On the comparison of PBIB designs with two associate classes

  • Ching-Shui Cheng


A method to compare two-associate-class PBIB designs is discussed. As an application, it is shown that ifd* is a group-divisible design withλ21+1, a group divisible design with group size two andλ21+1>1, a design based on a triangular scheme andv=10 andλ12+1, a design with anL2 scheme andλ21+1, a design with anLs scheme,v=(s+1)2, andλ21+1, wheres is a positive integer, or a design with a cyclic schemev=5, andλ12±1, thend* is optimum with respect to a very general class of criteria over all the two-associate-class PBIB designs with the same values ofv, b andk asd*. The best two-associate-class PBIB design, however, is not necessarily optimal over all designs.


Generalize Type Adjacency Matrix Small Eigenvalue Association Scheme Balance Incomplete Block Design 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bose, R. C. and Nair, K. R. (1939). Partially balanced incomplete block design,Sankhyā,4, 337–372.Google Scholar
  2. [2]
    Cheng, C. S. (1978). Optimality of certain asymmetrical experimental designs,Ann. Statist.,6, 1239–1261.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Cheng, C. S. (1980). On theE-optimality of some block designs,J. R. Statist. Soc., B,42, 199–204.Google Scholar
  4. [4]
    Clatworthy, W. H. (1973).Tables of Two-Associate-Class Partially Balanced Designs, Nat. Bur. Stand. (U.S.), Appl. Math. Ser.63.Google Scholar
  5. [5]
    Connor, W. S. and Clatworthy, W. H. (1954). Some theorems for partially balanced designs,Ann. Math. Statist.,25, 100–112.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Kiefer, J. (1958). On the nonrandomized optimality and randomized nonoptimality of symmetrical designs,Ann. Math. Statist.,29, 675–699.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Kiefer, J. (1975). Construction and optimality of generalized Youden designs, inA Survey of Statistical Designs and Linear Models (ed. J. N. Srivastava), North-Holland, Amsterdam, 333–353.Google Scholar
  8. [8]
    Mesner, D. M. (1965). A note on the parameters of PBIB association schemes,Ann. Math. Statist.,36, 331–336.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Mirsky, L. (1963). Results and problems in the theory of doubly stochastic matrices,Zeit. Wahrscheinlichkeitsth.,1, 319–334.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Mitchell, T. J. and John, J. A. (1976). Optimal incomplete block designs, ORNL/CSD-8, Oak Ridge National Laboratory Report.Google Scholar
  11. [11]
    Raghavarao, D. (1971).Constructions and Combinatorial Problems in Design of Experiments, Wiley, New York.MATHGoogle Scholar
  12. [12]
    Takeuchi, K. (1961). On the optimality of certain type of PBIB designs,Rep. Statist. Appl. Res. Un. Japan Sci. Engrs.,8, 140–145.MathSciNetMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1981

Authors and Affiliations

  • Ching-Shui Cheng
    • 1
  1. 1.University of CaliforniaBerkeley

Personalised recommendations