Exact and approximate D-optimal designs in polynomial regression

If the sample sizen is large enough, then the exact polynomial regression designs obtained by rounding the weights of the approximate D-optimal design to integral multiples of 1/n are D-optimal. This was shown by alaevskiî (1966) and Gaffke (1987). In this note, an efficient algorithm to determine the minimum sample sizen _d for a polynomial model of degreed is derived from a condition given by Huang (1987). Under an additional assumption we show that the conditions of Gaffke and Huang are equivalent; we verify the additional assumption for polynomial degreed40.     

Multi-level k-circulant Supersaturated Designs

In this paper we present a new method for constructing multi-level supersaturated designs with n rows, m columns and the equal occurrence property. We investigate the existence of multi-level supersaturated designs using a single generator vector and its k-cyclic permutations as rows. We find the conditions needed, in order this vector to generate a balanced supersaturated design. These conditions are simplified for the case of three level supersaturated designs. By using the proposed method three level supersaturated designs are constructed and explored. Moreover, many new, optimal and near optimal, multi-level supersaturated designs are provided as well.     

Optimal extended complete block designs for dependent observations

Optimal designs under general dependence structures are usually difficult to specify theoretically or find algorithmically. However, they can sometimes be found for a specific dependence structure and a particular parameter value. In this paper, a class of generalized binary block designs with t treatments and b blocks of size k>t is considered. Each block consists of h consecutive complete blocks and, at the end, an incomplete block of size k−ht (if k > ht). For a suitable number of blocks, a universally optimal design is found for a first-order stationary autoregressive process with positive correlations. Optimal generalized binary designs and balanced block designs are also considered. Some constructions for a universally optimal design are described. A negative dependence parameter, and some other dependence structures, are also considered.     

Efficiencies of Rounded Optimal Approximate Designs for Small Samples

Optimal exact designs are notoriously hard to study and only a few of them are known for polynomial models. Using recently obtained optimal exact designs (I mhof , 1997), we show that the efficiency of the frequently used rounded optimal approximate designs can be sensitive if the sample size is small. For some criteria, the efficiency of the rounded optimal approximate design can vary by as much as 25% when the sample size is changed by one unit. The paper also discusses lower efficiency bounds and shows that they are sometimes the best possible bounds for the rounded optimal approximate designs.     

Mixed Methods Studies in Environmental Management Research: Prevalence,Purposes and Designs

Mixed methods research, that is, the combined use of quantitative and qualitative methods in the same study, is a popular approach in several fields. However, its use in the field of environmental management and sustainable development has not been examined. The authors review the application of mixed methods research in the journal Business Strategy and the Environment in the last ten years. A total of 340 articles have been read and reviewed, and 26 mixed methods studies have been identified. The main purposes and mixed methods designs are examined. Suggestions on why and how to use mixed methods research are offered, and recommendations are provided to guide future mixed methods studies to advance our understanding of environmental management topics. Copyright © 2014 John Wiley & Sons, Ltd and ERP Environment     

TypeA operating characteristic functions of defects sampling plans

Investigations on acceptance sampling have attached rather few attention to defects inspection. As to sampling models and the corresponding operating characteristic (OC) function of single defects sampling plans, generally only typeB (process model) has been considered: sampling from a production process where the OC is conceived as a function of sample sizen, acceptance numberc, and process average number of defects per itemλ. For modern quality control, both the steadily increasing complexity of products and the need for differentiated cost calculation involve a clear demand for defects sampling in its practically most relevant form: lot-by-lot single sampling plans, where the OC (typeA, lot OC) is considered as a function of lot sizeN, sample sizen, acceptance numberc, number of defects in the lotK. The present paper develops two lot OC functions from suitable assumptions on the arrangements of the total numberK of defects over theN elements of the lot. Limiting theorems on these OC functions are used to justify to some extent the customary assumption that the Poisson process OC can serve as an approximation for typeA. The dependence of the OC functions on sample sizen, acceptance numberc, total number of defects in the lotK is described by simple formulae.     

A note on the robustness of Box-Behnken designs to the unavailability of data

Box-Behnken designs and central composite designs are efficient designs for fitting second order polynomials to response surfaces, because they use relatively small numbers of observations to estimate the parameters. In this paper we investigate the robustness of Box-Behnken designs to the unavailability of observations, in the sense of finding t _max , the maximum number of arbitrary rows in the design matrix that can be removed and still leave all of the parameters of interest estimable. The results are compared to the known results for the central composite designs found in MacEachern, Notz, Whittinghill & Zhu (1995). The blocked Box-Behnken designs are equally as robust as those that are not blocked. Received December 1997     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

Robust row-column designs for complete diallel cross experiments

This paper investigates the robustness of variance-balanced row-column designs for complete diallel cross experiments for estimating the comparisons among the general combining ability parameters against the loss of observations. A necessary and sufficient condition of robustness as per connectedness criterion is obtained. The robustness of optimal row-column designs of Gupta and Choi (1998) has been investigated for the loss of any m(≥1) observations in a column and for the loss of any two observations in the design. The study of robustness has also been conducted as per A-efficiency criterion.     

