Advertisement

Journal of Statistical Theory and Practice

, Volume 11, Issue 3, pp 393–401 | Cite as

How robust is the modified sequential triangular test of a correlation coefficient against nonnormality of the basic variables?

  • Dieter Rasch
  • Takuya Yanagida
Article

Abstract

There is a big lack of knowledge as concerns a test of the null hypothesis H0: 0 < ρρ0. Usually a test applies by some z-statistic according to Fisher (1921), which is approximately normally distributed. However, there is no evidence of whether the approximation is actually good enough—that is, it is of interest how the factual distribution of the respective test statistic holds the type-I risk—and which type-II risk results. Because this question cannot be answered theoretically at present, we try a simulation study in order to gain respective knowledge. For this, we even investigate the case of variables that are not (at all) normally distributed. Moreover, we consider variables not normally distributed but test the simple case of the exact t-test of H0: ρ = ρ0. The results show, in particular, that the test tracing back to R. A. Fisher does not hold the type-I risk if severe nonnormality of the variables’ distributions is given.

Keywords

Triangular sequential test correlation coefficient robustness nonnormality Fleishman system of distributions 

AMS Subject Classification

62 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cramér, H. 1946. Mathematical methods of statistics. Princeton, NJ: Princeton Press.zbMATHGoogle Scholar
  2. Demirtas, H., Y. Shi, and R. Allozi. 2016. PoisNonNor: Simultaneous generation of count and continuous data. R package version 1.1. https://doi.org/cran.r-project.org/web/packages/PoisNonNor/.
  3. Fisher, R. A. 1915. Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika 10:507–21.Google Scholar
  4. Fisher, R. A. 1921. On the “probable error” of a coefficient of correlation deduced from a small sample. Metron 1:3–32.Google Scholar
  5. Fleishman, A. J. 1978. A method for simulation non-normal distributions. Psychometrika 43:521–32. doi:10.1007/BF02293811.CrossRefGoogle Scholar
  6. Headrick, T. C, and S. S. Sawilowksy. 1999. Simulating correlated multivariate nonnormal distributions: Extending the Fleishman power method. Psychometrika 64:25–35. doi:10.1007/BF02294317.MathSciNetCrossRefGoogle Scholar
  7. Kubinger, K. D., D. Rasch, and M. Šimečkova. 2007. Testing a correlation coefficient’s significance: Using H0: 0 < ρ ≤ λ is preferable to H0: ρ = 0. Psychology Science 49:74–87.Google Scholar
  8. R Core Team. 2016. R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. https://doi.org/www.R-project.org/.Google Scholar
  9. Rasch, D., K. D. Kubinger, and T. Yanagida. 2011. Statistics in psychology—Using R and SPSS. Chichester, UK: Wiley.CrossRefGoogle Scholar
  10. Rasch, D., and D. Schott. 2016. Mathematische statistik [Mathematical statistics]. Weinheim, Germany: Wiley-VCH.zbMATHGoogle Scholar
  11. Schneider, B., D. Rasch, K. D. Kubinger, and T. Yanagida. 2015. A sequential triangular test of a correlation coefficient’s null-hypothesis: 0 < ρ ≤ ρ 0. Statistical Papers 56:689–99.MathSciNetCrossRefGoogle Scholar
  12. Vale, C. D., and V. A. Maurelli. 1983. Simulating multivariate nonnormal distributions. Psychometrika 48:465–71. doi:10.1007/BF02293687.CrossRefGoogle Scholar
  13. Yanagida, T. 2016. Miscor: Miscellaneous functions for the correlation coefficient. R package version 0.1-0. https://doi.org/cran.r-project.org/web/packages/miscor/.

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of Natural SciencesViennaAustria
  2. 2.School of Applied Health and Social SciencesUniversity of Applied Sciences Upper AustriaLinzAustria
  3. 3.RostockGermany

Personalised recommendations