ON THE ESTIMATION OF TOTAL WEIGHT IN SINGULAR SPRING BALANCE WEIGHING DESIGNS UNDER THE CO VARIANCE MATRIX OF ERRORS s2G

The problem of the estimation of the linear combination of weights, c′w, in a singular spring balance weighing design when the error structure takes the form E(ee′) =s?²G has been studied. A lower bound for the variance of the estimated linear combination of weights is obtained and a necessary and sufficient condition for this lower bound to be attained is given. The general results are applied to the case of the total of the weights. For a specified form for G, some optimum spring balance weighing designs for the estimated total weight are found.     

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.     

On the Construction of Nearest–Neighbour Balanced Row–Column Designs

Nearest–neighbour balance is considered a desirable property for an experiment to possess in situations where experimental units are influenced by their neighbours. This paper introduces a measure of the degree of nearest–neighbour balance of a design. The measure is used in an algorithm which generates nearest–neighbour balanced designs and is readily modified to obtain designs with various types of nearest–neighbour balance. Nearest–neighbour balanced designs are produced for a wide class of parameter settings, and in particular for those settings for which such designs cannot be found by existing direct combinatorial methods. In addition, designs with unequal row and column sizes, and designs with border plots are constructed using the approach presented here.     

Recent developments in D-optimal designs

In the paper, some issues concerned with the determining spring balance weighing designs satisfying the criterion of D-optimality under the assumption measurement errors are uncorrelated and they have the same variances are discussed. In addition, highly D-efficient spring balance weighing designs are also considered. Some conditions under which any spring balance weighing design is regular D-optimal or highly D-efficient are proved. What is more, new construction methods of regular D-optimal and highly D-efficient spring balance weighing designs are presented.     

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.     

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.     

Use of balanced bipartite weighing designs as chemical balance designs

This paper deals with the use of balanced bipartite weighing designs as chemical balance designs when there Is a restriction on the number of objects that can be placed on either pan..The variance of each estimated weight and the covarlance of each pair of estimated weights are worked out.The use of additional weighing when the design matrix is singular is discussed.     

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2^p)+O(3k³)+O(2^k) operations to check two 2^k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.     

D-optimal weighing designs and optimal chemical balance weighing designs for estimating the total weight

This paper provides D-optimal spring balance designs for estimating individual weights when the number of objects to be weighed in each weighing, B, is fixed. D-optimal chemical balance designs for estimating total weight under both homogeneous and nonhomogeneous error variances are found when the number of objects weighed in each weighing is ≥ B, a fixed number.

We indicate the restriction used in Chacko & Dey(1978) and Kageyama(1988), i.e. that chemical designs X be restricted to designs in which exactly “a” objects are replaced on the left pan and exactly “b” on the right pan in each of the weighings for a, b > 0, is unnecessary.     

Regular E-Optimal Spring Balance Weighing Designs with Correlated Errors

The problems linked with an E-optimal spring balance weighing design with correlated errors are discussed. The topic is focus on the determining the maximal eigenvalue of the inverse of the information matrix of estimators. The constructing method of the E-optimal design, based on the incidence matrices of balanced incomplete block designs, is presented.     

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.     

Row-column designs with minimal units

A new class of row-column designs is proposed. These designs are saturated in terms of eliminating two-way heterogeneity with an additive model. The proposed designs are treatment-connected, i.e., all paired comparisons of treatments in the designs are estimable in spite of the existence of row and column effects. The connectedness of the designs is justified from two perspectives: linear model and contrast estimability. Comparisons with other designs are studied in terms of A-, D-, E-efficiencies as well as design balance.     

A note on the connectivity of generalized cyclic designs

A necessary and sufficient condition is given to ensure that a generating blook of a generalized cyclic design will give rise to a connected design. The use of disconnected designs is briefly discussed.     

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.     

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

A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2^m 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.     

On the construction of a class of efficient row-column designs

In this paper a method for the construction of a class of row-column designs with good statistical properties and high efficiency is presented. The class of designs produced is shown to exhibit balance, orthogonality and adjusted orthogonality. The efficiencies of these designs are investigated in detail, and they are shown to be very high, and possibly maximal in some cases.     

Multiple comparisons for an unbalanced a × b design under mixed models with interaction

Consider a two-way factorial experiment involving a “treatment” factor A with fixed effects, a “blocking” factor B with random effects, and interaction effects perhaps non-negligible. The degree of balance required for multiple comparison procedures to be applicable for the comparison of the treatment effects using ordinary least-squares estimates is investigated. For main effects to be estimated independently of MSAB, a sufficient condition is that the design consist of identical blocks, a strong condition of proportional frequencies. Surprisingly, under this condition of proportional frequencies, MSAB does not provide an appropriate variance estimate for inferences on each treatment contrast, even though the statistics F = MSA/MSAB is appropriate for testing equality of the treatment effects. In short, when factor B is random, standard methods of multiple comparisons apply using the interaction mean square MSAB as a variance estimator only when the treatment-block incidences n_n are constant. Nevertheless, for designs with identical blocks, appropriate variance estimates can be identified to allow for conservative or approximate multiple comparisons. This is illustrated for certain treatment balanced designs for comparisons with a control.     

On nested block designs geometry

Nested block designs and block designs properties such as orthogonality, orthogonal block structure and general balance are examined using the concept of a commutative quadratic subspace and standard properties of orthogonal projectors. In this geometrical context conditions for existence of the best linear unbiased estimators of treatment contrasts are also discussed.     

DT-optimum designs for model discrimination and parameter estimation

The paper introduces DT-optimum designs that provide a specified balance between model discrimination and parameter estimation. An equivalence theorem is presented for the case of two models and extended to an arbitrary number of models and of combinations of parameters. A numerical example shows the properties of the procedure. The relationship with other design procedures for parameter estimation and model discrimination is discussed.     

Balanced block designs

In the present paper an attempt has been made to characterize and unify the three different concepts of balancing in incomplete block designs, namely (i) variance balance, (ii) efficiency balance and (iii) pairwise balance. Simple characterizations of variance balance and efficiency balance have been given using the P matrix. A method of constructing efficiency balanced (EB) and variance balanced designs has also been presented.