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Nonparametric tests for independence: a review and comparative simulation study with an application to malnutrition data in India

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Abstract

The detection of dependence structures within a set of random variables provides a valuable basis for a detailed subsequent investigation of their relationships. Nonparametric tests for independence require only basic assumptions on the marginal or joint distribution of the involved variables. In this paper, we review nonparametric tests of independence in bivariate as well as multivariate settings which are throughout ready-to-use, i.e., implemented in R packages. Highlighting their distinct empirical size and power properties in various small sample settings, our analysis supports an analyst in deciding for a most adequate test conditional on underlying distributional settings or data characteristics. Avoiding restrictive moment conditions, the copula based Cramér-von Mises distance of Genest and Rémillard (Test 13:335–370, 2004) is remarkably robust under the null hypothesis and powerful under diverse settings that are in line with the alternative hypothesis. Based on distinguished test outcomes in small samples, we detect nonlinear dependence structures between childhood malnutrition indices and possible determinants in an empirical application for India.

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Notes

  1. Tests for serial dependence in time series are not explicitly considered here. An overview of corresponding approaches is given in Diks (2009).

  2. More detailed descriptions of covered distributions and the theoretical background are given in Oja (2010). For the multivariate extension of Spearman’s rho we estimate the shape matrix by means of a rank based covariance matrix [see also the documentation of the corresponding R package SpatialNP (Sirkia et al. 2018)].

  3. For instance, Shih and Emura (2016) study the properties of Spearman’s rho and Kendall’s tau under specific copula structures.

  4. In this study, the considered nominal significance level is \(\alpha =0.05.\) Similar results obtain with respect to other conventional levels, for instance \(\alpha =0.1\).

  5. Siqueira Santos et al. (2013) consider alternative choices for the distribution of \(x_1\) as, for instance, equidistant points or the uniform distribution. Additionally, they study further nonmonotonic and nonlinear dependence structures, i.e., alternative choices of the function f.

  6. The asymptotic properties of Wilks’ Lambda have been shown under the multivariate Gaussian distribution. Thus, the comparison with the nonparametric tests is informative on the trade-off between efficient dependence detection within the Gaussian model, and robustness under more general distributional conditions.

  7. The whole set of characteristics, i.e. possible covariates in a regression model, are listed in Klein and Kneib (2016) and the references therein. Klein and Kneib (2016) further describe the nonlinear effects of the covariates on the bivariate distribution using nutrition data from all over India.

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Acknowledgements

We thank two anonymous referees and the Associate Editor Christine H. Müller for helpful comments and suggestions. Financial support from the Academy of Finland (308628) is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Economics, University of Goettingen, Göttingen, Germany

    Helmut Herwartz

  2. Department of Political and Economic Studies, University of Helsinki, P. O. Box 17, 00014, Helsinki, Finland

    Simone Maxand

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  1. Helmut Herwartz
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  2. Simone Maxand
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Correspondence to Simone Maxand.

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Herwartz, H., Maxand, S. Nonparametric tests for independence: a review and comparative simulation study with an application to malnutrition data in India. Stat Papers 61, 2175–2201 (2020). https://doi.org/10.1007/s00362-018-1026-9

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