Minimum Aberration Regular Two-Level Designs in the Presence of Dispersion Factors

In practice, the variance of the response variable may change as some specific factors change from one setting to another in a factorial experiment. These factors affecting the variation of the response are called dispersion factors, which can violate the usual assumption of variance homogeneity. In this study, we modify the conventional minimum aberration criterion to take the impact of dispersion factors into account. The situations of one or two dispersion factors are investigated. As a result, we present regular 2^{n ? p} designs with run sizes equal to 16 and 32 using the modified minimum aberration criterion.     

Characterization of Balanced Fractional 3 m Factorial Designs of Resolution III

We consider a fractional 3 ^m factorial design derived from a simple array (SA), which is a balanced array of full strength, where the non negligible factorial effects are the general mean and the linear and quadratic components of the main effect, and m ≥ 2. In this article, we give a necessary and sufficient condition for an SA to be a balanced fractional 3 ^m factorial design of resolution III. Such a design is characterized by the suffixes of indices of an SA.     

Minimum Aberration Two-Level Fractional Factorial Split-Plot Designs with Hard to Change Factors

In this article we will consider industrial experiments in which some experimental factors have hard to change levels and others have levels which are easy to change. In such situations, fractional factorial split plot designs are often used where the hard to change factors are included as a subset of the whole plot factors and the easy to change factors make up the subplot factors. Here we consider the problem of finding two-level split plot designs which have minimum aberration among those designs which also minimize the number of level changes for the hard to change factors.     

Some Results on Fractional Factorial Split-Plot Designs with Multi-Level Factors

Fractional factorial split-plot (FFSP) designs have received much attention in recent years. In this article, the matrix representation for FFSP designs with multi-level factors is first developed, which is an extension of the one proposed by Bingham and Sitter (1999b Bingham , D. , Sitter , R. R. ( 1999b ). Some theoretical results for fractional factorial split-plot designs . Ann. Statist. 27 : 1240 – 1255 . [Google Scholar]) for the two-level case. Based on this representation, periodicity results of maximum resolution and minimum aberration for such designs are derived. Differences between FFSP designs with multi-level factors and those with two-level factors are highlighted.     

Bayesian Analysis of Two-Level Fractional Factorial Experiments with Non-Normal Responses

An intractable issue on screening experiments is to identify significant effects and to select the best model when the number of factors is large, especially for fractional factorial experiments with non-normal responses. In such cases, a three-stage Bayesian approach based on generalized linear models (GLMs) is proposed to identify which effects should be included in the target model where the principles of effect sparsity, hierarchy, and heredity are successfully considered. Three simulation experiments with non-normal responses which follow Poisson, binomial, and gamma distributions are presented to illustrate the performance of the proposed approach.     

Existence Conditions for Balanced Fractional 2m Factorial Designs of Resolution 2l + 1 Derived from Simple Arrays

We consider a fractional 2^m factorial design derived from a simple array (SA) such that the (? + 1)-factor and higher-order interactions are negligible, where 2? ? m. The purpose of this article is to give a necessary and sufficient condition for an SA to be a balanced fractional 2^m factorial design of resolution 2? + 1. Such a design is concretely characterized by the suffixes of the indices of an SA.     

Fractional Factorial Designs for the Detection of Interactions between Design and Noise Factors

In industrial experiments on both design (control) factors and noise factors aimed at improving the quality of manufactured products, designs are needed which afford independent estimation of all design×noise interactions in as few runs as possible, while allowing aliasing between those factorial effects of less interest. An algorithm for generating orthogonal fractional factorial designs of this type is described for factors at two levels. The generated designs are appropriate for experimenting on individual factors or for experimentation involving group screening of factors.     

Fractional Factorial Designs for Model-Based Evaluation of Customer Preferences

In this article, we explore the connection between Conjoint Analysis (CA) and a recent theory for minimum size orthogonal fractional factorial design generation (Fontana, 2013 Fontana , R. ( 2013 ). Algebraic generation of minimum size orthogonal fractional factorial designs: an approach based on integer linear programming . Computat. Statist. 28 : 241 – 253 .[Crossref], [Web of Science ®] , [Google Scholar]).

We show how searching for a minimum size OFFD that satisfies a set of constraints, expressed in terms of orthogonality between simple and interaction effects, is equivalent to solving an integer linear programming problem. It is worth noting that the methodology puts no restriction either on the number of levels of each factor or on the orthogonality constraints and so it can be applied to a very wide range of designs, including mixed orthogonal arrays. An algorithm, that has been implemented in SAS/IML, is briefly described.

The use of this algorithm during the design stage of a generic CA is shown in two applications.     

