Optimal design results for 2n factorial paired comparison experiments

Four general classes of partially balanced designs for 2ⁿ factorials, corresponding to four different forms of a general null hypothesis H on factorial effects, are presented. For the typical design in each class, the simplified form of the non-centrality parameter λ² of the asymptotic chi-square distribution of the likelihood ratio statistic for testing the corresponding form of H₀ is derived under defined local alternatives. Optimal designs d₁ maximizing λ² in the i-th class and minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix, i =1,…,4, are determined.     

Optimal 2n-p fractional factorial designs for dispersion effects under a location-diapersion model

We consider main effects models for 2-level experiments that also include. Parameters characterizing potential dispersion effects due to specified factors. One special case is considered. In this case only a single specified factor is responsible for the dispersion effects. We determine the connection between alias relations and Optimality of a design for estimation of dispersion effects in the class of regu!ar fractional Y - P factorial designs of resolution III or higher. This rmectioil heips US identify those designs that are A-optimal for estimating dispersion effects by a suitable choice of defining contrasts. in particuiar, we show that an increase in efficiency with respect to dispersion effects is accompanied by a loss iii efficiency for estimating the location effects. In practice, one mmt thcrcfcre accept a trade& between the efficiencies associated with estirnates of location effects and dispersion effects.     

A simple method for constructing confounded designs for mixed factorial experiments

This paper presents the sinplesr procedure that uses wodular aryithmetic for constructing confounded designs for mixed factorial experiments. The present procedure and the classical one for confounding in symmetrical factorial experiments are both at the same mathema.tical level. The present procedure is written for

practitioners and is lllustrared with several examples.     

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.     

Using the chinese remainder theorem in constructing confounded designs for mixed factorial experiments

A procedure for constructing confounded designs for mixed factorial experiments derived from the Chinese Remainder Theorem is presented. The present procedure as well as others, all through use of modular arithmetic, are compared.     

Saturated main-effect designs for factorial experiments

The results of a computer search for saturated designs for 2ⁿ factorial experiments with n runs is reported, (where n = 2 mod 4). A complete search of the design space is avoided by focussing on designs constructed from cyclic generators. A method of searching quickly for the best generators is given. The resulting designs are as good as, and sometimes better than, designs obtained via search algorithms reported in the literature. The addition of a further factor having three levels is also considered. Here, too, a complete search is avoided by restricting attention to the most efficient part of the design space under p-efficiency.     

Block designs for incomplete factorial treatment structures with two factors

Abstract

In this paper, the problem of obtaining efficient block designs for incomplete factorial treatment structure with two factors excluding one treatment combination for estimation of dual versus single treatment contrasts is considered. The designs have been obtained using the A-optimal completely randomized designs and modified strongest treatment interchange algorithm. A catalog of efficient block designs has been prepared for m₁?=?3, 4 and m₂?=?2, b?≤?10 and k?≤?9 and for m₁?=?3,4 and m₂?=?3, 4, b?≤?10 and k?≤?10.     

Optimal paired comparison designs for first-order interactions

In many fields of applications paired comparisons are used in which either full or partial profiles of the alternatives are presented. For this situation we introduce an appropriate model and derive optimal designs in the presence of interactions when all attributes have the same number of levels.     

Numerical solution of likelihood equations for a first-order response surface model for mixture paired comparisons

Summary A simple procedure for numerical solution of the likelihood equations for estimating the regression parameters of a first-order response surface model for the treatment parameters of mixture paired comparison experiments is developed. It is demonstrated that, for defined rotatable designs, those regression parameters are simple functions of the main effect parameters of a corresponding factorial model with no interactions. The maximum likelihood estimators of those main effect parameters, and hence of their corresponding regression parameters, are obtained through using procedures of treatment contrasts, factorial and iterations. A numerical example is given to illustrate applications of the procedures developed in this paper.     

Defining equations for two-level factorial designs

Defining equations are introduced in the context of two-level factorial designs and they are shown to provide a concise specification of both regular and nonregular designs. The equations are used to find orthogonal arrays of high strength and some optimal designs. The latter optimal designs are formed in a new way by augmenting notional orthogonal arrays which are allowed to have some runs with a negative number of replicates before augmentation. Defining equations are also shown to be useful when the factorial design is blocked.     

