# Construction of effect-wise orthogonal factorial designs

01 Jan 1981-Journal of Statistical Planning and Inference (North-Holland)-Vol. 5, Iss: 3, pp 221-229

TL;DR: In this article, two distinct methods have been proposed for the construction of effect-wise orthogonal factorial designs, which ensure desirable properties with respect to main effects and require a smaller number of replications than any of the existing methods.

Abstract: Generalizing the concept of Kronecker products of designs, two distinct methods have been suggested for the construction of effect-wise orthogonal factorial designs. The methods described ensure desirable properties with respect to main effects, cover almost all cases of factorial designs and require, in most cases, a smaller number of replications than any of the existing methods.

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TL;DR: In this paper, the concept of factorial structure in changeover designs is defined and shown to be a desirable requirement for these types of designs, and different classes of designs are examined in the light of this concept.

Abstract: SUMMARY Existing changeover designs given in the literature are not necessarily suitable when the treatments come from a factorial experiment. The concept of factorial structure in changeover designs is defined and shown to be a desirable requirement for these types of designs. Different classes of designs are then examined in the light of this concept. In particular, it is shown that all generalized cyclic (GC/n) changeover designs have factorial structure, thereby providing a flexible class of useful designs.

17 citations

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TL;DR: In this article, the use of Kronecker designs for factorial experiments is considered and the efficiency of the interaction is shown to be at least as large as the product of the efficiency factors of the two main effects.

Abstract: In this paper the use of Kronecker designs for factorial experiments is considered. The two-factor Kronecker design is considered in some detail and the efficiency factors of the main effects and interaction in such a design are derived. It is shown that the efficiency factor of the interaction is at least as large as the product of the efficiency factors of the two main effects and when both the component designs are totally balanced then its efficiency factor will be higher than the efficiency factor of either of the two main effects. If the component designs are nearly balanced then its efficiency factor will be approximately at least as large as the efficiency factor of either of the two main effects. It is argued that these designs are particularly useful for factorial experiments. Extensions to the multi-factor design are given and it is proved that the two-factor Kronecker design will be connected if the component designs are connected.

9 citations

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TL;DR: In this article, the authors considered block designs having orthogonal factorial structure, and some useful methods of constructing such designs are described, which can be used to construct designs for a wide range of parameter specifications.

Abstract: SUMMARY Block designs having orthogonal factorial structure are considered and some useful methods of constructing such designs are described. These methods can be used to construct designs for a wide range of parameter specifications. They also produce designs which are smaller than those obtainable through some of the existing methods.

8 citations

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TL;DR: On montre que: 1) l'equilibre general relativement a structure factorielle and 2) la structure factoriele orthogonale, sont essentiellement les memes for des plans factoriels avec un systeme unique de blocs.

Abstract: On montre que: 1) l'equilibre general relativement a une structure factorielle et 2) la structure factorielle orthogonale, sont essentiellement les memes pour des plans factoriels avec un systeme unique de blocs

7 citations

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7 citations

##### References

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TL;DR: In this article, the Kronecker product of matrices of known designs is used to obtain new designs, and a general discussion of the application of the concept of the Krnoecker product is given.

Abstract: By a statistical design (or simply, a design) we mean an arrangement of a certain number of "treatments" in a certain number of "blocks" in such a way that some prescribed combinatorial conditions are fulfilled. With every design is associated a unique matrix called the incidence matrix of the design (definitions, etc., in subsequent sections). In many instances, e.g., [7] [8], [10], [12], [16], information regarding certain kinds of designs such as BIB, PBIB designs is obtained from properties of the matrix $NN'$ or of its determinant $|NN'|$ where $N$ is the incidence matrix of the design under consideration. On the other hand in a few cases, such as [4], [5], [11], [14], [15], the incidence matrix $N$ itself has been to investigate properties of designs. This paper gives a method of using incidence matrices of known designs to obtain new designs. In Section 2 we have defined the Kronecker product of matrices. This definition and some properties of the Kronecker product of matrices are given in [1]. Section 3 is devoted to a general discussion of an application of the concept of the Kronecker product of matrices to define the Krnoecker product of designs. This section also contains two theorems which illustrate the use of the method of obtaining Kronecker products of designs. Definitions of some well-known designs are given in Section 4, which also contains a number of results giving explicit forms of certain Kronecker products. Finally some illustrations of a few results of Section 4 are given in Section 5.

88 citations

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TL;DR: In this paper, generalized cyclic designs are used for partial confounding in factorial experiments, so as to provide an orthogonal analysis of main effects and interactions and the efficiencies of the designs are examined and it is shown how good designs for estimating main effects can be easily obtained.

Abstract: SlUMMARY The main purpose of the paper is to show how generalized cyclic designs can be used for partial confounding in factorial experiments, so as to provide an orthogonal analysis of main effects and interactions. The efficiencies of the designs are examined and it is shown how good designs for estimating main effects can be easily obtained.

39 citations

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TL;DR: In this article, the concepts of intcr-effect-orthogonality and regularity in factorial experiments have been generalised to the concept of inter effect-ortho-onality of order t. and corresponding necessary and sufficient conditions have been given.

Abstract: The concepts of intcr-effect-orthogonality and regularity in factorial experiments as introduced by Mukerjee (1979) have been generalised to the concepts of inter-effect-orthogonality and regularity of order t. and corresponding necessary and sufficient conditions have been given. The problems of balancing, retaining full information and calculation of sums of squares have also been considered.

19 citations

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TL;DR: In this article, a set of necessary and sufficient conditions for best linear estimates of estimable treatment contrasts belonging to different facotrial effects to be mutually orthogonal is established.

Abstract: A set of necessary and sufficient conditions has been established for best linear estimates of estimable treatment contrasts belonging to different facotrial effects to be mutually orthogonal. The cases of both connected and disconnected designs have been considered. The analysis of connected designs satisfying the orthogonality condition has been derived. The orthogonality condition obtained is seen to throw new light on some of the existing methods of construction of symmetric and asymmetric factorial designs. Extension of the conditions to designs eliminating heterogeneity in several directions is immediate

19 citations

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