Catalogue of Group Structures for Three-level Fractional Factorial Designs

Taguchi (1959) introduced the concept of split-unit design to sort the factors into different groups depending upon the difficulties involved in changing the levels of factors. Li et al. (1991) renamed it as split-plot design. Chen et al. (1993) have given a catalogue of small designs for two- and three-level fractional factorial designs pertaining to a single type of factors. Aggarwal et al. (1997) have given a catalogue of group structure for two-level fractional factorial designs developed under the concept of split-plot design. In this paper, an algorithm has been developed for generating group structure and possible allocations for various 3^n-k fractional factorial designs.     

Bounds on the Maximum Number of Factors in Designs with Two Distinct Groups of Factors

This article focuses on designs involving two distinct groups of factors. In particular, we assume that between-group interactions are more important than within-group interactions. Under this assumption, a new word-length pattern is proposed to characterize the aliasing severity of a design, and the concepts of resolution and aberration are defined accordingly. Furthermore, we have obtained various bounds on the maximum number of factors that a design with given resolution can accommodate.     

Two-Level Designs in Blocks of Size Two in Which All Main Effects and Two-Factor Interactions Are Estimable

There are many problems in the real world for which it is necessary to perform two points in one block. In this situation, information about certain treatments may be indistinguishable from, or confounded with, blocks. If the experimenter wants to estimate all main effects and two-factor interactions in which two points are in one block, then how to perform the blocking schemes is an important topic to study. In this article, we combine several 2 ^k?p fractions from same or different family to obtain better combined designs requiring fewer runs than those appearing in the literature.     

On the Choice of a Prior for Bayesian D-Optimal Designs for the Logistic Regression Model with a Single Predictor

The Bayesian design approach accounts for uncertainty of the parameter values on which optimal design depends, but Bayesian designs themselves depend on the choice of a prior distribution for the parameter values. This article investigates Bayesian D-optimal designs for two-parameter logistic models, using numerical search. We show three things: (1) a prior with large variance leads to a design that remains highly efficient under other priors, (2) uniform and normal priors lead to equally efficient designs, and (3) designs with four or five equidistant equally weighted design points are highly efficient relative to the Bayesian D-optimal designs.     

Nonparametric analysis of ordinal categorical response data with factorial structure

A simple nonparametric method of analysis for contingency tables with an ordinal response and factorial treatment structure is described. The method involves a partition of Pearson's X ²_P-statistic by using orthogonal polynomials so that location and dispersion effects are estimated for each level of the explanatory variable. Analyses of variance are then performed on these effects to determine the important factors. The methods are applied to two examples, where consumers rate their liking for a product on an ordered categorical scale, one of which highlights the need to look at dispersion as well as location effects.     

Analysis of Linear Models with Two Factors Having Both Fixed and Random Levels

A general theory for a case where some factors have both fixed and random effect levels is developed under a two-way treatment structure model. This is an extension of a one factor with both fixed and random levels (Njuho and Milliken, 2005 Njuho , P. M. , Milliken , G. A. ( 2005 ). Analysis of linear models with one factor having both fixed and random levels . Commun. Statist. Theor. Meth. 34 : 1979 – 1989 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We consider several alternative approaches for estimating the fixed effects and the variance components using mixed models. We propose conducting the analysis in stages depending on the hypothesis being tested. The computational procedures are illustrated using two numerical examples.     

The Performance of Two Data-Generation Processes for Data with Specified Marginal Treatment Odds Ratios

Monte Carlo simulation methods are increasingly being used to evaluate the property of statistical estimators in a variety of settings. The utility of these methods depends upon the existence of an appropriate data-generating process. Observational studies are increasingly being used to estimate the effects of exposures and interventions on outcomes. Conventional regression models allow for the estimation of conditional or adjusted estimates of treatment effects. There is an increasing interest in statistical methods for estimating marginal or average treatment effects. However, in many settings, conditional treatment effects can differ from marginal treatment effects. Therefore, existing data-generating processes for conditional treatment effects are of little use in assessing the performance of methods for estimating marginal treatment effects. In the current study, we describe and evaluate the performance of two different data-generation processes for generating data with a specified marginal odds ratio. The first process is based upon computing Taylor Series expansions of the probabilities of success for treated and untreated subjects. The expansions are then integrated over the distribution of the random variables to determine the marginal probabilities of success for treated and untreated subjects. The second process is based upon an iterative process of evaluating marginal odds ratios using Monte Carlo integration. The second method was found to be computationally simpler and to have superior performance compared to the first method.     

Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT

Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a “projection design”. In this article, we have identified all the geometric non isomorphic projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k = 3,4, and 5 factors when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.