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Biometrika 1979 66(1):115-123; doi:10.1093/biomet/66.1.115
© 1979 by Biometrika Trust
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Generalized N-ary balanced block designs

M. SHAFIQ and W. T. FEDERER

Department of Statistics University of Karachi
Biometrics Unit, Cornell University Ithaca, New York

The concept of N-ary balanced incomplete block designs where the incidence matrix n contains the N values 0, 1, ..., N–1, is extended to generalized N-ary balanced block designs, where the incidence matrix n* contains the N values ma for a = 0, 1, ..., N–1, for ma = am1–(a–1)m0, and for any m0 and m1 satisfying 0≤m0<m1. For ternary designs, m2 = 2m1–m0, and thus 0≤m1<m1<m2. Definitions, parameters and necessary conditions for 2* are presented. Given a fixed number v of treatments and a fixed total number N* of experimental units, a class of N-ary balanced block designs with a different set of ma is possible. Criteria are developed to select the designs in the class with the smallest variance of a contrast. Efficiency factors and a generalization of Fisher's inequality are also presented.

Key Words: Efficiency factor • Incomplete block design • Optimality of design


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