Dependence measures

Proceedings of ASMDA, 2007

Classical association criteria used for measuring statistical independence between categorical variables were initially defined using contingency tables. There is another way to represent categorical variables : Relational Data Analysis which uses binary pairwise comparison matrices. There exists correspondance formulas that enable to get from one representation to the other. By using these formulas, and these two representations, we can have a better understanding of the differences between some famous association criteria. In fact, several types of independences, namely statistical, geometrical and logical, appear using one representation or the other. The aim of this paper is to present these different kinds of independences and their relationships by studying the expressions of the following association criteria in the two different representations : Belson, Lerman, χ 2 of Tchuprow, Jordan, Rand and Janson and Vegelius.

Introduction. In this work we propose some features for analysing cross-classification multidimensional tables, and test their effectiveness on articially constructed and on real data. Methods. Measures of similarity between two levels of one variable by their profiles along all combinations of levels of the remaining variables, and coefficient that measures the strenght of dependence of the levels of one variable, and of the whole variable as well, are proposed based on Hellinger distance. Tests base on these coefficients are propose for reduction of complexity in the array through its collapsability along an independent variable, and test for total dependence in the array as well. Results. The tests when applied on artificial data behaved as it was expected to, for all the analysed situations were constructed and we knew what the association structures were, and when analysing the real data the whole methodology work well permitting new insights on already analysed data.

Journal of Applied Statistics, 2015

The multiple non-symmetric correspondence analysis (MNSCA) is a useful technique for analysing the prediction of a categorical variable through two or more predictor variables placed in a contingency table. In MNSCA framework, for summarizing the predictability between criterion and predictor variables, the Multiple-TAU index has been proposed. But it cannot be used to test association, and for overcoming this limitation, a relationship with C-Statistic has been recommended. Multiple-TAU index is an overall measure of association that contains both main effects and interaction terms. The main effects represent the change in the response variables due to the change in the level/categories of the predictor variables, considering the effects of their addition. On the other hand, the interaction effect represents the combined effect of predictor variables on the response variable. In this paper, we propose a decomposition of the Multiple-TAU index in main effects and interaction terms. In order to show this decomposition, we consider an empirical case in which the relationship between the demographic characteristics of the American people, such as race, gender and location (column variables), and their propensity to move (row variable) to a new town to find a job is considered.

The purpose of this paper is the study of dependence for r×s tables in the class of all bivariate tables with given margins. First of all, a dependence partial ordering will be defined, based on the signs and the absolute values of r×s contingencies. Then, to increase the number of couples of comparable tables, a new partial ordering, called ″intensity of dependence ordering″ and based only on the absolute value of contingencies, is introduced. To allow the comparison of all pairs of tables, a dependence total ordering is needed, therefore a broad class of functions - able to induce total synthetic dependence orderings – is introduced, their arguments being the absolute values of relative contingencies. Among them, Mortara’s and Pearson’s indexes can be identified. Finally, the dependence degree of a given table can be expressed by its relative position in a chosen dependence ordering of all the tables in the same class. This methodology is applied, as an example, to four classes of tables.

Springer Optimization and Its Applications, 2009

HAL (Le Centre pour la Communication Scientifique Directe), 2018

Measuring dependence among random variables can be an important aspect of empirical data analysis and, statistical modeling and inference. Our recent proposal of a generalization of well-known Pearson's correlation coefficient for measuring non-linear dependencies is reviewed. We emphasize that if one wants to have a general measure of degree of dependence then, ideally, the conceptual definition of the Pearson's correlation coefficient should be followed. It is expressed as a ratio of dependence of interest to full linear dependence, in terms of Euclidean type distances among probability distributions/densities. A generalization of it should use a well-behaved distance among probability distributions/densities, such as a metric in the continuum of them, along with consideration of non-linear dependencies. We have proposed to use socalled Hellinger distance metric for the purpose. Along with some future research directions, we discuss some of the advantages of our proposed general measure. Here the discussion is without any mathematical details, thus reaching a wider readership. Our objective is to present some arguments of how to measure dependencies generally, and to make some remarks on our generalization, that may lead to new turns in the research directions of the dependence measures in the future.

Computational Statistics & Data Analysis, 2006

The generalisation of the Goodman-Kruskal statistic that is used for the measurement of the strength of dependence (association) between two categorical variables with ordered categories is presented. The case when some data are not precise, and observations are described by possibility distributions over a set of categories of one variable is considered. For such data the fuzzy version of statistic has been defined.

Information and control, 1971

Linfoot (1957) introduced an informational measure r! of correlation between two random variables X and Y. The measure r 1 is based on the information gain r0 in knowing that X and Y are mutually dependent with a given bivariate density function as compared with the original knowledge that X and Y are statistically independent. In the present paper, an asymptotic form of the information measure rl, denoted by rl, is derived in terms of Pearson's (1904) chi-square for contingency tables. Hence ~l is suggested as an information measure of association in contingency tables. On comparing fl with Pearson's classical coefficient of contingency P, it is found that ~I /> P. This is a desirable property of rI, since Lancaster and Hamdan (1964) demonstrated that P underestimates the product-moment correlation coefficient in contingency tables with broad categories. The asymptotic variance of rl is derived and compared with the asymptotic variance of P.

Computer Methods and Programs in Biomedicine, 2008

Triangular contingency tables are a special class of incomplete contingency tables. Association and independence models are used to analyze such tables. This paper presents and compares some methods including the uniform association model and the quasiindependence model. These models can be described in terms of the association parameters for the analysis of triangular contingency tables having ordered categories. A computer program is developed for the analysis of quasi-independence model for positive (negative) likelihood dependence. The sign test, which is a nonparametric test of the independence against the likelihood ratio dependence, is also examined. These methods are applied to the disability ratings of stroke patients data. Effects of the structural zeros on the results are also discussed.

2024

In the article An entropy-based measure of dependence between two groups of random variables, the K-dependence coefficient or dependence coefficient is defined with entropy and conditional entropy to measure the degree of dependence of one group of categorical random variables on another group of categorical random variables. Also, the concept of the K-dependence coefficient is extended by defining the partial K-dependence coefficient and the semi-partial K-dependence coefficient with which the K-dependence coefficient decomposition expression is established mathematically. The K-Dependence coefficient decomposition expression shows that K-dependence coefficient is of additive structure and therefore, according to Measure Theory, K-dependence coefficient is a measure of the degree of dependence of one group of categorical random variables on another group of categorical random variables. In this commentary, first we introduce the Entropy and Information graphical representation to help us understand K-dependence coefficient. Next, we introduce the concept of measure followed by a briefly discussion on Kdependence coefficient additive structure together with several other measures, such as coefficient of determination, Entropy and Probability. Finally, we introduce some ideas to extend K-dependence coefficient.