Measures of Association for Cross Classifications III: Approximate Sampling Theory
01 Jun 1963-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 58, Iss: 302, pp 310-364
TL;DR: In this paper, the authors derived large sample normal distributions with their associated standard errors for various measures of association and various methods of sampling and explained how the large sample normality may be used to test hypotheses about the measures and about differences between them, and to construct corresponding confidence intervals.
Abstract: The population measures of association for cross classifications, discussed in the authors' prior publications, have sample analogues that are approximately normally distributed for large samples. (Some qualifications and restrictions are necessary.) These large sample normal distributions with their associated standard errors, are derived for various measures of association and various methods of sampling. It is explained how the large sample normality may be used to test hypotheses about the measures and about differences between them, and to construct corresponding confidence intervals. Numerical results are given about the adequacy of the large sample normal approximations. In order to facilitate extension of the large sample results to other measures of association, and to other modes of sampling, than those treated here, the basic manipulative tools of large sample theory are explained and illustrated.
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TL;DR: A general statistical methodology for the analysis of multivariate categorical data arising from observer reliability studies is presented and tests for interobserver bias are presented in terms of first-order marginal homogeneity and measures of interob server agreement are developed as generalized kappa-type statistics.
Abstract: This paper presents a general statistical methodology for the analysis of multivariate categorical data arising from observer reliability studies. The procedure essentially involves the construction of functions of the observed proportions which are directed at the extent to which the observers agree among themselves and the construction of test statistics for hypotheses involving these functions. Tests for interobserver bias are presented in terms of first-order marginal homogeneity and measures of interobserver agreement are developed as generalized kappa-type statistics. These procedures are illustrated with a clinical diagnosis example from the epidemiological literature.
64,109 citations
TL;DR: In this paper, a combination of seven surveys of blood parasites in North American passerines reveals weak, highly significant association over species between incidence of chronic blood infections (five genera of protozoa and one nematode) and striking display (three characters: male "brightness", female "brights", and male song).
Abstract: Combination of seven surveys of blood parasites in North American passerines reveals weak, highly significant association over species between incidence of chronic blood infections (five genera of protozoa and one nematode) and striking display (three characters: male "brightness," female "brightness," and male song). This result conforms to a model of sexual selection in which (i) coadaptational cycles of host and parasites generate consistently positive offspring-on-parent regression of fitness, and (ii) animals choose mates for genetic disease resistance by scrutiny of characters whose full expression is dependent on health and vigor.
3,537 citations
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01 Jan 1979TL;DR: In this article, a number of alternative measures are considered, almost all based upon a probabilistic model for activity to which the cross-classification may typically lead, and only the case in which the population is completely known is considered, so no question of sampling or measurement error appears.
Abstract: When populations are cross-classified with respect to two or more classifications or polytomies, questions often arise about the degree of association existing between the several polytomies. Most of the traditional measures or indices of association are based upon the standard chi-square statistic or on an assumption of underlying joint normality. In this paper a number of alternative measures are considered, almost all based upon a probabilistic model for activity to which the cross-classification may typically lead. Only the case in which the population is completely known is considered, so no question of sampling or measurement error appears. We hope, however, to publish before long some approximate distributions for sample estimators of the measures we propose, and approximate tests of hypotheses. Our major theme is that the measures of association used by an empirical investigator should not be blindly chosen because of tradition and convention only, although these factors may properly be g...
2,672 citations
TL;DR: A self-selection bias in satisfaction measures used commonly for QFD and for corporate incentive programs is demonstrated, demonstrating how a product-development team used the voice of the customer to create a successful new product.
Abstract: In recent years, many U.S. and Japanese firms have adopted Quality Function Deployment QFD. QFD is a total-quality-management process in which the "voice of the customer" is deployed throughout the R&D, engineering, and manufacturing stages of product development. For example, in the first "house" of QFD, customer needs are linked to design attributes thus encouraging the joint consideration of marketing issues and engineering issues. This paper focuses on the "Voice-of-the-Customer" component of QFD, that is, the tasks of identifying customer needs, structuring customer needs, and providing priorities for customer needs.
