Breaking alias chains in fractional factorials

Some experimenters carry out their investigation in stages. They begin with an initial 2^n-p fraction of resolution IV, in which the main effects are clean and the interactions are aliased in chains, Then, having analyzed the initial experiment, they plan further runs to isolate certain interactions by breaking the chains. In this paper a method called semifolding, for choosing the points in the second experiment, is presented.     

On the problem of augmented fractional factorial designs

Let D be a saturated fractional factorial design of the general K₁ x K₂ ...x K_t factorial such that it consists of m distinct treatment combinations and it is capable of providing an unbiased estimator of a subvector of m factorial parameters under the assumption that the remaining k-m,t (k = H it ) factorial parameters are negligible. Such a design will not provide an unbiased estimator of the varianceσ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatmentcombinations with the aim to estimate ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatment combinations with the aim to estimate σ² unbiasedly. The problem then is how to select the c treatment combinations such that the augmented design D retains its optimality property. This problem, in all its generality is extremely complex. The objective of this paper is to provide some insight in the problem by providing a partial answer in the case of the 2^tfactorial, using the d-optimality criterion.     

On aliasing and generalized defining relationships of factorial arrangements

The concepts of defining contrast (DC), generalized defining relationship (GDR) and aliasing structure (AS) are now well established in the terminology of regression analysis and factorial design theory. There is no complete agreement in the literature about the meaning of regular and irregular fractional factorial designs. This paper provides a workable definition of a regular fraction from a symmetrial prime-powered factorial. It characterizes the uniqueness of the GDR for fractions from the most general factorial. Results are also présentés on the uniqueness of the GDR for regular designs, on orthogonality aspects of regular and irregular designs, and on group-theoretic generation of the complete aliasing structure. Examples are provided to illustrate the developments.     

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.     

Graph based isomorph-free generation of two-level regular fractional factorial designs

We provide a new necessary and sufficient check for testing the isomorphism of two 2-level regular fractional factorial designs. The approach is based on modeling fractional factorial designs as bipartite graphs. We employ an efficient canonical graph labeling approach to compare two designs for isomorphism. We then improve upon the existing non-isomorphic fractional factorial design generation algorithm by reducing the number of candidate designs from which isomorphs need to be removed. Not only does our method generate non-isomorphic designs much faster, it is also able to generate designs with run sizes of 2048 and 4096 runs, which were not generated by the existing methods.     

The robustness of chain block designs and coat-of-mail designs

Chain block designs are relatively vulnerable to loss of information when missing values or outliers may occur. An alternative class of designs, coat-of-mail designs, are proposed and the relative robustness of the two types of design are compared.     

Algebraic statistics of level codings for fractional factorial designs

We discuss the applications of algebraic statistics to fractional factorial design with special emphasis on the choice of level coding. In particular, we deal with the theory of Bayley's level codings in that framework.     

An optimal selection of two-level regular single arrays for robust parameter experiments

Efforts have been made in the literature to find optimal single arrays which work best for the robust parameter experiments. However, examples show that in many cases the optimal designs obtained by the existing criteria cloud not attain the maximum number of clear interested effects for robust parameter experiments. In this paper, through a similar way of Zhang et al. (2008) (ZLZA, in short), an aliasing pattern to measure the confounding between the interested effects and other effects for the case of robust parameter designs is introduced. A new criterion for selecting optimal two-level regular single arrays is proposed. In the consideration of the criterion, two rank-orders of effects are suggested: one is based on the interest of experimenters and the other is under the usual effect hierarchy principle. The optimal designs are tabulated in the appendix.     

A New Conjoint Analysis Procedure with Application to Marketing Research

Abstract

In this article we propose some extensions and applications of the nonparametric combination of dependent rankings (see Pesarin, F., Lago, A. (2000). Nonparametric combination of department rankings with applications to the quality assessment of industrial products. Metron LVIII (1–2):39–52.) This methodology is applied to Conjoint Analysis in order to aggregate (ex ante) preferences from a group of individuals. Furthermore, a new global association test (GAT) is introduced in order to test for the association of the global ranking with all attributes of interest. The GAT procedure allows the experimenter to have clear indications on significant attributes by considering the intensity of the optimal weights given by the procedure itself. This may help the experimenter in interpreting the usual analysis involving the normal plot for detecting active effects.     