Nonparametric multiple comparison procedures for unbalanced one-way factorial designs

In this paper, we present several nonparametric multiple comparison (MC) procedures for unbalanced one-way factorial designs. The nonparametric hypotheses are formulated by using normalized distribution functions and the comparisons are carried out on the basis of the relative treatment effects. The proposed test statistics take the form of linear pseudo rank statistics and the asymptotic joint distribution of the pseudo rank statistics for testing treatments versus control satisfies the multivariate totally positive of order two condition irrespective of the correlations among the rank statistics. Therefore, in the context of MCs of treatments versus control, the nonparametric Simes test is validated for the global testing of the intersection hypothesis. For simultaneous testing of individual hypotheses, the nonparametric Hochberg stepup procedure strongly controls the familywise type I error rate asymptotically. With regard to all pairwise comparisons, we generalize various single-step and stagewise procedures to perform comparisons on the relative treatment effects. To further compare with normal theory counterparts, the asymptotic relative efficiencies of the nonparametric MC procedures with respect to the parametric MC procedures are derived under a sequence of Pitman alternatives in a nonparametric location shift model for unbalanced one-way layouts. Monte Carlo simulations are conducted to demonstrate the validity and power of the proposed nonparametric MC procedures.     

A comparison of recent nonparametric methods for testing effects in two-by-two factorial designs

The two-way two-levels crossed factorial design is a commonly used design by practitioners at the exploratory phase of industrial experiments. The F-test in the usual linear model for analysis of variance (ANOVA) is a key instrument to assess the impact of each factor and of their interactions on the response variable. However, if assumptions such as normal distribution and homoscedasticity of errors are violated, the conventional wisdom is to resort to nonparametric tests. Nonparametric methods, rank-based as well as permutation, have been a subject of recent investigations to make them effective in testing the hypotheses of interest and to improve their performance in small sample situations. In this study, we assess the performances of some nonparametric methods and, more importantly, we compare their powers. Specifically, we examine three permutation methods (Constrained Synchronized Permutations, Unconstrained Synchronized Permutations and Wald-Type Permutation Test), a rank-based method (Aligned Rank Transform) and a parametric method (ANOVA-Type Test). In the simulations, we generate datasets with different configurations of distribution of errors, variance, factor's effect and number of replicates. The objective is to elicit practical advice and guides to practitioners regarding the sensitivity of the tests in the various configurations, the conditions under which some tests cannot be used, the tradeoff between power and type I error, and the bias of the power on one main factor analysis due to the presence of effect of the other factor. A dataset from an industrial engineering experiment for thermoformed packaging production is used to illustrate the application of the various methods of analysis, taking into account the power of the test suggested by the objective of the experiment.     

Minimum cost linear trend-free 12-run fractional factorial designs

Time trend resistant fractional factorial experiments have often been based on regular fractionated designs where several algorithms exist for sequencing their runs in minimum number of factor-level changes (i.e. minimum cost) such that main effects and/or two-factor interactions are orthogonal to and free from aliasing with the time trend, which may be present in the sequentially generated responses. On the other hand, only one algorithm exists for sequencing runs of the more economical non-regular fractional factorial experiments, namely Angelopoulos et al. [1 P. Angelopoulos, H. Evangelaras, and C. Koukouvinos, Run orders for efficient two-level experimental plans with minimum factor level changes robust to time trends, J. Statist. Plann. Inference 139 (2009), pp. 3718–3724. doi: 10.1016/j.jspi.2009.05.002[Crossref], [Web of Science ®] , [Google Scholar]]. This research studies sequential factorial experimentation under non-regular fractionated designs and constructs a catalog of 8 minimum cost linear trend-free 12-run designs (of resolution III) in 4 up to 11 two-level factors by applying the interactions-main effects assignment technique of Cheng and Jacroux [3 C.S. Cheng and M. Jacroux, The construction of trend-free run orders of two-level factorial designs, J. Amer. Statist. Assoc. 83 (1988), pp. 1152–1158. doi: 10.1080/01621459.1988.10478713[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the standard 12-run Plackett–Burman design, where factor-level changes between runs are minimal and where main effects are orthogonal to the linear time trend. These eight 12-run designs are non-orthogonal but are more economical than the linear trend-free designs of Angelopoulos et al. [1 P. Angelopoulos, H. Evangelaras, and C. Koukouvinos, Run orders for efficient two-level experimental plans with minimum factor level changes robust to time trends, J. Statist. Plann. Inference 139 (2009), pp. 3718–3724. doi: 10.1016/j.jspi.2009.05.002[Crossref], [Web of Science ®] , [Google Scholar]], where they can accommodate larger number of two-level factors in smaller number of experimental runs. These non-regular designs are also more economical than many regular trend-free designs. The following will be provided for each proposed systematic design:

