Clear Two-factor Interactions Research Articles

An algorithm for blocking regular fractional factorial 2-level designs with clear two-factor interactions

Regular fractional factorial designs with 2-level factors are among the most frequently used experimental plans. In many cases, designs should be blocked for dealing with inhomogeneity of experimental units. At the same time, the research question at hand may imply a focus on specified sets of two-factor interactions, while it is not justified to assume negligibility of other low order effects. An algorithm is provided for blocking a regular fraction into – possibly small – blocks while keeping specified two-factor interactions clear from confounding with main effects or other two-factor interactions. The proposed algorithm is implemented in the R package FrF2 and combines an estimability algorithm by the author with an automated implementation of a recent proposal for blocking fractions by hand.

Mixed-level designs with multi block variables containing clear two-factor interaction components

This paper studies mixed-level designs with multi block variables, denoted by . Some necessary and sufficient conditions on the existence of clear treatment two-factor interaction components (2fic’s) in designs with resolution III, IV and IV are obtained. We also discuss how to find designs with the maximum number of clear 2fic’s with resolution at least III, and tabulate some 32-run designs containing the maximum number of clear 2fic’s in the appendix.

Some results on two-level regular designs with multi block variables containing clear effects

In this paper, the idea of clear effects is extended to the case of blocked regular $$2^{n{-}m}$$ designs with multi block variables. Some necessary and sufficient conditions on the existence of clear treatment main effects and clear treatment two-factor interactions (2fi’s) in $$2^{n{-}m}:2^l$$ designs with resolution III, IV $$^- $$ and IV are obtained, where the notation $$2^{n-m}:2^l$$ means that there are n treatment factors, m treatment defining words and l block variables. We discuss how to find $$2^{n-m}:2^l$$ designs with the maximum number of clear 2fi’s with resolution III, IV $$^-$$ or IV, and present some 16-, 32-, 64-run designs containing the maximum number of clear 2fi’s in tables.

Mixed-level designs with resolution III or IV containing clear two-factor interaction components

Mixed-level designs are widely used in factorial experiments. Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs. It is highly desirable to know when mixed-level designs with resolution III or IV can have clear two-factor interaction components. This paper considers mixed-level designs with one or two high-level factors and some two-level factors, denoted as \((2^{r})\times 2^n\) and \((2^{r_1})\times (2^{r_2})\times 2^n\), respectively, and gives a complete classification of the existence of clear two-factor interaction components in such designs with resolution III or IV. The results reveal the structures of these designs.

Generalized variable resolution designs

In this paper, the concept of generalized variable resolution is proposed for designs with nonnegligible interactions between groups. The conditions for the existence of generalized variable resolution designs are discussed. Connections between different generalized variable resolution designs and compromise plans, clear compromise plans and designs containing partially clear two-factor interactions are explored. A general construction method for the proposed designs is also discussed.

A Note on Dominating Fractional Factorial Two-Level Designs With Clear Two-Factor Interactions

This note builds on results from Wu, Mee, and Tang's article (henceforth WMT) on admissible fractional factorial two-level designs, specifically concentrating on the “dominating designs” that were introduced but not further pursued in WMT. WMT's work has been used for increasing the efficiency of the author's graph-based algorithm for creation of minimum aberration designs that keep a requirement set of two-factor interactions clear: That algorithm originally searched aberration-sorted complete catalogs of nonisomorphic resolution IV designs; according to WMT's proposal, it suffices to search the dominating designs. For implementing this reduction of search space, aberration-sorted catalogs of dominating resolution IV regular fractional factorial two-level designs with up to 128 runs were created; this was achieved with the help of a recent subgraph isomorphism algorithm by Solnon that proved to be particularly efficient for this search task. The online supplementary materials provide the code used for identifying the dominating designs and ASCII files with the design generators for all resolution IV and higher dominating designs in 64 and 128 runs.

On blocked resolution IV designs containing clear two-factor interactions

Blocking is an effective method for reducing systematic variations and increasing the precision of effect estimation when the experimental units are not homogeneous. Clear two-factor interactions (2fi’s) can be estimated under the weak assumption that interactions involving three or more factors can be ignored. Some theoretical results on the existence of clear 2fi’s in 2IVm−p:2l designs are obtained, where 2IVm−p:2l denotes a regular resolution IV blocked design with m treatment factors, l block factors, and 2m−p runs. An algorithm is proposed for constructing 2IVm−p:2l designs with the maximum number of clear 2fi’s. Based on this algorithm, the 16-, 32-, and 64-run 2IVm−p:2l designs containing the maximum number of clear 2fi’s are constructed and tabulated.

Conditions of 2<sup>(<italic>n</italic><sub>1</sub>+<italic>n</italic>2)-(<italic>k</italic><sub>1</sub>+<italic>k</italic><sub>2</sub>)</sup>4<sup>2</sup> mixed-level split-plot designs containing clear effects

Mixed-level fractional factorial split-plot designs are widely used in various experiments.Under the weak assumption that the interactions involving three or more factors can be ignored,the experimenter can obtain the unbiased estimations of the clear main effects or clear two-factor interaction components.The paper gives the conditions of a mixed-level split-plot design with two four-level factors and some two-level factors to have various clear main effects or clear two-factor interaction components.The methods of constructing the corresponding designs are also given.

