# Designs with Partial Factorial Balance

01 Aug 1972-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 43, Iss: 4, pp 1333-1341

TL;DR: In this paper, a class of multidimensional experimental designs with partial factorial balance is introduced and the analysis of these designs is given in detail and several series of three, four and five dimensional designs are presented.

Abstract: In this paper a class of multidimensional experimental designs said to have partial factorial balance is introduced. These designs are shown to belong to the more general class of multidimensional partially balanced designs. The analysis of designs with partial factorial balance is given in detail and several series of three, four and five dimensional designs are presented.

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TL;DR: In this article, the inverse of a circulant matrix having only three nonzero elements in each row (located in cyclically adjacent columns) is derived analytically from the solution of a recurrence equation.

Abstract: The elements of the inverse of a circulant matrix having only three non-zero elements in each row (located in cyclically adjacent columns) are derived analytically from the solution of a recurrence equation. Expressing any circulant as a product containing these three-element-type circulants then provides an algorithm for inverting circulants in general. Extension is also made to deriving generalized inverses of certain singular circulants.

25 citations

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01 Jan 1976TL;DR: The subject is reviewed from a combinatorial viewpoint, and unsolved problems are indicated.

Abstract: Statisticians are interested in designs for two non-interacting sets of treatments. These designs present many interesting combinatorial problems. The subject is reviewed from a combinatorial viewpoint, and unsolved problems are indicated. An extensive bibliography is appended.

21 citations

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15 citations

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TL;DR: A general construction for m-way completely variance balanced designs where each factor has v levels, m is any integer less than or equal to k, and N = vk, where k = 2λ+1 and v = 4λ3 is a prime power is given in this article.

Abstract: A general construction is given for m-way completely variance balanced designs where each factor has v levels, m is any integer less than or equal to k, and N = vk, where k = 2λ+1 and v = 4λ3 is a prime power. The construction gives rise to a variety of designs, easily enumerated, with the same parameters pairwise but with differing variance properties. For m = 3 there are only two distinct designs possible, and their relative efficiency is shown to be .

6 citations

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TL;DR: In the United States, any part of the Internet is available for any purpose of the United states Government as discussed by the authors, i.e., search, download, upload, and share.

Abstract: in whole or in part is permitted for any purpose of the United states Government.

250 citations

01 Jan 1963

TL;DR: In this paper, the authors introduce a class of multidimensional designs under the model of additivity of factorial effects which involve partial balance, and apply the theory so developed to the problem of analysing irregular fractions of oertain kinds.

Abstract: The purpose of this paper is three fold. The first one is to introduce a class of multidimensional designs (under the model of additivity of factorial effects) which involve partial balance. The next purpose is to examine closely the pattern in the matrices which one has to invert in order to carry out the analysis. The third aim is to apply the theory so developed to the problem of analysing irregular fractions of oertain kinds.

48 citations

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TL;DR: In this paper, the authors considered balanced incomplete block designs for two-way elimination of heterogeneity and showed that they can be converted to Youden's squares by ordering the varieties in the blocks such that each variety occurs once in each position.

Abstract: Sometimes in a design the position within the block is important as a source of variation, and the experiment gains in efficiency by eliminating the positional effect. The classical example is due to Youden in his studies on the tobacco mosaic virus [1]. He found that the response to treatments also depends on the position of the leaf on the plant. If the number of leaves is sufficient so that every treatment can be applied to one leaf of a tree, then we get an ordinary Latin square, in which the trees are columns and the leaves belonging to the same position constitute the rows. But if the number of treatments is larger than the number of leaf positions available, then we must have incomplete columns. Youden used a design in which the columns constituted a balanced incomplete block design, whereas the rows were complete. These designs are known as Youden's squares, and can be used when two-way elimination of heterogeneity is desired. In Fisher and Yates statistical tables [2] balanced incomplete block designs in which the number of blocks $b$ is equal to the number of treatments $v$ have been used to obtain Youden's squares, and the authors state that "in all cases of practical importance" it has been found possible to convert balanced incomplete blocks of the above kind to a Youden's square by so ordering the varieties in the blocks that each variety occurs once in each position. F. W. Levi noted ([3], p. 6) that this reordering can always be done, in virtue of a theorem given by Konig [4] which states that an even regular graph of degree $m$ is the product of $m$ regular graphs of degree 1. Smith and Hartley [5] give a practical procedure for converting balanced incomplete blocks with $b = v$ into Youden's squares. In this paper I have considered some general classes of designs for two-way elimination of heterogeneity. In Section 3 balanced incomplete block designs for which $b = mv$ have been used to obtain two-way designs in which each treatment occurs in a given position $m$ times. The case $m = 1$ gives Youden's squares. In Section 4 it has been shown that balanced incomplete block designs for which $b$ is not an integral multiple of $v$ can be used to obtain designs for two-way elimination of heterogeneity in which there are two accuracies (i.e., some pairs of treatments are compared with one accuracy, while other pairs are compared with a different accuracy) as in the case of lattice designs for one-way elimination of heterogeneity. In Sections 5 and 6 partially balanced designs have been used to obtain two-way designs with two accuracies. In every case the method of analysis and tables of actual designs have been given.

42 citations

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TL;DR: In this paper, a class of designs for the case of three factors at levels m, n and p were described for which the total number of experiments required is a fraction of mnp.

Abstract: A class of designs are described for the case of three factors at levels m, n and p respectively for which the total number of experiments required is a fraction of mnp.

33 citations