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.     

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.     

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.     

Balanced treatment incomplete block designs through integer programming

An algorithm is presented to construct balanced treatment incomplete block (BTIB) designs using a linear integer programming approach. Construction of BTIB designs using the proposed approach is illustrated with an example. A list of efficient BTIB designs for 2 ? v ? 12, v + 1 ? b ? 50, 2 ? k ? min(10, v), r ? 10, r₀ ? 20 is provided. The proposed algorithm is implemented as part of an R package.     

D-Optimal Designs for Covariate Parameters in Block Design Set Up

The problem considered is that of finding D-optimal design for the estimation of covariate parameters and the treatment and block contrasts in a block design set up in the presence of non stochastic controllable covariates, when N = 2(mod 4), N being the total number of observations. It is clear that when N ≠ 0 (mod 4), it is not possible to find designs attaining minimum variance for the estimated covariate parameters. Conditions for D-optimum designs for the estimation of covariate parameters were established when each of the covariates belongs to the interval [?1, 1]. Some constructions of D-optimal design have been provided for symmetric balanced incomplete block design (SBIBD) with parameters b = v, r = k = v ? 1, λ =v ? 2 when k = 2 (mod 4) and b is an odd integer.     

Mixture designs with orthogonal blocks

Constructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.     

BALANCED ARRAYS AND WEIGHING DESIGNS

Dey (19711, Saha (1975), Kageyama & Saha (1983) and others have shown how optimum chemical balance weighing designs can be constructed from the incidence matrices of balanced incomplete block (BIB) designs. In this paper, it is shown that weighing designs can be constructed from some suitably chosen two-symbol balanced arrays of strength two, which need not always be incidence matrices of BIB designs. The findings lead us to construct new optimum chemical balance weighing designs from incidence matrices of BIB designs.     

OPTIMUM SINGULAR SPRING BALANCE WEIGHING DESIGNS WITH NON-HOMOGENEITY OF THE VARIANCES OF ERRORS FOR ESTIMATING THE TOTAL WEIGHT

The problem of estimation of the total weight of objects using a singular spring balance weighing design with non-homogeneity of the variances of errors has been dealt with in this paper. Based on a theorem by Katulska (1984) giving a lower bound for the variance of the estimated total weight, a necessary and sufficient condition for this lower bound to be attained is obtained. It is shown that weighing designs for which the the lower bound is attainable, can be constructed from the incidence matrices of (α₁,.,α_t)-resolvable block designs, α-resolvable block designs, singular group divisible designs, and semi-regular group divisible designs.     

Row-column designs for comparing treatments with a control

This paper considers the problem of the design and analysis of experiments for comparing several treatments with a control when heterogeneity is to be eliminated in two directions. A class of row-column designs which are balanced for treatment vs. control comparisons (referred to as the balanced treatment vs. control row-column or BTCRC designs) is proposed. These designs are analogs of the so-called BTIB designs proposed by Bechhofer and Tamhane (Technometrics 23 (1981) 45–57) for eliminating heterogeneity in one direction. Some methods of analysis and construction of these designs are given. A measure of efficiency of BTCRC designs in terms of the A-optimality criterion is derived and illustrated by several examples.     

Resistant BTIB desings

Hedayat and John (1974) defined the concept of a resistant BIB design as one that remains variance-balanced upon loss of all the observations on a treatment. In this paper we consider the designs with supplemented balance (Pearce (1960)), or BTIB designs (Bechhofer and Tamhane (1981)). We investigate a subclass of these designs to determine which of these retain their ‘balance’ upon loss of all observations on a test treatment. We study the properties of such designs, give various methods of construction and examine their efficiencies with respect to the A-criterion.     

D-optimal designs for covariate models

The problem of finding D-optimal or D-efficient designs in the presence of covariates is considered under a completely randomized design set-up with v treatments, k covariates and N experimental units. In contrast to Lopes Troya [Lopes Troya, J., 1982, Optimal designs for covariates models. Journal of Statistical Planning and Inference, 6, 373–419.], who considered this problem in the equireplicate case, we do not assume that N/v is an integer, and this allows us to study situations where no equireplicate design exists. Even when N/v is an integer, it is seen quite counter-intuitively that there are situations where a non-equireplicate design outperforms the best equireplicate design under the D-criterion.     

D-1-Partially efficiency-balance designs with at most two efficiency classes

A wide class of block designs admitting a simple analysis has been considered. The statistical properties of such designs have been indicated and the problems relating to their characterization and construction have been investigated.     

