Some sufficient conditions for the type 1 optimality of block designs

In this paper, we investigate the problem of determining block designs which are optimal under type 1 optimality criteria within various classes of designs having υ treatments arranged in b blocks of size k. The solutions to two optimization problems are given which are related to a general result obtained by Cheng (1978) and which are useful in this investigation. As one application of the solutions obtained, the definition of a regular graph design given in Mitchell and John (1977) is extended to that of a semi-regular graph design and some sufficient conditions are derived for the existence of a semi-regular graph design which is optimal under a given type 1 criterion. A result is also given which shows how the sufficient conditions derived can be used to establish the optimality under a specific type 1 criterion of some particular types of semi- regular graph designs having both equal and unequal numbers of replicates. Finally,some sufficient conditions are obtained for the dual of an A- or D-optimal design to be A- or D-optimal within an appropriate class of dual designs.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

On Sampling Designs with Ordered Conditional Inclusion Probabilities

In this paper we consider a family of sampling designs for which increasing first‐order inclusion probabilities imply, in a specific sense, increasing conditional inclusion probabilities. It is proved that the complementary Midzuno, the conditional Poisson, and the Sampford designs belong to this family. It is shown that designs of the family are more efficient than a comparable with‐replacement design. Furthermore, the efficiency gain is explicitly given for these designs.     

Balanced Treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k = 3, p = 3(1)10

Bechhofer and Tamhane (1981) proposed a new class of incomplete block designs called BTIB designs for comparing p ≥ 2 test treatments with a control treatment in blocks of equal size k < p + 1. All BTIB designs for given (p,k) can be constructed by forming unions of replications of a set of elementary BTIB designs called generator designs for that (p,k). In general, there are many generator designs for given (p,k) but only a small subset (called the minimal complete set) of these suffices to obtain all admissible BTIB designs (except possibly any equivalent ones). Determination of the minimal complete set of generator designs for given (p,k) was stated as an open problem in Bechhofer and Tamhane (1981). In this paper we solve this problem for k = 3. More specifically, we give the minimal complete sets of generator designs for k = 3, p = 3(1)10; the relevant proofs are given only for the cases p = 3(1)6. Some additional combinatorial results concerning BTIB designs are also given.     

Block-circulant matrices for constructing optimal Latin hypercube designs

Computer simulations are usually needed to study a complex physical process. In this paper, we propose new procedures for constructing orthogonal or low-correlation block-circulant Latin hypercube designs. The basic concept of these methods is to use vectors with a constant periodic autocorrelation function to obtain suitable block-circulant Latin hypercube designs. A general procedure for constructing orthogonal Latin hypercube designs with favorable properties and allowing run sizes being different from a power of 2 (or a power of 2 plus 1), is presented here for the first time. In addition, an expansion of the method is given for constructing Latin hypercube designs with low correlation. This expansion is useful when orthogonal Latin hypercube designs do not exist. The properties of the generated designs are further investigated. Some examples of the new designs, as generated by the proposed procedures, are tabulated. In addition, a brief comparison with the designs that appear in the literature is given.     

Lower bounds of the wrap-around -discrepancy and relationships between MLHD and uniform design with a large size

The wrap-around (WD) L₂-discrepancy has been commonly used in experimental designs. In this paper, some lower bounds of the WD L₂-discrepancy for asymmetrical U-type designs are given and the expectation and variance of midpoint Latin hypercube designs (LHD) are also obtained. Relationships between midpoint LHD and uniform designs for symmetrical and asymmetrical cases are discussed in the sense of comparing the lower bound and the expectation of squared wrap-around L₂-discrepancy of U-type designs. Some comparisons between simple random sampling and the lower bounds of U-type designs are given.     

Screening designs for model discrimination

We introduce new criteria for model discrimination and use these and existing criteria to evaluate standard orthogonal designs. We show that the capability of orthogonal designs for model discrimination is surprisingly varied. In fact, for specified sample sizes, number of factors, and model spaces, many orthogonal designs are not model discriminating by the definition given in this paper, while others in the same class of orthogonal designs are. We also use these criteria to construct optimal two-level model-discriminating designs for screening experiments. The efficacy of these designs is studied, both in terms of estimation efficiency and discrimination success. Simulation studies indicate that the constructed designs result in substantively higher likelihoods of identifying the correct model.     

Orthogonal designs for computer experiments

Computer experiments are becoming increasingly popular surrogates for physical experiments in recent years. In this paper, we propose a general procedure for constructing orthogonal designs with many levels and distinct runs. These designs are suitable for computer experiments. The basic idea of the method is to fold-over suitable orthogonal matrices. The properties of the derived designs are studied and a brief comparison with the designs in the literature is given.     

Single replicate orthogonal block designs for circulant partial diallel crosses

Some incomplete block designs for partial diallel crosses have been given in the literature. These designs are obtained by regarding the number of crosses as treatments, and consequently require several replications of each cross. The need for resorting to a partial diallel cross itself implies that it is desired to have fewer crosses. A method for constructing single replicate incomplete block designs for circulant partial diallel crosses is provided in this paper. The designs are orthogonal, and thus they retain full efficiency for estimation of the contrasts of interest.     

