Abstract
Spearman’s rank correlation coefficient might be the best known nonparametric measure of association currently in use. It assesses the linear relationships between the ranks of monotonically related variables even if the relationship between the variables is not linear. This study presents a new method for calculating two-dimensional (horizontal and vertical) rank correlation coefficients between matrices composed of variables which are not necessarily in linear association. The matrices contain the ranks of measurements instead of raw values. The averages of all rows are used for calculating the horizontal rank correlation and the averages of all columns are used for calculating the vertical rank correlation instead of considering the averages of the whole matrices. This approach enables a separate determination of the degree of horizontal and vertical relationships between the compared data matrices by using the horizontal and vertical variance and covariance values that constitute the base of the two-dimensional correlation method. The presented method is first applied on 5 simple hypothetical matrices and then on the monthly total precipitation records of 6 stations in southwest Turkey. The results have shown that the presented rank correlation approach successfully assesses the two-dimensional associations between the hypothetical matrices and time series data like precipitation, provides a measure for exactly determining the monotonic relationships not determined by the conventional Pearson’s and Spearman’s correlation approaches, it is much more robust to outliers and normality is not a prerequisite. The software developed for calculating two-dimensional rank correlation coefficients is freely provided together with this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad I, Tang D, Wang T, Wang M, Wagan B (2015) Precipitation trends over time using Mann-Kendall and Spearman’s Rho tests in swat river basin, Pakistan. Adv Meteorol 2015:431860. https://doi.org/10.1155/2015/431860
Amirataee B, Montaseri M, Sanikhani H (2016) The analysis of trend variations of reference evapotranspiration via eliminating the significance effect of all autocorrelation coefficients. Theor Appl Climatol 126:131–139. https://doi.org/10.1007/s00704-015-1566-z
Bland JM, Altman DG (1986) Statistical methods for assessing agreement between two methods of clinical measurement. Lancet 1:307–310
Brett MT (2004) When is a correlation between non-independent variables "spurious"? Oikos 105:647–656. https://doi.org/10.1111/j.0030-1299.2004.12777.x
de Winter JCF, Gosling SD, Potter J (2016) Comparing the Pearson and Spearman correlation coefficients across distributions and sample sizes: a tutorial using simulations and empirical data. Psychol Methods 21:273–290. https://doi.org/10.1037/met0000079.supp
Dikbas F (2016) Three-dimensional imputation of missing monthly river flow data. Sci Iranica 23:45–53
Dikbas F (2017a) Frequency based imputation of precipitation. Stoch Env Res Risk A 31:2415–2434. https://doi.org/10.1007/s00477-016-1356-x
Dikbas F (2017b) A novel two-dimensional correlation coefficient for assessing associations in time series data. Int J Climatol 37:4065–4076. https://doi.org/10.1002/joc.4998
Gauthier TD (2001) Detecting trends using Spearman’s rank correlation coefficient. Environ Forensic 2:359–362. https://doi.org/10.1006/enfo.2001.0061
Hauke J, Kossowski T (2011) Comparison of values of Pearson’s and Spearman’s correlation coefficients on the same sets of data. Quaestiones Geographicae 30:87–93. https://doi.org/10.2478/v10117-011-0021-1
Iqbal MA, Penas A, Cano-Ortiz A, Kersebaum KC, Herrero L, del Río S (2016) Analysis of recent changes in maximum and minimum temperatures in Pakistan. Atmos Res 168:234–249. https://doi.org/10.1016/j.atmosres.2015.09.016
Kahya E, Kalayci S (2004) Trend analysis of streamflow in Turkey. J Hydrol 289:128–144. https://doi.org/10.1016/j.jhydrol.2003.11.006
Li J, Tan S (2015) Nonstationary flood frequency analysis for annual flood peak series, adopting climate indices and check dam index as covariates. Water Resour Manag 29:5533–5550. https://doi.org/10.1007/s11269-015-1133-5
Lobbes MBI, Nelemans PJ (2013) Good correlation does not automatically implygood agreement: the trouble with comparingtumour size by breast MRI versus histopathology. Eur J Radiol 82:e906–e907. https://doi.org/10.1016/j.ejrad.2013.08.025
Lovie AD (1995) Who discovered Spearman’s rank correlation? Br J Math Stat Psychol 48:255–269. https://doi.org/10.1111/j.2044-8317.1995.tb01063.x
Seibert J, Bishop K, Rodhe A, McDonnell JJ (2003) Groundwater dynamics along a hillslope: a test of the steady state hypothesis. Water Resour Res 39:SWC21–SWC29
Shadmani M, Marofi S, Roknian M (2012) Trend analysis in reference evapotranspiration using Mann-Kendall and Spearman’s Rho tests in arid regions of Iran. Water Resour Manag 26:211–224. https://doi.org/10.1007/s11269-011-9913-z
Spearman C (1904) The proof and measurement of association between two things. Am J Psychol 15:72–101. https://doi.org/10.2307/1412159
Székely GJ, Rizzo ML, Bakirov NK (2007) Measuring and testing dependence by correlation of distances. Ann Stat 35:2769–2794
Tabari H, Abghari H, Hosseinzadeh Talaee P (2012) Temporal trends and spatial characteristics of drought and rainfall in arid and semiarid regions of Iran. Hydrol Process 26:3351–3361. https://doi.org/10.1002/hyp.8460
Xiong L, KX Y, Gottschalk L (2014) Estimation of the distribution of annual runoff from climatic variables using copulas. Water Resour Res 50:7134–7152. https://doi.org/10.1002/2013WR015159
Yilmaz AG, Perera BJC (2015) Spatiotemporal trend analysis of extreme rainfall events in Victoria, Australia. Water Resour Manag 29:4465–4480. https://doi.org/10.1007/s11269-015-1070-3
Yue S, Pilon P, Cavadias G (2002) Power of the Mann-Kendall and Spearman’s Rho tests for detecting monotonic trends in hydrological series. J Hydrol 259:254–271. https://doi.org/10.1016/S0022-1694(01)00594-7
Zamani R, Mirabbasi R, Abdollahi S, Jhajharia D (2017) Streamflow trend analysis by considering autocorrelation structure, long-term persistence, and Hurst coefficient in a semi-arid region of Iran. Theor Appl Climatol 129:33–45. https://doi.org/10.1007/s00704-016-1747-4
Zarei AR, Moghimi MM, Mahmoudi MR (2016) Parametric and non-parametric trend of drought in arid and semi-arid regions using RDI Index. Water Resour Manag 30:5479–5500. https://doi.org/10.1007/s11269-016-1501-9
Acknowledgements
The author would like to thank to The General Directorate of State Hydraulic Works of Turkey for providing the data used in this study. The complete data set is restricted for explicit publication and is commercially available at http://www2.dsi.gov.tr/duyuru/bedel.htm (in Turkish). The author would also like to thank the editors and reviewers for their valuable contributions and comments. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
ESM 1
(DOCX 371 kb)
Rights and permissions
About this article
Cite this article
Dikbaş, F. A New Two-Dimensional Rank Correlation Coefficient. Water Resour Manage 32, 1539–1553 (2018). https://doi.org/10.1007/s11269-017-1886-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11269-017-1886-0