On the Properties of Proper $(M, S)$ Optimal Block Designs

Conjectures for Optimal Block Designs

“…of D* are given by er where er = l-IX(1-ei) (i= 1,2, ...,v-l) (4) with IX = rk/(r* k*). It follows from (2), (3) and (4) that if D is (M, S) (or E)-optimal then D* is also (M,S) (or E)-optimal; see also Jacroux (1978).…”

“…Takeuchi (1961) showed that a group divisible PBIB/2 design with tt 2 = }'I + 1 is E-optimal. Other results on PBIB/2 designs have been given by Cheng (1978Cheng ( , 1980and Jacroux (1980). Cheng (1980) has also considered the 209 parameter sets studied by John and Mitchell (1977).…”

“…For 176 sets he shows that the designs are E-optimal; the remaining 33 sets for which the E-optimal designs are stillunknown are listed in the paper. Jacroux (1980) also considers the E-optimality of RG-designs. Support for the conjecture is also given by Conniffe andStone (1974,1975) and Shah et al (1976).…”

Definitions and Properties for M-Concurrence Designs

“…If Xi is an eigenvector of I~(rk)-l NN' corresponding to the eigenvalue e.; we have L'x, = 0 and (NN')xj = rk(l-ej)x j from which the result is readily obtained. It certainly follows, as shown by Jacroux (1978), that the complement design is (M, S)optimal if and only if the design is (M, S)-optimaI. However, we are also led to the observations that (i) The parameters P~k for the design and its complement are identical, since P~k is defined in terms of the matrices B 1 , ••• , B m and the parameters n 1 , ••• , n m , v. We note also that the bounds (3.3) and (3.4) are unchanged.…”

“…(ii) E-optimality: C 2 = emin minimizes the maximum variance of any treatment contrast (Wald, 1943;Ehrenfeld, 1955;Cheng, 1980;Jacroux, 1980); (iii) D-optimality: C 3 = (TI i ey /(v-1 l , the geometric mean, minimizes the generalized variance (Wald, 1943);…”