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Showing papers in "Biometrika in 1952"









Journal ArticleDOI

188 citations



Journal ArticleDOI

141 citations




Journal ArticleDOI

Journal ArticleDOI
TL;DR: In this paper, a simple model for representing the effects of two or more factors is proposed, where the main effects of one factor are assumed to be proportional, rather than equal, at different levels of the other factors.
Abstract: Where the joint effects of two or more factors are not additive, a simple model is proposed for representing the effects. The effects of one factor are assumed to be proportional, rather than equal, at different levels of the other factors. The main effects of the first factor are given as weighted averages of the simple effects at the different levels of the other factors, the weights being the estimated factors of proportionality. The weights are given as the latent vector of a matrix of sums of squares and products corresponding to the largest latent root of the matrix; the sum of squares for the weighted main effect is a multiple of this latent root, and the other latent roots correspond to a partition of the interaction sum of squares. The analysis is closely related to the canonical analysis of a set of variates. Tests of significance of (a) the residual interactions and (b) the adequacy of a proposed set of weights are discussed. For the case where the matrix has only two non-vanishing latent roots, the approach of the joint distribution of the roots to its limiting form is discussed. The joint probability density is expanded as a series of Bessel functions of imaginary argument. Asymptotic formulae for the moments and product-moments of the roots are derived. Exact tests for the adequacy of a proposed set of weights, when there are only two non-vanishing latent roots, are presented. The methods of analysis are illustrated with a numerical example.

Journal ArticleDOI

Journal ArticleDOI
TL;DR: In this article, a systematic discussion of the general properties and enumeration of possible experimental designs is presented, in which treatments are applied to the experimental material in a number of successive periods, each experimental unit receiving a different treatment in each period.
Abstract: This paper is concerned with the type of experimental design in which treatments are applied to the experimental material in a number of successive periods, each experimental unit receiving a different treatment in each period. Although some of the simpler examples of these designs have been in practical use in at least one field of research for several years, a systematic discussion of their general properties and the enumeration of possible designs does not appear to have been previously attempted. In order that estimates of the effects of treatments and the errors can be estimated by a reasonably simple statistical analysis certain elements of balance, set out below, are required. The requirements depend largely upon which residual effects are to be estimated. For example, any Latin square arrangement, in which the rows represent the different periods so that a column of symbols refers to a sequence of treatments, is suitable for the case in which residual effects are negligible. When first residual effects (i.e. the effects of treatments in the period after application) are to be estimated the class of available designs is more restricted. A simple example has been described by Cochran, Autrey & Cannon (1 94 1) in connexion with feeding experiments on dairy cows. This arrangement, shown in Fig. 1, consists of two particular Latin squares.




Journal ArticleDOI
TL;DR: In this paper, the authors formulate mathematically some of the properties of structure constituting style, so that for a given text the application of a simple mathematical criterion allows its attribution to a particular author at a certain period of his mental development.
Abstract: Every significant text of a grammatical exposition consists of a certain material, the vocabulary, and some structural properties, the style, of its author. The passive vocabulary is formed by the totality of all words of that language, s, the author writes in, the active vocabulary is formed by a certain set, s', of that totality, the selection of which is determined essentially by the sort of literature the text belongs to and depends only in a lower degree on the peculiarity of the author. Style, however, is characteristic of the author at a certain period of his personal development. The aim of the following investigation is to formulate mathematically some of the properties of structure constituting style, so that for a given text the application of a simple mathematical criterion allows its attribution to a particular author at a certain period of his mental development.








Journal ArticleDOI
TL;DR: In this paper, the 6th and 8th orders of the moment coefficients of the distribution of the k-statistics in samples from a finite population were extended to all moment coefficients evaluated to this order.
Abstract: 1. Tukey (1950) has shown how it is possible to simplify the presentation of the moment coefficients of the distribution of certain of the k-statistics in samples from a finite population, illustrating in particular by calculating the variances of k, and k2, and their covariance. His method is to introduce a new sample statistic krs. , and to work out once and for all certain non-linear functions of these as linear functions of themselves. This is much of the labour of calculating the sampling moment coefficients; the subsequent work consists of selecting those that are required and putting them together. It should be mentioned that Dressel (1940), following the work of Dwyer (1938), introduced statistics L (rs, 1) as unbiased estimates of products ArAs... of the Thiele seminvariants. These are the same quantities as Tukey's kr8 although, because of the stated limitation to the values of r, 8, ..., they do not form a complete set. It is of some interest to carry the calculations further and derive new results. In this paper the formulae are extended to the 6th order, and all moment coefficients evaluated to this order. This will carry us a good way beyond the results of Irwin & Kendall (1944). Finally, certain basic formulae are given for the 7th and 8th orders. 2. Using the notation of David & Kendall (1949) for the symmetric functions of the n sample observations, we may write krs in general as