Characterization of Balanced Fractional 3 m Factorial Designs of Resolution III

We consider a fractional 3 ^m factorial design derived from a simple array (SA), which is a balanced array of full strength, where the non negligible factorial effects are the general mean and the linear and quadratic components of the main effect, and m ≥ 2. In this article, we give a necessary and sufficient condition for an SA to be a balanced fractional 3 ^m factorial design of resolution III. Such a design is characterized by the suffixes of indices of an SA.     

Existence Conditions for Balanced Fractional 2m Factorial Designs of Resolution 2l + 1 Derived from Simple Arrays

We consider a fractional 2^m factorial design derived from a simple array (SA) such that the (? + 1)-factor and higher-order interactions are negligible, where 2? ? m. The purpose of this article is to give a necessary and sufficient condition for an SA to be a balanced fractional 2^m factorial design of resolution 2? + 1. Such a design is concretely characterized by the suffixes of the indices of an SA.     

2 m 4 n designs with resolution III or IV containing clear two-factor interaction components

The orthogonal arrays with mixed levels have become widely used in fractional factorial designs. It is highly desirable to know when such designs with resolution III or IV have clear two-factor interaction components (2fic’s). In this paper, we give a complete classification of the existence of clear 2fic’s in regular 2 ^m 4 ⁿ designs with resolution III or IV. The necessary and sufficient conditions for a 2 ^m 4 ⁿ design to have clear 2fic’s are given. Also, 2 ^m 4 ⁿ designs of 32 runs with the most clear 2fic’s are given for n = 1,2.      

COMPUTER AIDED-CONSTRUCTION OF D-OPTIMAL 2m FRACTIONAL DESIGNS OF RESOLUTION V

A new exchange algorithm for construction of 2^mD-optimal fractional factorial design (FFD) is devised. This exchange algorithm is a modification of the one due to Fedorov (1969, 1972) and is an improvement over similar algorithm due to Mitchell (1974) and Galil & Kiefer (1980). This exchange algorithm is then used to construct 54 D-optimal 2^m-FFD's of resolution V for m = 4,5,6.     

On some optimal fractional 2m factorial designs of resolution V

This paper presents the trace of the covariance matrix of the estimates of effects based on a fractional 2^m factorial (2^m-FF) design T of resolution V for the following two cases: One is the case where T is constructed by adding some restricted assemblies to an orthogonal array. The other is one where T is constructed by removing some restricted assemblies from an orthogonal array of index unity. In the class of 2^m-FF designs of resolution V considered here, optimal designs with respect to the trace criterion, i.e. A-optimal, are presented for m = 4, 5, and 6 and for a range of practical values of N (the total number of assemblies). Some of them are better than the corresponding A-optimal designs in the class of balanced fractional 2^m factorial designs of resolution V obtained by Srivastava and Chopra (1971b) in such a sense that the trace of the covariance matrix of the estimates is small.     

Reverse Foldovers for Blocked 2 m−k Fractional Factorial Designs

Two-level regular fractional factorial designs are often used in industry as screening designs to help identify early on in an experimental process those experimental or system variables which have significant effects on the process being studied. When the experimental material to be used in the experiment is heterogenous or the experiment must be performed over several well-defined time periods, blocking is often used as a means to improve experimental efficiency by removing the possible effects of heterogenous experimental material or possible time period effects. In a recent article, Li and Jacroux (2007 Li , F. , Jacroux , M. (2007). Optimal foldover plans for blocked 2 ^m?k fractional factorial designs. J. Statsist. Plann. Infer 137:2434–2452. [Google Scholar]) suggested a strategy for constructing optimal follow-up designs for blocked fractional factorial designs using the well-known foldover technique in conjunction with several optimality criteria. In this article, we consider the reverse foldover problem for blocked fractional factorial designs. In particular, given a 2^(m+p)?(p+k) blocked fractional factorial design D, we derive simple sufficient conditions which can be used to determine if there exists a 2^{(m+p?1)?(p?1+k+1)} initial fractional factorial design d which yields D as a foldover combined design as well how to generate all such d. Such information is useful in developing an overall experimental strategy in situations where an experimenter wants an overall blocked fractional factorial design with “desirable” properties but also wants the option of analyzing the observed data at the halfway mark to determine if the significant experimental variables are obvious (and the experiment can be terminated) or if a different path of experimentation should be taken from that initially planned.     

On the number of 2n-p fractional factorials of resolution III

An algorithm is specified and demonstrated which will compute the total number of ways a 2ⁿ factorial design may be partitioned into 2^p mutually exclusive 2^n-p fractional factorial designs, each having resolution III. The results of its application to all designs possessing resolution III fractions for n=5,…,20 are also given.     