Some orthogonal arrays with 32 runs and their projection properties

Screening designs are useful for situations where a large number of factors are examined but only a few, k, of them are expected to be important. Traditionally orthogonal arrays such as Hadamard matrices and Plackett Burman designs have been studied for this purpose. It is therefore of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties. In this paper we present 15 inequivalent Hadamard matrices of order n=32 constructed from circulant cores. We study their projection properties using several well-known statistical criteria and we provide minimum generalized aberration 2 level designs with 32 runs and up to seven factors that are embedded into these Hadamard matrices. A concept of generalized projectivity and design selection of such designs is also discussed.AMS Subject Classification: Primary 62K15, Secondary 05B20     

α-optimal sampling plans for lot-by-lot defects inspection

Investigations on acceptance sampling have attached rather few importance to defects inspection. For modern quality control, both the steadily increasing complexity of products and the need for differentiated cost calculation involve a clear demand for economic defects sampling in its practically most relevant form: lot-by-lot single sampling plans, where the OC (lot OC) is considered as a function of lot sizeN, sample sizen, acceptance numberc, number of defects in the lotK. Starting from an appropriate lot OC function, the present paper develops an economic cost and control model and an economic objective function for single defects sampling plans by adapting theα-optimal sampling scheme, introduced by E. von Collani for defectives inspection, to the purposes of defects inspection. A simple and accurate approximation ofα-optimal defects plans is derived by means of a Poisson approximation of the lot OC function.     

Bayesian U-type design for nonparametric response surface prediction

This paper deals with Bayesian design over U-type designs of n runs and s factors with q levels for nonparametric response surface prediction. The criterion is developed in terms of the asymptotic approach of Mitchell et al. (Ann Statist 22: 634–651, 1994) for a specific covariance kernel. An optimal design is given in approximate design theory over the all level combinations. A connection with orthogonality and aberration is established. A lower bound for the criterion is provided, and numerical results show that this lower bound is tight.     

Optimal block designs for diallel crosses

Dey and Midha (Biometrika 83(2):484–489, 1996) constructed optimal block designs for complete diallel cross experiment using triangular partially balanced incomplete block designs with two associate classes. They listed optimal block designs for the lines in the range from 5 ≤ v ≤ 10. In this paper, we are also proposing additional optimal block designs for complete diallel cross experiment using two associate class partially balanced block designs for some additional values of v. Our method yields designs for proper and non-proper settings for complete diallel cross experiments. The proper and non proper designs are optimal in the sense of Kempthorne (Genetics 41:451–459, 1956) and non-proper designs are universally optimal in the sense of Kiefer (A survey of statistical design and linear models, North Holland, Amsterdam, 1975). The list of practically important designs is also given.     

On the construction of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">s</Emphasis><Superscript>2</Superscript>)-optimal supersaturated designs

Supersaturated designs are an important class of factorial designs in which the number of factors is larger than the number of runs. These designs supply an economical method to perform and analyze industrial experiments. In this paper, we consider generalized Legendre pairs and their corresponding matrices to construct E(s ²)-optimal two-level supersaturated designs suitable for screening experiments. Also, we provide some general theorems which supply several infinite families of E(s ²)-optimal two-level supersaturated designs of various sizes.      

Optimal allocation of measurements in a linear calibration process

A problem of allocation of measurements for a linear calibration process is considered in this article. It is assumed that a total of N measurements are made some of which may be measurements on two distinct standards, while the remaining measurements are on m different unknown specimens. We discuss allocation of the N measurements for the two standards and m unknown specimens based on A-optimality criterion, which is applied to asymptotic variances of maximum likelihood estimators for the true values of unknown specimens. It can be shown that the optimal allocation depends on the true values of unknown specimens. Hence, the user may resort to locally or Bayesian A-optimal measurement designs. Some practical solution is presented. Furthermore, the impact of prior on the allocation is also discussed.     

Zur Definition von Zufallsstichproben aus endlichen Grundgesamtheiten

Summary Suppose there is a universe which containsN elements,M of which are marked by a special property. From this universe a sample of sizen is drawn without replacement. From the Laplace-condition about the ( _n ^N ) possible outcomes of this experiment it follows that the number of marked elements in the sample is distributed hypergeometrically. It is shown that an inverse of this theorem is true, whereas the analogue inverse for the with-replacement-case is not true.Further some other evident properties of samples are derived from the Laplace-condition for both cases of the drawing scheme. For all cases of possible implications between these properties and the Laplace-condition it is decided if they are true or not.     

α-Optimal sampling plans for large acceptance numbers

Acceptance sampling by attributes is an important tool of industrial quality control. In this paper the sampling plan is selected according to a given cost situation using the assumption that one knows the probability that the number of defective items in a lot does not exceed break even quality. An approximate solution is derived for the case that either the sampling cost is small or the lot sizeN large, which both imply a large acceptance numberc.     

Extending design-optimality from an initial model to augmented models

We consider an initial linear model E(Y)=B a and an augmented model E(Y)=B a+C b. We show that, if a design is optimal for estimating K a in the initial model, then there exists a matrix L such that this design is also optimal for estimating K a+L b in the augmented model. Received: December 1998     

Designs for estimating an extremal point of quadratic regression models in a hyperball

This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The problem of estimating an extremal point is reduced to that of estimating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Müller (1997) for onedimensional quadratic regression function in the unit segment. Received February 2002     

Optimality and efficiency of circular neighbor balanced designs for correlated observations

The purpose of this paper is to apply the Kunert and Martins (2000b) method for finding optimal designs into the case of dependence. Using this method we study optimality of circular neighbor balanced designs at distances 1 and 2 under the one-dimensional interference model with errors correlated according to a circular autoregressive process. We determine the efficiency of binary designs for specified values of correlation coefficient, for which these designs are not optimal.Research partially supported by the KBN Grant Number 5 P03A 041 21.