Asymmetric Factorials Research Articles

A lower bound of average mixture discrepancy for row augmented designs

Follow-up experimental designs are frequently employed in a wide range of scientific studies and industries. Lower bounds of average mixture discrepancy for row augmented designs are determined in this paper utilizing the level permutation technique, which is also used to analyse the uniformity of row augmented designs with asymmetrical factorials. Between the row augmented designs and its initial designs or follow-up designs, quantitative relationships of lower bounds of average mixture discrepancy are constructed. Additionally, a proposal is made for the construction algorithm of asymmetrical uniform row augmented designs. Numerical examples indicate that the asymmetrical uniform row augmented designs can be constructed with high efficiency and recommended for use in practice.

Influence of Seed Priming and Growing Medium on Germination and Seedling Vigour of Tamarind (Tamarindus indica L.)

A field experiment was conducted in 2020–21 at the Fruit Research Station, Imaliya, Department of Horticulture, JNKVV, Jabalpur. The AFRBD (Asymmetrical Factorial Randomized Block Design) was set up with 20 distinct treatment combinations of soil media Soil + Sand, Soil + Vermicompost, Soil + Sand + Vermicompost, Soil +Vermicompost + Biofertisol, Soil + Vermicompost + Azatobactor + PSB + KSB and plant growth regulator i.e. GA3 0 ppm (control), GA3 100 ppm, GA3 200 ppm, GA3 300 ppm. The vigour and germination of seeds were significantly impacted by the type of media used and the usage of plant growth regulators. In regard to the growth parameter, better values for seedling height, number of leaves and stem girth at 30, 60 and 90 days after sowing were found with treatment GA3 200 ppm. Growth metrics, such as root length, fresh shoot weight, dry shoot weight and dry root weight at 120 days, revealed improved results with a higher vigour index when the combination of GA3 200 ppm + growing medium (soil+ vermicompost+ Biofertisol) were applied. The GA3 200 ppm with growing medium (Soil+ Vermicompost+ Biofertisol) were proven to be better in terms of the seed germination parameters. Minimum time was taken for seed germination and increased seed germination percentage had been recorded at 30 days after planting.

Projection Uniformity of Asymmetric Fractional Factorials

The objective of this paper is to study the issue of the projection uniformity of asymmetric fractional factorials. On the basis of level permutation and mixture discrepancy, the average projection mixture discrepancy to measure the uniformity for low-dimensional projection designs is defined, the uniformity pattern and minimum projection uniformity criterion are presented for evaluating and comparing any asymmetric factorials. Moreover, lower bounds to uniformity pattern have been obtained, and some illustrative examples are also provided.

New lower bound for Lee discrepancy of asymmetrical factorials

Lee discrepancy has wide applications in design of experiments, which can be used to measure the uniformity of fractional factorials. An improved lower bound of Lee discrepancy for asymmetrical factorials with mixed two-, three- and four-level is presented. The new lower bound is more accurate for a lot of designs than other existing lower bound, which is a useful complement to the lower bounds of Lee discrepancy and can be served as a benchmark to search uniform designs with mixed levels in terms of Lee discrepancy.

Optimum mechanism for breaking the confounding effects of mixed-level designs

Fractional factorial designs (FFD’s) are no doubt the most widely used designs in the experimental investigations due to their efficient use of experimental runs to study many factors simultaneously. One consequence of using FFD’s is the aliasing of factorial effects. Follow-up experiments may be needed to break the confounding. A simple strategy is to add a foldover of the initial design, the new fraction is called a foldover design. Combining a foldover design with the original design converts a design of resolution r into a combined design of resolution $$r+1$$ . In this paper, we take the centered $$L_2$$ -discrepancy $$({\mathcal {CD}})$$ as the optimality measure to construct the optimal combined design and take asymmetrical factorials with mixed two and three levels, which are most commonly used in practice, as the original designs. New and efficient analytical expressions based on the row distance of the $${\mathcal {CD}}$$ for combined designs are obtained. Based on these new formulations, we present new and efficient lower bounds of the $${\mathcal {CD}}$$ . Using the new formulations and lower bounds as the benchmarks, we may implement a new algorithm for constructing optimal mixed-level combined designs. By this search heuristic, we may obtain mixed-level combined designs with low discrepancy.