Fractional Factorial Designs With Admissible Sets of Clear Two-Factor Interactions

We consider the problem of selecting two-level fractional factorial designs that allow joint estimation of all main effects and some specified two-factor interactions (2fi’s) without aliasing from other 2fi’s. This problem is to find, among all 2 m−p designs with given m and p, those resolution IV designs whose sets of clear 2fi’s contain the specified 2fi’s as subsets. We use a linear graph to represent the set of clear 2fi’s for a resolution IV design, where each line connecting two vertexes represents a clear 2fi between the factors represented by the two vertexes. We call a 2 m−p resolution IV design admissible if its graph is not isomorphic to any proper subgraph of the graph of any other 2 m−p resolution IV design. We show that all even resolution IV designs are inadmissible. We then use a classical subgraph-isomorphism algorithm to determine all admissible designs of 32, 64, and 128 runs. This leads to a concise catalog of all admissible designs of 32 and 64 runs, and a lengthy but substantially reduced list (compared to the number of non-isomorphic designs) for 128 runs. All admissible designs of 128 runs are included in supplementary materials that are available online. A file of R code is available on the first author’s website for generating the actual design tables for any 2 m−p designs of 32, 64, and 128 runs.

Robust designs through partially clear two-factor interactions

SUMMARY Orthogonal arrays with clear two-factor interactions provide a class of designs that are robust to nonnegligible effects. If certain prior knowledge is available, then robust designs allow additional factors to be studied. This is done through partially clear two-factor interactions. We study the existence and construction of such robust designs and present an upper bound on the maximum number of clear two-factor interactions.

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O ( 2 p ) + O ( 3 k 3 ) + O ( 2 k ) operations to check two 2 k − p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.

Designing fractional factorial split-plot experiments using integer programming

Split-plot designs are commonly used in industrial experiments when there are hard-to-change and easy-to-change factors. Due to the number of factors and resource limitations, it is more practical to run a fractional factorial split-plot (FFSP) design. These designs are variations of the fractional factorial (FF) design, with the restricted randomisation structure to account for the whole plots and subplots. We discuss the formulation of FFSP designs using integer programming (IP) to achieve various design criteria. We specifically look at the maximum number of clear two-factor interactions and variations on this criterion.

On weak minimum aberration and number of clear two-factor interaction components in [formula omitted] designs

This article obtains some theoretical results on the number of clear two-factor interaction components and weak minimum aberration in an s IV m - p design, by considering the number of not clear two-factor interaction components of the design.

Compromise 4 m 2 n plans with clear two-factor interactions

This paper gets some necessary conditions for the existence of some kinds of clear 4m2n compromise plans which allow estimation of all main effects and some specified two-factor interactions without assuming the remaining two-factor interactions being negligible. Some methods for constructing clear 4m2n compromise plans are introduced.

Bound on the maximum number of clear two-factor interactions for 2 n-(n-k) designs

Clear effects criterion is an important criterion for selecting fractional factorial designs [1]. Tang et al. [2] derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi's) in 2 n-(n-k) designs of resolution III and IV by constructing 2 n-(n-k) designs. But the method in [2] does not perform well sometimes when the resolution is III. This article modifies the construction method for 2 n-(n-k) designs of resolution III in [2]. The modified method is a great improvement on that used in [2].

Some results on 4 m 2 n designs with clear two-factor interaction components

Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4m2n designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4m2n designs under the clear effects criterion.

Grid representations for [formula omitted] join designs containing maximum number of clear two-factor interactions

This paper shows that the join designs [Butler, N.A., 2005. Classification of efficient two-level fractional factorial design of resolution IV or more. J. Statist. Plann. Inference 131, 145–159] include all 2 IV m - ( m - k ) designs with the maximum number of clear two-factor interactions (2FIs) for k = 5 , 6 , 7 . The designs are shown to have concise grid representations which provide simple interpretations of the 2FIs in a 2 IV m - ( m - k ) design. In addition, a better upper bound on the maximum number of clear 2FIs in a 2 IV m - ( m - k ) design than the one obtained by Tang et al. [2002. Bounds of the maximum members of clear factor interactions for 2 m - p designs of resolution III and IV. Canad. J. Statist. 30, 127–136] is given.

Construction of $$ 2^{m - (m - k)}_{{\rm{III}}} $$ Designs with the Maximum Number of Clear Two-factor Interactions

It is useful to know the maximum number of clear two-factor interactions in a $$ 2^{m - (m - k)}_{{\rm{III}}} $$ design. This paper provides a method to construct a $$ 2^{m - (m - k)}_{{\rm{III}}} $$ design with the maximum number of clear two-factor interactions. And it is proved that the resulting designs have more clear two-factor interactions than those constructed by Tang et al. [6]. Moreover, the designs constructed are shown to have concise grid representations.

Bounds on the maximum numbers of clear two-factor interactions for $$2^{(n_1 + n_2 ) - (k_1 + k_2 )} $$ fractional factorial split-plot designs

Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures. There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors, which can form three types of two-factor interactions: WP2fi, WS2fi and SP2fi. This paper considers FFSP designs with resolution III or IV under the clear effects criterion. It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs, and gives some methods for constructing the desired FFSP designs. It further examines the performance of the construction methods.

Nonregular two-level designs of resolution IV or more containing clear two-factor interactions

In this paper, the concepts of clear effects, alias sets and grid representations are generalized to nonregular two-level designs. Many good generalized join designs of n runs with resolution IV or more containing many clear two-factor interactions are given for n = 48 up to 192 and n being a multiple of 16. The designs constructed are shown to have concise generalized two-factor interaction grid representations. Finally, theoretical results for the necessary and sufficient conditions under which there exist nonregular two-level designs of resolution IV or more containing clear two-factor interactions are proved.