Teruhiro Shirakura

Search designs for 2m factorial experiments

Abstract Theorems 5, 6 and 10, and Tables 1–2 in Ghosh (1981) are corrected. These are concerned with search designs which permit the estimation of the general mean and main effects, and allow the search and estimation of one possibly unknown nonzero effect among the two- and three-factor interactions in 2m factorial experiments. Some new results are presented.

Main effect + one or two plans for 2m factorials

Abstract Under the assumption that the four-factor and higher-order interactions are negligible, we consider the problem of finding a design T2 so that for the design T1 of Ghosh (1980), T = T1 + T2 is a main effect plus k (= 1, 2) plan for 2m factorials (m = 2h - 1, h ⩾ 3). It is shown that the minimum number of treatments of T2 is 7 for m = 7 and k = 1. In particular, when the effects to be searched and estimated lie in the two-factor interactions only, a main effect + two plan is also constructed by utiliz- ing some BIB design.

Search designs for 2m factorials derived from balanced arrays of strength 2(l+1) and ad-optimal search designs

Abstract We consider a search design for the 2m type such that at most knonnegative effects can be searched among (l+1)-factor interactions and estimated along with the effects up to l- factor interactions, provided (l+1)-factor and higher order interactions are negligible except for the k effects. We investigate some properties of a search design which is yielded by a balanced 2m design of resolution 2l+1 derived from a balanced array of strength 2(l+1). A necessary and sufficient condition for the balanced design of resolution 2l+1 to be a search design for k=1 is given. Optimal search designs for k=1 in the class of the balanced 2m designs of resolution V (l=2), with respect to the AD-optimality criterion given by Srivastava (1977), with N assemblies are also presented, where the range of (m,N) is (m=6; 28≤N≤41), (m=7; 35≤N≤63) and (m=8; 44≤N≤74).

More precise tables of Srivastava-Chopra balanced optimal 2m fractional factorial designs of resolution V, m≤6

Abstract More precise tables of balanced trace-optimal 2m fractional factorial designs of resolution V for m=4,5,6, which were originally obtained by Srivastava and Chopra, are presented.

Fractional factorial designs of two and three levels

Abstract This paper consists of a survey of the results on balanced fractional designs of the 2 m and 3 m types established by Chopra, Kuwada, Shirakura, Srivastava and Yamamoto, new results on search designs of 2 m type, and open problems on the construction of fractional designs. Combinatorial and algebraic properties of these designs are discussed. Basic criteria based on eigenvalues on the information matrices for selecting fractional designs are also discussed. Unsolved problems on the construction of fractional designs are presented.

Weighted A-optimality for fractional 2m factorial designs of resolution V

Abstract A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2m factorial design of resolution V. A WA-optimality criterion having one weight may be considered for designs. It is shown that designs derived from orthogonal arrays are WA-optimal for any weight. From a WA-optimal design, a procedure for finding WA-optimal designs for various weights is given. WA-optimal balanced designs are presented for 4 ⩽ m ⩽ 7 and for the values of n assemblies in certain ranges. It is pointed out that designs for m = 7 and for n = 41, 42 given in Chopra and Srivastava (1973a) or in the corrected paper by Chopra et al. (1986), are not A-optimal.

A series of search designs for 2m factorial designs of resolution V which permit search of one or two unknown extra three-factor interactions

In the absence of four-factor and higher order interactions, we present a series of search designs for 2m factorials (m≥6) which allow the search of at most k (=1,2) nonnegligible three-factor interactions, and the estimation of them along with the general mean, main effects and two-factor interactions. These designs are derived from balanced arrays of strength 6. In particular, the nonisomorphic weighted graphs with 4 vertices in which two distinct vertices are assigned with integer weight ω (1≤ω≤3), are useful in obtaining search designs for k=2. Furthermore, it is shown that a search design obtained for each m≥6 is of the minimum number of treatments among balanced arrays of strenth 6. By modifying the results for m≥6, we also present a search design for m=5 and k=2.

On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1

The norm ‖A‖ = {tr(A′A)}12 of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for ‖A‖ will be given for a balanced fractional 2m factorial design of resolution 2l + 1 which obtained from a simple array with parameters (m; λ0, λ1,…, λm). This array is identical with a balanced array of strength m, m constraints and index set {λ0, λ1,…, λm}. In the class of the designs of resolution V (l = 2) obtained from S-arrays, ones which minimize ‖A‖ will be presented for any fixed N assemblies satisfying (i) m = 4, 11 ⩽ N ⩽ 16, (ii) m = 5, 16 ⩽ N ⩽ 32, and (iii) m = 6, 22 ⩽ N ⩽ 40.

Enumeration of unlabelled bicolored graphs by degree parities

Abstract This paper gives an ordinary generating function for unlabelled bicolored graphs with a given number of odd vertices, where the cardinalities of the bipartite sets are equal. Moreover, the generating functions for the cardinality of each bipartite set from 1 to 8 are listed.

Enumeration of digraphs with given number of vertices of odd out-degree and vertices of odd in-degree

Abstract In a digraph, a vertex of odd out(in)-degree is called an odd out(in)-vertex. This paper will give the ordinary generating functions for labelled digraphs and unlabelled withgiven numbers of odd out-vertices and odd in-vertices.