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.     

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.     

Robustness of Optimal Chemical Balance Weighing Designs for Estimation of Total Weight

The subject of our study is the chemical balance weighing design in which the errors are correlated and have equal variances. A lower bound for the variance of estimated total weight is obtained and the necessary and sufficient conditions for the attainability of this lower bound are given. Some designs for which the lower bound is attainable are considered.     

BALANCED CHANGE-OVER DESIGNS FOR AUTOCORRELATED OBSERVATIONS

Some aspects of design of change-over experiments with autocorrelated observations are investigated. Williams' designs (1949, 1950) are seen to be balanced for autocorrelated observations.     

A-Optimal Spring Balance Weighing Designs Under Some Conditions

In this article, the estimation problem of individual weights of objects in spring balance weighing design using the criterion of A-optimality is discussed. It is assumed that the measurement errors have different variances. The lowest bound of the trace of the dispersion matrix is obtained and the conditions when this lowest bound is achieved are given. A new construction method of an A-optimal design is presented.     

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.     

Notes about A-optimal spring balance weighing design

In the paper, the investigation of the A-optimal spring balance weighing design are dealt with under the assumption of normality of errors. The topics are concerned with determining the lower bound of the trace of inverse of information matrix. Moreover, a new method of construction for optimal design is presented.     

D-Optimal Chemical Balance Weighing Designs with N ≡ 0 (mod 4) and 3 Objects

In this article, the problem of estimation of the individual weights of three objects using a chemical balance weighing design is considered. We use the criterion of D-optimality. We assume that the covariance matrix of errors is the matrix of first-order autoregressive process. Such problems were discussed in Li and Yang (2005 Li , C. H. , Yang , S. Y. ( 2005 ). On a conjecture in D-optimal designs with n ≡ 0 (mod 4) . Lin. Alg. Applic. 400 : 279 – 290 .[Crossref], [Web of Science ®] , [Google Scholar]) and also in Yeh and Lo Huang (2005 Yeh , H. G. , Lo Huang , M. N. ( 2005 ). On exact D-optimal designs with 2 two-level factors and n autocorrelated observations . Metrika 61 : 261 – 275 .[Crossref], [Web of Science ®] , [Google Scholar]). We present some results of D-optimal designs in certain class of designs with the design matrix X  ∈ M _n×3(±1) such that each column of matrix X has at least one 1 and one ?1.     

ROW AND COLUMAN DESIGNS WITH CONTIGUOUS REPLICATES

A method is given for constructing row and column designs for situations where replicates are contiguous. Designs of this type are needed in cotton variety trials. A table of generating arrays is given from which a series of resolvable designs can be constructed; these designs are called latinized α-designs. Some results from cotton variety trials are presented.     

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.     

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.     

On a paper by kageyama and mohan

Kageyama Mohan (1984) have presented three methods of constructing new incomplete block designs from balanced incomplete block designs, They raise questions about the designs which come from each of their methods, These questions are answered, Another series of group divisible designs is derived as a special case of their second method.     

Statistical properties of Rechtschaffner designs

Rechtschaffner designs are saturated designs of resolution V   in which main effects and two-factor interactions are estimable if three-factor and higher order interactions are negligible. Statistical properties of Rechtschaffner designs are studied in this paper. Best linear unbiased estimators of main effects and two-factor interactions are given explicitly and asymptotic properties of correlations between these estimators are studied as well. It is shown that designs recommended by Rechtschaffner [1967. Saturated fractions of 2ⁿ$2^n$ and 3ⁿ$3^n$ factorial designs, Technometrics 9, 569–576] are not only A-optimal but also D-optimal. Comparisons of Rechtschaffner designs with other A- and D-optimal designs of resolution V are also discussed.     

VARIATION REDUCTION AND ROBUST DESIGNS

In this paper we consider two strategies for variation reduction. One of them is the exploitation of interactions. We also discuss the role of experiments in discovering interactions and in particular the use of robust designs to obtain the interaction between control and noise factors. Then we attempt to reduce the variation in a measurement system using a robust design.     

NESTING OPTIMAL MAIN EFFECTS PLANS AND OPTIMAL FOLDOVER DESIGNS

ABSTRACT

Optimal main effects plans (MEPs) and optimal foldover designs can often be performed as a series of nested optimal designs. Then, if the experiment cannot be completed due to time or budget constraints, the fraction already performed may still be an optimal design. We show that the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEP for 4t factors in 4t + 2 points nested within it. In general, the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEPs for 4t factors in 4t + 1, 4t + 2, or 4t + 3 points and the optimal MEP for 4t + 1 factors in 4t + 4 points does not contain the optimal MEPs for 4t + 1 factors in 4t + 2 or 4t + 3 points. We also show that the runs in an orthogonal design for 4t factors in 4t + 4 points, and the optimal foldover designs obtained by folding, should be performed in a certain sequence in order to avoid the possibility of a singular X'X matrix.     

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.     

Pseudo-Youden designs balanced for intersection

If the row-column intersections of a row-column design A form a balanced incomplete block design, then A is said to be balanced for intersection. This property was originally defined for triple arrays by McSorley et al. (2005a), Section 8, where an example was presented and questions of existence were raised and discussed. We give sufficient conditions for the class of balanced grids in order to be balanced for intersection, and prove that a family of binary pseudo-Youden designs has this property.     

Balanced arrays and main-effect plans from regular group-divisible designs

A method of constructing balanced arrays of strength two and corresponding main-effect fractional factorial plans from some regular group-divisible designs is presented. Main-effect fractional factorial plans for three-level factorial experiments are constructed. The plans have reasonably high efficiencies for different single degree of freedom effects.     

THE ROBUSTNESS OF BINARY AND NON-BINARY BALANCED NESTED ROW-COLUMN DESIGNS UNDER THE UNAVAILABILITY OF BLOCKS: A COMPARISON

In cases where both exist, the balanced, binary nested row-column designs are known to be inferior to a class of balanced non-binary designs. However, if it is possible for blocks of observations to become unavailable after an experiment has commenced, a binary nested row-column design may possibly be better than a non-binary one. This paper investigates the robustness of binary and non-binary variance-balanced nested row-column designs to the unavailability of one or more blocks of observations. Robustness is measured through the C-matrices of the designs resulting from removing blocks, using optimality criteria such as A-, D-, E- and MV-optimality.     

THE CONSTRUCTION OF EQUILEVERAGE DESIGNS FOR MULTIPLE LINEAR REGRESSION

The hat matrix is widely used as a diagnostic tool in linear regression because it contains the leverages which the independent variables exert on the fitted values. In some experiments, cases with high leverage may be avoided by judicious choice of design for the independent variables. A variety of methods for constructing equileverage designs for linear regression are discussed. Such designs remove one of the factors, namely large leverage points, which can lead to nonrobust estimators and tests. In addition, a method is given for combining equileverage designs to test for lack of fit of the linear model.