Characteristic polynomials of the information matrices of balanced fractional 3m factorial designs of resolution v

An explicit expression for the characteristic polynomial of the information matrix M_T of a balanced fractional 3^m factorial (3^m-BFF) design T of resolution V is obtained by utilizing the algebraic structure of the underlying multidimentional relationship. Also by using of the multidimensional relationship algebra, the trace and the determinant of the covariance matrix of the estimates of effects are derived.     

Balanced arrays of strength 4 and balanced fractional 3m factorial designs

A connection between a balanced fractional 2^m factorial design of resolution 2l + 1 and a balanced array of strength 2l with index set {μ₀, μ₁,…, μ_2l} was established by Yamamoto, Shirakura and Kuwada (1975). The main purpose of this paper is to give a connection between a balanced fractional 3^m factorial design of resolution V and a balanced array of strength 4, size N, m constraints, 3 levels and index set {λ_l0l1l2}.     

2m fractional factorial designs of resolution V with high A-efficiency, 7 ⩽ m ⩽ 10

We present 111 2^m fractional factorial designs of resolution V for 7 ⩽ m ⩽ 10. These designs are the best known to the authors with respect to the A-optimality criterion (as of October 1995).     

On the characteristic polynomial of the information matrix of balanced fractional sm factorial designs of resolution Vp,q

An explicit expression for the characteristic polynomial of the information matrix for a balanced fractional s^m factorial design of resolution V_{p, q} (in particular, when p = q = s − 1, of resolution V) is obtained by utilizing the decomposition of a multidimensional relationship algebra into its four two-sided ideals. Furthermore, by use of the algebraic structure of the underlying multidimensional relationship, the trace and the determinant of the covariance matrix of the estimates of effects to be interest are derived.     

Balanced fractional factorial designs of resolution 2l+1 for interesting effects orthogonal to some nuisance parameters: 2m1+m2 series

For 2^m₁+m₂ factorial designs, this paper investigates balanced fractional 2^m₁ factorial designs of resolution 2l+1 with some nuisance parameters concerning the second factors. They are derivable from partially balanced arrays and further permit estimation of the effects up to the l-factor interactions concerning the first factors orthogonally to the nuisance parameters.     

Characterization of information matrices for balanced two-level fractional factorial designs of odd resolution derived from two-symbol simple arrays

We consider a balanced fractional 2^m factorial design of resolution 2?+1 which permits estimation of all factorial effects up through ?-factor interactions under the situation in which all (?+1)-factor and higher order interactions are to be negligible for an integer satisfying [m/2]<lE;?m, where [x] denotes the greatest integer not exceeding x. This paper investigates algebraic structure of the information matrix of such a design derived from a simple array through that of an atomic array. We obtain an explicit expression for the irreducible matrix representation based on the above design and its algebraic properties. The results in this paper will be useful to characterize the designs under consideration.     

A-optimal partially balanced fractional 2m1+m2 factorial designs of resolution V,with 4≤m1&gt;+m2≤6

By use of the algebraic structure, we obtain an explicit expression for the characteristic polynomial of the information matrix of a partially balanced fractional 2^m₁+m₂ factorial design of resolution V derived from a partially balanced array. For 4≤m₁+m₂≤6, A-optimal designs considered here are also presented for reasonable number of assemblies.     

Analysis of variance of balanced fractional 2n factorial designs of resolution 2l+1

By use of the algebraic structure of the triangular multidimensional partially balanced association scheme, we present the analysis of variance and the hypotheses testing of a balanced fractional 2ⁿfactorial design of resolution 2l+1, which is derived from a balanced array of strength 2l.     

On the use of partial confounding for the construction of alternative regular two-level blocked fractional factorial designs

In this paper, we consider experimental situations in which a regular fractional factorial design is to be used to study the effects of m two-level factors using n=2^m−k experimental units arranged in 2^p blocks of size 2^m−k−p. In such situations, two-factor interactions are often confounded with blocks and complete information is lost on these two-factor interactions. Here we consider the use of the foldover technique in conjunction with combining designs having different blocking schemes to produce alternative partially confounded blocked fractional factorial designs that have more estimable two-factor interactions or a higher estimation capacity or both than their traditional counterparts.     

Modifying and using Yates' algorithm

It is well known that Yates' algorithm can be used to estimate the effects in a factorial design. We develop a modification of this algorithm and call it modified Yates' algorithm and its inverse. We show that the intermediate steps in our algorithm have a direct interpretation as estimated level-specific mean values and effects. Also we show how Yates' or our modified algorithm can be used to construct the blocks in a 2 ^k factorial design and to generate the layout sheet of a 2 ^k−p fractional factorial design and the confounding pattern in such a design. In a final example we put together all these methods by generating and analysing a 2^6-2 design with 2 blocks.