiments. In Proc. Second Meeting of the Biometric Society East/Central/Southern African Network. Harare, Zimbabwe.
  • Kempton, R. A. (1997). Interference between plots. In Statistical Methods for Plant Variety Evaluation (R. A. Kempton and P. N. Fox, eds.) 101–116. Chapman and Hall, London.
  • Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In A Survey of Statistical Design and Linear Models (J. N. Srivastava, ed.) 333–353. North-Holland, Amsterdam.
  • Kunert, J. (1983). Optimal design and refinement of the linear model with applications to repeated measurements designs. Ann. Statist. 11 247–257.
  • Kunert, J. (1984a). Designs balanced for circular residual effects. Comm. Statist. A–-Theory Methods 13 2665–2671.
  • Kunert, J. (1984b). Optimality of balanced uniform repeated measurements designs. Ann. Statist. 12 1006–1017.
  • Kunert, J. and Martin, R. J. (2000). On the determination of optimal designs for an interference model. Ann. Statist. 28 1728–1742.
  • Kushner, H. B. (1997). Optimal repeated measurements designs: The linear optimality equations. Ann. Statist. 25 2328–2344.
  • Langton, S. (1990). Avoiding edge effects in agroforestry experiments: The use of neighbour-balanced designs and guard areas. Agroforestry Systems 12 173–185.
  • Magda, C. G. (1980). Circular balanced repeated measurements designs. Comm. Statist. A–-Theory Methods 9 1901–1918.
  • Matthews, J. N. S. (1988). Recent developments in crossover designs. Internat. Statist. Rev. 56 117–127.
  • McGilchrist, C. A. (1965). Analysis of competition experiments. Biometrics 21 975–985.
  • McGilchrist, C. A. and Trenbath, B. R. (1971). A revised analysis of competition experiments. Biometrics 27 659–671.
  • Patterson, H. D. (1950). The analysis of change-over trials. J. Agricultural Sci. 40 375–380.
  • Patterson, H. D. (1951). Change-over trials (with discussion). J. Roy. Statist. Soc. Ser. B 13 256–271.
  • Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York.
  • Shah, K. R. and Sinha, B. K. (1989). Theory of Optimal Designs. Lecture Notes in Statist. 54. Springer, New York.
  • Speckel, D., Vincourt, P., Azaï s, J.-M. and Kobilinsky, A. (1987). Etude de la compétition interparcellaire chez le tournesol. Biom. Praxim. 27 21–43.