A survey and evaluation of methods for determination of combinatorial equivalence of factorial designs

Equivalent factorial designs have identical statistical properties for estimation of factorial contrasts and for model fitting. Non-equivalent designs, however, may have the same statistical properties under one particular model but different properties under a different model. In this paper, we describe known methods for the determination of equivalence or non-equivalence of two-level factorial designs, whether they be regular factorial designs, non-regular orthogonal arrays, or have no particular structure. In addition, we evaluate a number of potential fast screening methods for detecting non-equivalence of designs. Although the paper concentrates mainly on symmetric designs with factors at two levels, we also evaluate methods of determining combinatorial equivalence and non-equivalence of three-level designs and indicate extensions to larger numbers of levels and to asymmetric designs.     

Designs for cropping systems research

The present article establishes equivalence between extended group divisible (EGD) designs and designs for crop sequence experiments. This equivalence has encouraged the agricultural experimenters to use EGD designs for their experimentation. Some real life applications of EGD designs have been given. It has also been shown that several existing association schemes are special cases of EGD association scheme. Some methods of construction of EGD designs are also given. A catalogue of EGD designs obtainable through methods of construction along with efficiency factors of various factorial effects is also presented. In some crop sequence experiments that are conducted to develop suitable integrated nutrient supply system of a crop sequence, the treatments do not comprise of a complete factorial structure. The experimenter is interested in estimating the residual and direct effect of the treatments along with their cumulative effects. For such experimental settings block designs with two sets of treatments applied in succession are the appropriate designs. The correspondence established between row–column designs and block designs for two stage experiments by Parsad et al. [2003. Structurally incomplete row–column designs. Comm. Statist. Theory Methods 32(1), 239–261] has been exploited in obtaining designs for such experimental situations. Some open problems related to designing of crop sequence experiments are also given.     

Characterization of minimum aberration mixed factorials in terms of consulting designs

In this paper, by introducing a concept of consulting design and based on the connection between factorial design theory and coding theory, we obtain combinatorial identities that relate the wordlength pattern of a regular mixed factorial design to that of its consulting design. According to these identities, we further-more establish the general and unified rules for identifying minimum aberration mixed factorial designs through their consulting designs. It is an improvement and generalization of the results in Mukerjee and Wu (2001). This paper is supported by NNSF of China grant No. 10171051 and RFDP grant No. 1999005512.     

A note on the connection between uniformity and generalized minimum aberration

Discrete discrepancy has been utilized as a uniformity measure for comparing and evaluating factorial designs. In this paper, for asymmetrical factorials, we give some linkages between uniformity measured by the discrete discrepancy and other criteria, such as generalized minimum aberration (Xu and Wu, 2001) and minimum projection variance (Ai and Zhang, 2004). These close linkages show a significant justification for the discrete discrepancy used to measure uniformity of factorial designs, and provide an additional rationale for using uniform designs. This research was partially supported by the NNSF of China (No. 10441001), the Key Project of Chinese Ministry of Education (No. 105119) and the Project-sponsored by SRF for ROCS (SEM) (No. [2004]176).     

Confounded row–column designs

Confounded row–column designs for factorial experiments are considered and a simple method of construction using the classical method of confounding is described. Partially confounded designs are also studied and a method for generating Rao's (1946) designs is presented.     

Generators for asymmetrical factorial experiments

The paper lists generators for asymmetrical factorial experiments consisting of 200 or fewer treatment combinations and up to 7 factors each having 7 or fewer levels. The generators define fractional or confounded single replicate factorial designs.     

D‐optimal minimax fractional factorial designs

The D‐optimal minimax criterion is proposed to construct fractional factorial designs. The resulting designs are very efficient, and robust against misspecification of the effects in the linear model. The criterion was first proposed by Wilmut & Zhou (2011); their work is limited to two‐level factorial designs, however. In this paper we extend this criterion to designs with factors having any levels (including mixed levels) and explore several important properties of this criterion. Theoretical results are obtained for construction of fractional factorial designs in general. This minimax criterion is not only scale invariant, but also invariant under level permutations. Moreover, it can be applied to any run size. This is an advantage over some other existing criteria. The Canadian Journal of Statistics 41: 325–340; 2013 © 2013 Statistical Society of Canada     

Asymptotic relative efficiency of designs for factorial paired comparison experiments

A general approach for comparing designs of paired comparison experiments on the basis of the asymptotic relative efficiencies, in the Bahadur sense, of their respective likelihood ratio tests is discussed and extended to factorials. Explicit results for comparing five designs of 2^q factorial paired comparison experiments are obtained. These results indicate that some of the designs which require comparison of fewer distinct pairs of treatments than does the completely balanced design are, generally, more efficient for detecting main effects and/or certain interactions. The developments of this paper generalize the work of Littell and Boyett (1977) for comparing two designs of R x C factorial paired comparison experiments.     

