By J. A. John, E. R. Williams (auth.)
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28) Another second moment bound has been given by Tjur (1990). 29) 2 If the design is efficiency-balanced then 8 2 = 0 and U1 = U2 = e. The second moment bounds U1 and U2 require knowledge of 8 2 , However, for equal replicate and equal block size binary designs it is possible to find a lower bound for 8 2 which then leads to an upper bound for E based solely on the design parameters; note that both U 1 and U2 are decreasing functions of 8 2 • It can be shown that r 2k 28 2 = L L (Aij - :\)2 i-Ij where Aij is the (ij)th element of NN and :\ = r(k - l)/(v - 1) is the mean of the Aij over all i f.
Thus the dual is obtained by interchanging the treatment and block symbols in the original design. 2 Consider the following binary design for v = 6, r b = 3, where again blocks are written in columns: = 2, k = 4 and 001 122 334 455 The incidence matrix N, with a row for each treatment and a column for each block, is N= 1 1 0 1 0 1 o 1 1 110 1 0 1 o 1 1 Consequently, the incidence matrix of the dual design is N' =( 1 1 011 1 0 1 1 0 o 1 1 0 1 40 EFFICIENCY FACTORS Hence, the dual design is o 0 1 001 122 1 2 2 and has parameters bd = v = 6.
Even so there will often be large numbers of designs for which the canonical efficiency factors have to be computed. Eigenvalue computations are relatively time consuming and, consequently, attempts have been made to find criteria based on quick and simple counting rules. The idea is to use such rules to identify a small group of designs which are near optimal and then, if necessary, to choose the best designs from this group using the canonical efficiency factors and appropriate optimality criteria.
Cyclic and Computer Generated Designs by J. A. John, E. R. Williams (auth.)