Uniformity Pattern of Asymmetric Fractional Factorials

The objective of this paper is to discuss the issue of the projection uniformity of asymmetric fractional factorials. On the basis of Lee discrepancy, the authors define the projection Lee discrepancy to measure the uniformity for low-dimensional projection designs. Moreover, the concepts of uniformity pattern and minimum projection uniformity criterion are proposed, which can be used to assess and compare two and three mixed levels factorials. Statistical justification of uniformity pattern is also investigated.

Generalized projection discrepancy and its applications in experimental designs

The objective of this paper is to study the issue of the generalized projection discrepancy along the line of Qin et al. (J Stat Plan Inference 142:1170–1177, 2012) based on generalized discrete discrepancy measure proposed by Chatterjee and Qin (J Stat Plan Inference 141:951–960, 2011). We shall study the projection properties for general asymmetric factorials and provide some analytic connections between minimum generalized projection uniformity and other optimality criteria. A new lower bound on the generalized projection discrepancy for asymmetric factorials is presented here.

Two-level Orthogonal and Saturated Designs for Estimating Main Effects and Certain Important Interactions

In a two-level factorial experiment, we consider orthogonal designs that allow joint estimation of the grand mean, all main effects, and certain classes of two-level interactions, assuming that the remaining effects are all negligible. Based on a judicious allocation of the factorial effects of interest to the columns of a Hadamard matrix, we propose some general classes of orthogonal and saturated designs which include some existing orthogonal main-effect plans of asymmetric factorials as special cases.

A lower bound for the centred L 2-discrepancy on combined designs under the asymmetric factorials

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L 2-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.

Lee discrepancy on asymmetrical factorials with two- and three-levels

Lee discrepancy has been employed to measure the uniformity of fractional factorials. In this paper, we further study the statistical justification of Lee discrepancy on asymmetrical factorials. We will give an expression of the Lee discrepancy of asymmetrical factorials with two- and three-levels in terms of quadric form, present a connection between Lee discrepancy, orthogonality and minimum moment aberration, and obtain a lower bound of Lee discrepancy of asymmetrical factorials with two- and three-levels.

Obtaining industrial experimental designs using a heuristic technique

There are several methods of obtaining orthogonal fractions of symmetrical and asymmetrical factorials, including Hadamard matrices, collapsing and replacing, and orthogonal arrays. The objective is to find the right permutations of levels of each factor in order to obtain uncorrelated main effects with a minimum number of runs. This leads to a combinatorial optimization problem. Simulated annealing is an optimization technique that has been successfully used to find a good solution for combinatorial optimization problems. By applying the computational technique of simulated annealing to orthogonal fractional factorial designs, we provide a method to obtain uncorrelated main effects for symmetrical and asymmetrical factorials.

A note on connections among criteria for asymmetrical factorials

Various optimal criteria have been proposed to rank asymmetrical frac- tional factorial designs. Among them, the generalized minimum aberration and the minimum moment aberration criteria are the most popular ones and have received much attention. Recently, Liu et al. (Stat Sin 16:1285-1297, 2006) proposed the mini- mum χ 2 criterion in terms of level-combinations. In this paper, the equivalency of the generalized minimum aberration and the minimum χ 2 criteria is reported, which not only provides another justification for each other but also develops some theoretical results for designs with generalized minimum aberration and some lower bounds for the generalized wordlength pattern. Besides, an analytic relationship between gener- alized minimum aberration and minimum moment aberration is obtained for asym- metrical fractional factorial designs.

Lower Bounds for the Uniformity Pattern of Asymmetric Fractional Factorials

The objective of this article is to study the issue of the projection discrepancy (Fang and Qin, 2005; Ma et al. 2003), which has wide application to the field of fractional factorials. Fang and Qin (2005) and Zhang and Qin (2006) derived some properties of the projection properties of two-level factorials. Here we extend their results to asymmetric factorials. Some lower bounds of projection discrepancy of asymmetrical factorials are also obtained.