In the identification stage, we address the questions of 1 how many customers need be interviewed, 2 how many analysts need to read the transcripts, 3 how many customer needs do we miss, and 4 are focus groups or one-on-one interviews superior? In the structuring stage the customer needs are arrayed into a hierarchy of primary, secondary, and tertiary needs. We compare group consensus affinity charts, a technique which accounts for most industry applications, with a technique based on customer-sort data. In the stage which provides priorities we present new data in which product concepts were created by product-development experts such that each concept stressed the fulfillment of one primary customer need. Customer interest in and preference for these concepts are compared to measured and estimated importances. We also address the question of whether frequency of mention can be used as a surrogate for importance. Finally, we examine the stated goal of QFD, customer satisfaction. Our data demonstrate a self-selection bias in satisfaction measures that are used commonly for QFD and for corporate incentive programs.
We close with a brief application to illustrate how a product-development team used the voice of the customer to create a successful new product.
1,867 citations
TL;DR: It is shown that the computational behaviour of a hierarchical sorting-strategy depends on three properties, which are established for five conventional strategies and four measures.
Abstract: It is shown that the computational behaviour of a hierarchical sorting-strategy depends on three properties, which are established for five conventional strategies and four measures. The conventional strategies are shown to be simple variants of a single linear system defined by four parameters. A new strategy is defined, enabling continuous variation of intensity of grouping by variation in a single parameter. An Appendix provides specifications of computer programs embodying the new principles.
1,697 citations
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01 Jan 1979TL;DR: In this article, a number of alternative measures are considered, almost all based upon a probabilistic model for activity to which the cross-classification may typically lead, and only the case in which the population is completely known is considered, so no question of sampling or measurement error appears.
Abstract: When populations are cross-classified with respect to two or more classifications or polytomies, questions often arise about the degree of association existing between the several polytomies. Most of the traditional measures or indices of association are based upon the standard chi-square statistic or on an assumption of underlying joint normality. In this paper a number of alternative measures are considered, almost all based upon a probabilistic model for activity to which the cross-classification may typically lead. Only the case in which the population is completely known is considered, so no question of sampling or measurement error appears. We hope, however, to publish before long some approximate distributions for sample estimators of the measures we propose, and approximate tests of hypotheses. Our major theme is that the measures of association used by an empirical investigator should not be blindly chosen because of tradition and convention only, although these factors may properly be g...
2,672 citations
TL;DR: In this article, the authors considered the problem of estimating a U-statistic of the population characteristic of a regular functional function, where the sum ∑″ is extended over all permutations (α 1, α m ) of different integers, 1 α≤ (αi≤ n, n).
Abstract: Let X 1 …, X n be n independent random vectors, X v = , and Φ(x 1 …, x m ) a function of m(≤n) vectors . A statistic of the form , where the sum ∑″ is extended over all permutations (α1 …, α m ) of different integers, 1 α≤ (αi≤ n, is called a U-statistic. If X 1, …, X n have the same (cumulative) distribution function (d.f.) F(x), U is an unbiased estimate of the population characteristic θ(F) = f … f Φ(x 1,…, x m ) dF(x 1) … dF(x m ). θ(F) is called a regular functional of the d.f. F(x). Certain optimal properties of U-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4)
2,439 citations
TL;DR: The three measures considered at length are the quadrant measure, Kendall's tau, and Spearman's rho as mentioned in this paper, with emphasis on the probabilistic and operational interpretations of their population values.
Abstract: Ordinally invariant, i.e., rank, measures of association for bivariate populations are discussed, with emphasis on the probabilistic and operational interpretations of their population values. The three measures considered at length are the quadrant measure, Kendall's tau, and Spearman's rho. Relationships between these measures are discussed, as are connections between these measures and certain measures of association for cross classifications. Sampling theory is surveyed with special attention to the motivation for sample values of the measures. The historical development of ordinal measures of association is outlined. * This research was supported in part by the Statistics Branch, Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the United States Government. A large part of the work leading to this paper was done at the Department of Statistics, University of California, Berkeley. I would like to thank the following persons for their suggestions and c...
684 citations
TL;DR: The central limit theorem has been extended to the case of dependent random variables by several authors (Bruns, Markoff, S. Bernstein, P. Levy, and Loeve) as mentioned in this paper.
Abstract: The central limit theorem has been extended to the case of dependent random variables by several authors (Bruns, Markoff, S. Bernstein, P. Levy, Loeve). The conditions under which these theorems are stated either are very restrictive or involve conditional distributions, which makes them difficult to apply. In the present paper we prove central limit theorems for sequences of dependent random variables of a certain special type which occurs frequently in mathematical statistics. The hypotheses do not involve conditional distributions.
456 citations