• (1) The run order in minimum number of factor-level changes.

• (2) The total number of factor-level changes between the 12 runs (i.e. the cost).

• (3) The closed-form least-squares contrast estimates for all main effects as well as their closed-form variance–covariance structure.

In addition, combined designs of each of these 8 designs that can be generated by either complete or partial foldover allow for the estimation of two-factor interactions involving one of the factors (i.e. the most influential).     

Optimal partially replicated parallel-flats designs for 2×3 mixed factorial experiments

Augmenting additional repeated runs to an unreplicated factorial design provides an economical scheme of obtaining an unbiased estimate for the error variance based on pure replicates. The augmented partially replicated design usually performs satisfactorily in identifying truly active effects regardless of whether the effect sparsity principle holds. Liao and Chai (2009) proposed a set of sufficient conditions for a partially replicated two-level design to be D-optimal over the class of parallel-flats designs. In this article, we generalize their result to the 2^n₁×3^n₂ mixed factorial under D-, A- and E-optimality, and include the 2ⁿ and 3ⁿ symmetric factorials as special cases. In addition, some examples are given to illustrate the main results.     

On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2^n-p$2^{n-p}$ fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.     

Incomplete row-column designs with factorial treatment structure for estimating main effects with full efficiency

In this paper, we propose two methods of constructing row-column designs for factorial experiments. The constructed designs have orthogonal factorial structure with balance and permits estimation of main effects with full efficiency.     

Chain pooling model selection for two-level fixed effects factorial experiments

As many as three iterated statistical model deletion procedures are considered for an experiment.Population model coeff cients were chosen to simulate a saturated 2⁴experiment having an unfavorable distribution of parameter values.Using random number studies, three model selection strategies were developed, namely, (1) a strategy to be used in anticipation of large coefficients of variation (neighborhood of 65 percent), (2) strategy to be used in anticipation of small coefficients of variation (4 percent or less), and (3) a security regret strategy to be used in the absence of such prior knowledge     

Search designs for 2m factorial experiments

Theorems 5, 6 and 10, and Tables 1–2 in Ghosh (1981) are corrected. These are concerned with search designs which permit the estimation of the general mean and main effects, and allow the search and estimation of one possibly unknown nonzero effect among the two- and three-factor interactions in 2^m factorial experiments. Some new results are presented.     

Optimal designs for stated choice experiments that incorporate ties

In 1970 Davidson generalised the Bradley–Terry model to allow respondents to say that the two options presented in a choice task were equally attractive. In this paper we extend this idea to the MNL model with m options in each choice set and we show that the optimal designs for the MNL model are also optimal in this setting.     

Optimal designs for experiments with mixtures: a survey

This is a survey article on known results about analytic solutions and numerical solutions of optimal designs for various regression models for experiments with mixtures. The regression models include polynomial models, models containing homogeneous functions, models containing inverse terms and ratios, log contrast models, models with quantitative variables, and mod els containing the amount of mixture, Optimality criteria considered include D-, A-, E-,φ_p- and I_λ-Optimalities. Uniform design and uniform optimal design for mixture components, and efficiencies of the {q,2} simplex-controid design are briefly discussed.