title: Dependence and independence: Structure and inference.
abstract: Evaluations of relationships between pairs of variables, including testing for independence, are increasingly important. Erich Leo Lehmann noted that "the study of the power and efficiency of tests of independence is complicated by the difficulty of defining natural classes of alternatives to the hypothesis of independence." This paper presents a general review, discussion and comparison of classical and novel tests of independence. We investigate a broad spectrum of dependence structures with/without random effects, including those that are well addressed in both the applied and the theoretical scientific literatures as well as scenarios when the classical tests of independence may break down completely. Motivated by practical considerations, the impact of random effects in dependence structures are studied in the additive and multiplicative forms. A novel index of dependence is proposed based on the area under the Kendall plot. In conjunction with the scatterplot and the Kendall plot, the proposed method provides a comprehensive presentation of the data in terms of graphing and conceptualizing the dependence. We also present a graphical methodology based on heat maps to effectively compare the powers of various tests. Practical examples illustrate the use of various tests of independence and the graphical representations of dependence structures.
authors.affiliation: Department of Biostatistics, The State University of New York at Buffalo, Buffalo, NY, USA.
authors.forename: Albert
authors.initials: A
authors.lastname: Vexler
authors.affiliation: Department of Biostatistics, The State University of New York at Buffalo, Buffalo, NY, USA.
authors.forename: Xiwei
authors.initials: X
authors.lastname: Chen
authors.affiliation: Department of Biostatistics, The State University of New York at Buffalo, Buffalo, NY, USA.
authors.forename: Alan D
authors.initials: AD
authors.lastname: Hutson
mesh_terms: D003627:Data Interpretation, Statistical; D016013:Likelihood Functions; D015233:Models, Statistical; D013223:Statistics as Topic; D018709:Statistics, Nonparametric
keywords: Data-driven test; Pearson’s correlation coefficient; dependence measure; empirical likelihood; random effect; rank test; test of independence