Bayes a-opttmal and efficient block designs for comparing test treatments with a standard treatment

A sufficient condition for the Bayes A-optimality of block designs when comparing a standard treatment with v test treatments is given by Majumdar. (In:Optimal Design and Analysis of Experiments, Y. Dodge, V. V. Fedorov and H. P. Wynn (Eds.), 15-27, North-Holland, 1988). The priors that he considers depend on a constant α ε [0, ∞), with α - 0 corresponding to no prior information at all. The given sufficient condition, consequently, also depends on a. Large families of optimal and highly efficient designs are only known for the case α - 0. We will show how some of the results for α - 0 can be extended to obtain large families of optimal and highly efficient designs for arbitrary values of α. In addition, these results are useful when considering design robustness against an improper choice of α.     

A method of selecting the levels of regressor variables in lognormal regression model

Kupper and Meydrech and Myers and Lahoda introduced the mean squared error (MSE) approach to study response surface designs, Duncan and DeGroot derived a criterion for optimality of linear experimental designs based on minimum mean squared error. However, minimization of the MSE of an estimator maxr renuire some knowledge about the unknown parameters. Without such knowledge construction of designs optimal in the sense of MSE may not be possible. In this article a simple method of selecting the levels of regressor variables suitable for estimating some functions of the parameters of a lognormal regression model is developed using a criterion for optimality based on the variance of an estimator. For some special parametric functions, the criterion used here is equivalent to the criterion of minimizing the mean squared error. It is found that the maximum likelihood estimators of a class of parametric functions can be improved substantially (in the sense of MSE) by proper choice of the values of regressor variables. Moreover, our approach is applicable to analysis of variance as well as regression designs.     

Monotony relations between distribution functions and their application to experimental design

In a linear regression model an estimator of the unknown coefficients is considered which, in special cases, includes the least squares estimator. In the ease of stable symmetric error distribution and by means of a certain monotony relation between distribution functions optimality of this estimator is proved and the designing problem is investigated. A robustness property of optimal designs against the designing criterion and some conclusions are given concerning the least squares estimator in the case of G- and C-optimality.     

A relaxation procedure for calculating (G–)minimax optimal designs

Summary: In nonlinear statistical models, standard optimality functions for experimental designs depend on the unknown parameters of the model. An appealing and robust concept for choosing a design is the minimax criterion. However, so far, minimax optimal designs have been calculated efficiently under various restrictive conditions only. We extend an iterative relaxation scheme originally proposed by Shimizu and Aiyoshi (1980) and prove its convergence under very general assumptions which cover a variety of situations considered in experimental design. Application to different specific design criteria is discussed and issues of practical implementation are addressed. First numerical results suggest that the method may be very efficient with respect to the number of iterations required.*Supported by a grant from the Deutsche Forschungsgemeinschaft. We are grateful to a referee for his constructive suggestions.     

An algorithm for the design of factorial experiments when the data are correlated

Since the development of methods for the analysis of experiments with dependent data, see for example Gleeson and Cullis (1987), the design of such experiments has been an area of active research. We investigate the design of factorial experiments, complete and fractional, for various dependency structures. An algorithm for generating optimal or near optimal designs is presented and shown to be useful across a wide range of dependency structures.     

A reference prior for the analysis of a response surface

Standard response surface methodology employs a second order polynomial model to locate the stationary point ξ$\mathit{\xi }$ of the true response function. To make Bayesian analysis more direct and simpler, we refer to an alternative and equivalent parametrization, which contains ξ$\mathit{\xi }$ as parameter of interest. The marginal reference prior of ξ$\mathit{\xi }$ is derived in its general form and particular cases are also given in detail, showing the Bayesian role of rotatability.