Some methods for constructing C-designs

Block designs having a property described by Caliński (1971) and Saha (1976), referred to as C-property, are further considered. A necessary and sufficient condition for a block design to have the C-property is given by Saha (1976) and another by Ceranka and Koz?owska (1983). In this paper some methods for constructing such block designs are presented.     

Optimal orthogonal block designs for four-mixture components in two blocks based on F-squares for Becker's models and K-model

Orthogonal block designs in mixture experiments have been extensively studied by various authors. Aggarwal et al. [M.L. Aggarwal, P. Singh, V. Sarin, and B. Husain, Mixture designs in orthogonal blocks using F-squares, METRON – Int. J. Statist. LXVII(2) (2009), pp. 105–128] considered the case of components assuming the same volume fractions and obtained mixture designs in orthogonal blocks using F-squares. In this paper, we have used the class of designs presented by Aggarwal et al. and have obtained D-, A- and E-optimal orthogonal block designs for four components in two blocks for Becker's mixture models and K-model, respectively. Orthogonality conditions for the considered models are also given.     

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.     

Selection and screening procedures to determine optimal product designs

To compare several promising product designs, manufacturers must measure their performance under multiple environmental conditions. In many applications, a product design is considered to be seriously flawed if its performance is poor for any level of the environmental factor. For example, if a particular automobile battery design does not function well under temperature extremes, then a manufacturer may not want to put this design into production. Thus, this paper considers the measure of a product's quality to be its worst performance over the levels of the environmental factor. We develop statistical procedures to identify (a near) optimal product design among a given set of product designs, i.e., the manufacturing design that maximizes the worst product performance over the levels of the environmental variable. We accomplish this by intuitive procedures based on the split-plot experimental design (and the randomized complete block design as a special case); split-plot designs have the essential structure of a product array and the practical convenience of local randomization. Two classes of statistical procedures are provided. In the first, the δ-best formulation of selection problems, we determine the number of replications of the basic split-plot design that are needed to guarantee, with a given confidence level, the selection of a product design whose minimum performance is within a specified amount, δ, of the performance of the optimal product design. In particular, if the difference between the quality of the best and second best manufacturing designs is δ or more, then the procedure guarantees that the best design will be selected with specified probability. For applications where a split-plot experiment that involves several product designs has been completed without the planning required of the δ-best formulation, we provide procedures to construct a ‘confidence subset’ of the manufacturing designs; the selected subset contains the optimal product design with a prespecified confidence level. The latter is called the subset selection formulation of selection problems. Examples are provided to illustrate the procedures.     

Neighbor-Balanced Row-column Designs

In this article, row-column designs incorporating directional neighbor effects have been studied. A row-column design is said to be neighbor balanced if every treatment has all other treatments appearing as a neighbor a constant number of times. We considered here three different situations under row-column setup incorporating neighbor effects viz., row-column design with one-sided neighbor effect, two-sided neighbor effect, and four-sided neighbor effect. The information matrices for all the situations for estimating the direct and neighbor effects of treatments have been derived. Methods of constructing neighbor-balanced row-column designs have been developed and its characterization properties have been studied.     

Orthogonally blocked mixture component–amount designs via projections of F-squares

Orthogonal block designs for Scheffé’s quadratic model have been considered previously by Draper et al. (1993), John (1984), Lewis et al. (1994) and Prescott, Draper, Dean, and Lewis (1993). Prescott and Draper (2004) obtained mixture component–amount designs via projections of standard mixture designs, viz., the simplex-lattice, the simplex-centroid and the orthogonally blocked mixture designs based on latin squares. Aggarwal, Singh, Sarin, and Husain (2009) considered the case of components assuming equal volume fractions and obtained mixture designs in orthogonal blocks using F-squares. In this paper, we construct orthogonal blocks of two and three mixture component–amount blends by projecting the class of four component mixture designs presented by Aggarwal et al. (2009).     

On a class of variance balanced incomplete block designs

In this paper variance balanced incomplete block designs have been constructed for situations when suitable BIB designs do not exist for a given number of treatments, because of the contraints bk=vr, λ(v-1) = r(k-l). These variance balanced designs are in unequal block sizes and unequal replications.     

An infinite class of PBIB designs

In this paper, an infinite class of partially balanced incomplete block (PBIB) designs of m+1 associate classes is constructed through the use of a series of row-orthogonal matrices known as partially balanced orthogonal designs (PBOD) of m-associate classes. For the purpose, a series of PBOD is obtained through a method described herein. An infinite class of regular GD designs is also reported.