Efficient Use of Two Decks of Cards in Randomized Response Sampling

This paper extends the work of Russell (1976). Proof Is given that, for many parameter sets, all O:XB designs belonging to the set are connected. It is shown how an (M,S)-optinal design nay be selected from the M-optimal designs of a given parameter set. The efficiencies of these (M,S)-optimal designs are displayed, and it is concluded that the (M,S)-optimality criterion is useful for selecting designs of high efficiency.     

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k + 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.     

Proper generalized neighbor designs in circular blocks

Generalized neighbor designs relax the condition of constancy on the number of times two treatments occur as neighbors in a design. In this paper, we constructed generalized neighbor designs in circular blocks of equal size. Catalogs of these designs for block size 5, 7 and 9 are also compiled. All the designs given here are binary.     

Minimal point second order designs

This paper deals with the problem of finding nearly D-optimal designs for multivariate quadratic regression on a cube which take as few observations as possible and still allow estimation of all parameters. It is shown that among the class of all such designs taking as many observations as possible on the corners of the cube there is one which is asymptotically efficient as the dimension of the cube increases. Methods for constructing designs in this class, using balanced arrays, are given. It is shown that the designs so constructed for dimensions ≤6 compare well with existing computer generated designs, and in dimensions 5 and 6 are better than those in literature prior to 1978.     

A basic lemma and the analysis of block and Kronecker product designs

In this paper the analysis of the class of block designs whose C matrix can be expressed in terms of the Kronecker product of some elementary matrices is considered. The analysis utilizes a basic result concerning the spectral decomposition of the Kronecker product of symmetric matrices in terms of the spectral decomposition of the component matrices involved in the Kronecker product. The property (A) of Kurkjian and Zelen (1963) is generalised and the analysis of generalised property (A) designs is given. It is proved that a design is balanced factorially if and only if it is a generalised property (A) design. A method of analysis of Kronecker product block designs whose component designs are equi-replicate and proper is also suggested.     

The construction of multi-factor crossover designs in animal husbandry studies

For ethical reasons it is important to try to obtain as much useful information as possible from an animal experiment whilst minimizing the number of animals used. Crossover designs, where applicable, provide an ideal framework for achieving this. If two or more treatment factors are included in the crossover design then the reduction in total animal usage can be considerable. In this paper we consider such designs, defined as multi-factor crossover designs. The designs are applicable when there are several different treatment factors, each at t levels, to be applied to the experimental units. The motivation for investigating these designs was a study conducted at GlaxoSmithKline to determine the preference of male and female dogs for t=5 different types of bed and t=5 different bedding conditions. A construction method is given for forming universally optimal designs for t not too large. Also given is an example for the special case where the number of treatment levels t=6.     

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.     

On an assumption of bandemer and ketzel in optimal experimental design theory

In bandemee et al. [1] and ketzel [4] assumptions in the construction of optimal experimental designs are given according to which the convergence of certain iteration procedures is stated. in this paper modifications of these well-known procedures are made allowing that the given theorems stand, although in slightly different versions.     

Efficient three-level screening designs using weighing matrices

Screening is the first stage of many industrial experiments and is used to determine efficiently and effectively a small number of potential factors among a large number of factors which may affect a particular response. In a recent paper, Jones and Nachtsheim [A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 2011;43:1–15] have given a class of three-level designs for screening in the presence of second-order effects using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 1995;37:60–69]. Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8] have used conference matrices to construct definitive screening designs with good properties. In this paper, we propose a method for the construction of efficient three-level screening designs based on weighing matrices and their complete foldover. This method can be considered as a generalization of the method proposed by Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8]. Many new orthogonal three-level screening designs are constructed and their properties are explored. These designs are highly D-efficient and provide uncorrelated estimates of main effects that are unbiased by any second-order effect. Our approach is relatively straightforward and no computer search is needed since our designs are constructed using known weighing matrices.     

Symmetric designs of types V and VI

In Butler (1984a) a semi-translation block was defined and a classification given of all symmetric 2-(υ,k,λ) designs with λ>1, which contain more than one such block. In this paper we consider symmetric designs of type V and VI. We show that symmetric designs of type V are also of type VI, and in addition we show that all such designs can be obtained from a P_n,q by a construction which we give. Finally examples of proper symmetric designs of type V which are not of type VI are given.     

General upper bound for the number of blocks having a given number of treatments common with a given block

The purpose of this paper is systematically to derive the general upper bound for the number of blocks having a given number of treatments common with a given block of certain incomplete block designs. The approach adopted here is based on the spectral decomposition of N′N for the incidence matrix N of a design, where N' is the transpose of the matrix N. This approach will lead us to upper bounds for incomplete block designs, in particular for a large number of partially balanced incomplete block (PBIB) designs, which are not covered with the standard approach (Shah 1964, 1966), Kapadia (1966)) of using well known relations between blocks of the designs and their association schemes. Several results concerning block structure of block designs are also derived from the main theorem. Finally, further generalizations of the main theorem are discussed with some illustrations.