Sage Journals: Discover world-class research

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.

Keywords

Data-driven test dependence measure empirical likelihood Pearson’s correlation coefficient random effect rank test test of independence

Get full access to this article

View all access options for this article.

References

Beeton

Yule

Pearson

. Data for the problem of evolution in man. v. on the correlation between duration of life and the number of offspring. Proc R Soc London 1900; 67: 159–179.

Kendall M and Stuart A. The Advanced Theory of Statistics. Vol. 2: Inference and Relationship. London: Griffin, 1961.

Pearson

. Notes on the history of correlation. Biometrika 1920; 13: 25–45.

Spearman

. The proof and measurement of association between two things. Am J Psychol 1904; 15: 72–101.

Embrechts

McNeil

Straumann

. Correlation and dependence in risk management: properties and pitfalls. Risk Management: Value at Risk and Beyond, New York: Cambridge University Press, 2002, pp. 176–223.

Kendall

. A new measure of rank correlation. Biometrika 1938; 30: 81–93.

Kendall

. Rank correlation methods, London: Griffin, 1948.

Mudholkar

Wilding

. On the conventional wisdom regarding two consistent tests of bivariate independence. J R Stat Soc Ser D 2003; 52: 41–57.

Harrell

Lee

Mark

. Tutorial in biostatistics multivariable prognostic models: issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors. Stat Med 1996; 15: 361–387.

10.

May

Bigelow

. Modeling nonlinear dose-response relationships in epidemiologic studies: statistical approaches and practical challenges. Dose-Response 2005; 3: 474–490.

11.

Kallenberg

Ledwina

. Data-driven rank tests for independence. J Am Stat Assoc 1999; 94: 285–301.

12.

Owen

. Empirical likelihood ratio confidence regions. Ann Stat 1990; 18: 90–120.

13.

Einmahl

McKeague

. Empirical likelihood based hypothesis testing. Bernoulli 2003; 9: 267–290.

14.

Vexler

Tsai

Hutson

. A simple density-based empirical likelihood ratio test for independence. Am Stat 2014; 68: 158–169.

15.

Hoeffding

. A non-parametric test of independence. Ann Math Stat 1948; 19: 546–557.

16.

Reshef

Finucane

et al.

Detecting novel associations in large data sets. Science 2011; 334: 1518–1524.

17.

Fisher

Switzer

. Graphical assessment of dependence: Is a picture worth 100 tests? Am Stat 2001; 55: 233–239.

18.

Wilk

Gnanadesikan

. Probability plotting methods for the analysis for the analysis of data. Biometrika 1968; 55: 1–17.

19.

Genest

Boies

. Detecting dependence with Kendall plots. Am Stat 2003; 57: 275–284.

20.

Vexler

Schisterman

Liu

. Estimation of roc curves based on stably distributed biomarkers subject to measurement error and pooling mixtures. Stat Med 2008; 27: 280–296.

21.

Pepe

. A regression modelling framework for receiver operating characteristic curves in medical diagnostic testing. Biometrika 1997; 84: 595–608.

22.

Dong

Zhang

et al.

Strong association between two polymorphisms on 15q25. 1 and lung cancer risk: a meta-analysis. PloS One 2012; 7: e37970–e37970.

23.

Rakovski

Weisenberger

Marjoram

et al.

Modeling measurement error in tumor characterization studies. BMC Bioinform 2011; 12: 284–284.

24.

Balakrishnan

Lai

. Continuous bivariate distributions, New York: Springer-Verlag, 2009.

25.

Schweizer

Wolff

. On nonparametric measures of dependence for random variables. Ann Stat 1981; 9: 879–885.

26.

Kendall

Buckland

. A dictionary of statistical terms, London: Longman, 1957.

27.

Neyman

Pearson

. On the problem of the most efficient tests of statistical hypotheses, New York: Springer-Verlag, 1992.

28.

Breiman

. Probability, Reading, MA: Addison-Wesley, 1968.

29.

Inglot

Kallenberg

Ledwina

. Data driven smooth tests for composite hypotheses. Ann Stat 1997; 25: 1222–1250.

30.

Owen

. Empirical likelihood, Boca Raton, FL: Chapman & Hall/CRC Press, 2001.

31.

Hollander

Wolfe

Chicken

. Nonparametric statistical methods 2013; Vol. 751, New Jersey: John Wiley & Sons.

32.

Elffers

. On interpreting the product moment correlation coefficient. Stat Neerl 1980; 34: 3–11.

33.

Simon N and Tibshirani R. Comment on “Detecting novel associations in large data sets” by Reshef et al., Science, Dec 16, 2011. ArXiv e-prints 2014; January.

34.

Bamber

. The area above the ordinal dominance graph and the area below the receiver operating characteristic graph. J Math Psych 1975; 12: 387–415.

35.

Johnson

. Multivariate statistical simulation: a guide to selecting and generating continuous multivariate distributions, New Jersey: John Wiley & Sons, 2013.

36.

Edwards

. An introduction to linear regression and correlation, New York: WH Freeman, 1976.

37.

Maor

King

. The story of a number 1994; Vol. 38, New York: Oxford University Press.

38.

Ball

. Shapes: Nature’s patterns: A tapestry in three parts 2011; Vol. 1, New York: Oxford University Press.

39.

Mardia

. Families of bivariate distributions 1970; Vol. 27, London: Griffin.

40.

Kotz

Balakrishnan

Johnson

. Continuous multivariate distributions, models and applications 2004; Vol. 1, New Jersey: John Wiley & Sons.

41.

Fang

Zhang

. Generalized multivariate analysis, Berlin: Springer-Verlag, 1990.

42.

Azzalini

Dalla Valle

. The multivariate skew-normal distribution. Biometrika 1996; 83: 715–726.

43.

Johnston

DiNardo

. Econometric methods, New York: McGraw-Hill Book Co, 1963.

44.

Cochran

. Errors of measurement in statistics. Technometrics 1968; 10: 637–666.

45.

Kulathinal

Kuulasmaa

Gasbarra

. Estimation of an errors-in-variables regression model when the variances of the measurement errors vary between the observations. Stat Med 2002; 21: 1089–1101.

46.

Peck

Beal

Sheiner

Nichols

. Extended least squares nonlinear regression: a possible solution to the choice of weights problem in analysis of individual pharmacokinetic data. J Pharmacokinet Biopharm 1984; 12: 545–558.

47.

Wood

. A simple test for random effects in regression models. Biometrika 2013; 100: 1005–1010.

48.

Armstrong

. Free radicals in diagnostic medicine: A systems approach to laboratory technology, clinical correlations, and antioxidant therapy, New York: Plenum Press, 1994.

49.

Schisterman

Faraggi

Browne

et al.

Tbars and cardiovascular disease in a population-based sample. J Cardiovasc Risk 2001; 8: 219–225.

50.

Stigler

. Do robust estimators work with real data? Ann Stat 1977; 5: 1055–1098.

51.

PYB

Donley

Hausmann

et al.

Candidate salivary biomarkers associated with alveolar bone loss: cross-sectional and in vitro studies. FEMS Immunol Med Microbiol 2007; 49: 252–260.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

1.87 MB

Dependence and independence: Structure and inference

Abstract

Keywords

Get full access to this article

References

Supplementary Material