Review Articles

Regression models of Pearson correlation coefficient

Abdisa G. Dufera ,

School of Statistics, East China Normal University, Shanghai, People's Republic of China

Tiantian Liu ,

School of Statistics, East China Normal University, Shanghai, People's Republic of China; Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa, Israel

Jin Xu

School of Statistics, East China Normal University, Shanghai, People's Republic of China; Key Laboratory of Advanced Theory and Application in Statistics and Data Science – MOE, East China Normal University, Shanghai, People's Republic of China

jxu@stat.ecnu.edu.cn

Pages | Received 06 Sep. 2022, Accepted 01 Jan. 2023, Published online: 11 Jan. 2023,
  • Abstract
  • Full Article
  • References
  • Citations

We propose two simple regression models of Pearson correlation coefficient of two normal responses or binary responses to assess the effect of covariates of interest. Likelihood-based inference is established to estimate the regression coefficients, upon which bootstrap-based method is used to test the significance of covariates of interest. Simulation studies show the effectiveness of the method in terms of type-I error control, power performance in moderate sample size and robustness with respect to model mis-specification. We illustrate the application of the proposed method to some real data concerning health measurements.

  • Anderson, T. (2003). An introduction to statistical multivariate analysis. 3rd ed. Wiley. 
  • Bartlett, R. F. (1993). Linear modelling of Pearson's product moment correlation coefficient: An application of Fisher's z-transformation. Journal of the Royal Statistical Society: Series D42(1), 45–53.
  • Fryback, D. G. (2009). United States National Health Measurement Study, 2005-2006. Inter-University Consortium for Political and Social Research. 
  • Kim, D. (2016). Correlation between physical function, cognitive function, and health-related quality of life in elderly persons. Journal of Physical Therapy Science28(6), 1844–1848. https://doi.org/10.1589/jpts.28.1844 
  • Liu, Q., Li, C., Wanga, V., & Shepherd, B. E. (2018). Covariate-adjusted Spearman rank correlation with probability-scale residuals. Biometrics74(2), 595–605. https://doi.org/10.1111/biom.v74.2
  • Newey, W. K., & McFadden, D. (1994). Large sample estimation and hypothesis testing. Handbook of Econometrics4, 2111–2245. https://doi.org/10.1016/S1573-4412(05)80005-4 
  • Paul, S. (1989). Test for the equality of several correlation coefficients. Canadian Journal of Statistics17(2), 217–227. https://doi.org/10.2307/3314850
  • Qaqish, B. F. (2003). A family of multivariate binary distributions for simulating correlated binary variables with specified marginal means and correlations. Biometrika90(2), 455–463. https://doi.org/10.1093/biomet/90.2.455 
  • Thomas, P., Jolanda, L., Antonius, J. M. d. C., Joris, P. J. S., & Rudi, G. J. W. (2016). Impact of physical and mental health on life satisfaction in old age: A population based observational study. BMC Geriatrics16(1), 194. https://doi.org/10.1186/s12877-016-0365-4 
  • Tsiatis, A. A. (2006). Semiparametric theory and missing data. Springer. 
  • Ware Jr, J. E., Kosinski, M., & Keller, S. D. (1994). SF-36 physical and mental summary scales: A user's manual. The Health Institute, New England Medical Center.
  • Wilding, G. E., Cai, X., Hutson, A., & Yu, Z. (2011). A linear model-based test for the heterogeneity of conditional correlations. Journal of Applied Statistics38(10), 2355–2366. https://doi.org/10.1080/02664763.2011.559201
  • Yu, M., & Dunn, O. (1982). Robust tests for the equality of two correlation coefficients: A monte carlo study. Educational and Psychological Measurement42(4), 987–1004. https://doi.org/10.1177/001316448204200407

To cite this article: Abdisa G. Dufera, Tiantian Liu & Jin Xu (2023) Regression models of Pearson correlation coefficient, Statistical Theory and Related Fields, 7:2, 97-106, DOI: 10.1080/24754269.2023.2164970

To link to this article: https://doi.org/10.1080/24754269.2023.2164970