Blocking Mixed-Level Factorials withSAS

Orthogonal array screening designs for mixed-level or asymmetric factorials have recently become popular. Tables of designs and software for creating these designs are readily available to practitioners. However, confounded block designs for mixed-level factorials are not as popular partly due to the fact that software for creating these designs has not been well publicized. Classical methods for creating confounded-block mixed-level factorials normally described in textbooks utilize modular arithmetic or finite fields. In the recent literature optimal design theory has also been proposed as method for creating these designs. Although no examples are shown in the online documentation, both classical and optimal confounded-block mixed-level factorials can be easily created using SAS data step programming in conjunction with proc plan, proc factex or proc optex. In this article we show examples of creating these designs in SAS, and we compare the properties of designs created by classical methods and optimal design theory.

Supersaturated Designs for Asymmetrical Factorial Experiments

This article describes a general method of construction of supersaturated designs for asymmetric factorials obtained by exploiting the concept of resolvable orthogonal arrays and Hadamard matrices. The supersaturated design constructed here has a restricted form of t:q z n, where z factors are at q-levels each and one factor is at t-levels and the number of runs is n. The designs obtained have the factor at t-levels always orthogonal to the z factors with q levels each in the symmetric design q z n. The method of construction is illustrated with the help of examples. A catalogue of designs obtained is prepared and fNOD-efficiency and c 2 -efficiency of the designs are given. Many designs are optimal while other designs have high efficiencies. The efficiency of the resulting design is better than that of the symmetric design q z n. AMS Subject Classification: 62K15, 62K10

Lower Bounds on the Symmetric L 2-Discrepancy and Their Application

The role of uniformity measured by the symmetric L 2-discrepancy given in Hickernell (1998) has been studied in fractional factorial designs. The issue of lower bounds on the symmetric L 2-discrepancy is crucial in the construction of uniform designs. This article reports some new lower bounds on the symmetric L 2-discrepancy for symmetric fractional factorials and for a set of asymmetric fractional factorials. It is valuable to use these lower bounds to measure uniformity of given designs.

A note on the connection between uniformity and generalized minimum aberration

Discrete discrepancy has been utilized as a uniformity measure for comparing and evaluating factorial designs. In this paper, for asymmetrical factorials, we give some linkages between uniformity measured by the discrete discrepancy and other criteria, such as generalized minimum aberration (Xu and Wu, 2001) and minimum projection variance (Ai and Zhang, 2004). These close linkages show a significant justification for the discrete discrepancy used to measure uniformity of factorial designs, and provide an additional rationale for using uniform designs.

Asymmetrical Factorial Designs Containing Clear Effects

The asymmetrical or mixed-level factorial design is a kind of important design in practice. There is a natural problem on how to choose an optimal (s2)sn design for the practical need, where s is any prime or prime power. This paper considers the clear effects criterion for selecting good designs. It answers the questions of when an (s2)sn design with fixed number of runs contains clear two-factor interaction (in brief 2fi) components and when it contains clear main effects or clear 2fis. It further gives the complete classification of (s2)sn designs according to the clear 2fi components, main effects and 2fis they have.

A Lower Bound for the Centered L 2-Discrepancy on Asymmetric Factorials and its Application

The role of uniformity measured by the centered L2-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L2-discrepancy is crucial in the construction of uniform designs. Fang and Mukerjee (2000) and Fang et al. (2002, 2003b) derived lower bounds for fractions of two- and three-level factorials. In this paper we report some new lower bounds for the centered L2-discrepancy for a set of asymmetric fraction factorials. Using these lower bounds helps to measure uniformity of a given design. In addition, as an application of these lower bounds, we propose a method to construct uniform designs or nearly uniform designs with asymmetric factorials.

Small asymmetric fractional factorial plans for main effects and specified two-factor interactions

Fractional factorial plans represented by orthogonal arrays of strength two are known to be optimal in a very strong sense under a model that includes the mean and all the main effects, when all interactions are assumed to be absent. When a fractional factorial plan given by an orthogonal array of strength two is not saturated, one might think of entertaining some two-factor interactions also in the model. In such a situation, it is of interest to examine which of the two-factor interactions can be estimated via a plan represented by an orthogonal array, as also to study the overall efficiency of the plan when some interactions are in the model alongwith the mean and all main effects. In this paper, an attempt has been made to examine these issues by considering some practically useful plans for asymmetric (mixed level) factorials with small number of runs.