Interaction balance for symmetrical factorial designs with generalized minimum aberration

There are two different systems of contrast parameterization when analyzing the interaction effects among the factors with more than two levels, i.e., linear-quadratic system and orthogonal components system. Based on the former system and an ANOVA model, Xu and Wu (2001) introduced the generalized wordlength pattern for general factorial designs. This paper shows that the generalized wordlength pattern exactly measures the balance pattern of interaction columns of a symmetrical design ground on the orthogonal components system, and thus an alternative angle to look at the generalized minimum aberration criterion is given. This work is partially supported by NNSF of China grant No. 10231030.     

On Kronecker block designs for factorial experiments

In this paper the use of Kronecker designs for factorial experiments is considered. The two-factor Kronecker design is considered in some detail and the efficiency factors of the main effects and interaction in such a design are derived. It is shown that the efficiency factor of the interaction is at least as large as the product of the efficiency factors of the two main effects and when both the component designs are totally balanced then its efficiency factor will be higher than the efficiency factor of either of the two main effects. If the component designs are nearly balanced then its efficiency factor will be approximately at least as large as the efficiency factor of either of the two main effects. It is argued that these designs are particularly useful for factorial experiments.Extensions to the multi-factor design are given and it is proved that the two-factor Kronecker design will be connected if the component designs are connected.     

Factorial-balance and the analysis of designs with factorial structure

Cotter, John and Smith (1973) have given conditions for an incomplete block design to have orthogonal factorial structure. Further results on the intra-block analysis of such designs are given. The concept of balance in factorial design is discussed and results are given which enable the degree of balance in generalised cyclic designs to be determined.     

Invariance of generalized wordlength patterns

The generalized wordlength pattern (GWLP) introduced by Xu and Wu [2001. Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29, 1066–1077] for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang [2004. Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285] defined the J$J$-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the J$J$-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the J$J$-characteristics are not. We briefly discuss some implications of these results.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

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.     

Optimum covariate designs in partially balanced incomplete block (PBIB) design set-ups

The use of covariates in block designs is necessary when the covariates cannot be controlled like the blocking factor in the experiment. In this paper, we consider the situation where there is some flexibility for selection in the values of the covariates. The choice of values of the covariates for a given block design attaining minimum variance for estimation of each of the parameters has attracted attention in recent times. Optimum covariate designs in simple set-ups such as completely randomised design (CRD), randomised block design (RBD) and some series of balanced incomplete block design (BIBD) have already been considered. In this paper, optimum covariate designs have been considered for the more complex set-ups of different partially balanced incomplete block (PBIB) designs, which are popular among practitioners. The optimum covariate designs depend much on the methods of construction of the basic PBIB designs. Different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of products of matrices viz. Khatri–Rao product, Kronecker product have been conveniently used to construct optimum covariate designs with as many covariates as possible.     

Some new factor screening designs using the search linear model

Factor screening designs for searching two and three effective factors using the search linear model are discussed. The construction of such factor screening designs involved finding a fraction with small number of treatments of a 2^m factorial experiment having the property P_2t (no 2t columns are linearly dependent) for t=2 and 3. A ‘Packing Problem’ is introduced in this connection. A complete solution of the problem in one case and partial solutions for the other cases are presented. Many practically useful new designs are listed.     

A unified theory for generalized cyclic designs

Properties of abelian groups are used to unify the construction of cyclic and generalized cyclic designs. Necessary and sufficient conditions are obtained for the generation of fractional sets of blocks, and also for the generation of resolvable designs. It is shown how to apply these results in conjunction with efficiency arguments to obtain useful designs.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block designs.     

Optimal saturated block designs when observations are correlated

A- and MV-optimal block designs are identified in the class of minimally connected designs when the observations within blocks are spatially correlated. All connected designs are shown to be D-equal regardless of the correlation structure, and a sufficient condition for E-optimality is presented. Earlier results for the uncorrelated case are strengthened.     

Incomplete row-column designs with factorial treatment structure for estimating main effects with full efficiency

In this paper, we propose two methods of constructing row-column designs for factorial experiments. The constructed designs have orthogonal factorial structure with balance and permits estimation of main effects with full efficiency.