# Effectiveness of rank correlations in curvilinear relationships

- 51 Downloads
- 1 Citations

## Abstract

In cases where only linear relationships are suspected, Pearson’s correlation is generally used to measure the strength of the association between variables. It is well-known, however, that when a non-linear or non-linearizable connection exists, the use of Pearson’s coefficient on original values can be deceiving. On the other hand, rank correlations should perform satisfactorily because of their properties and versatility. There are many coefficients of rank correlation, from simple ones to complicated definitions invoking one or more special transformations. Each of these methods is sensitive to a different feature of dependence between variables The purpose of this article is to find a coefficient, if one exists, that tends to be different from zero at least in a meaningful way more often than others when the relationship between two rankings is of a non linear type. In this regard, we analyze the behavior of a few well-known rank correlation coefficients by focusing on some frequently encountered non-linear patterns. We conclude that a reasonably robust answer to the special needs arising from non-linear relationships could be given by a variant of the Fisher–Yates coefficient, which has a more marked tendency to reject the hypothesis of independence between pairs of rankings connected by several forms of non-linear interaction.

## Keywords

Ordinal data Non-linear relationship Non-monotone correlation Rank-order association## References

- Amerise IL, Tarsitano A (2016) A new symmetrical test of bivariate independence (under review)Google Scholar
- Amerise IL, Marozzi M, Tarsitano A (2016) pvrank: Rank correlations. R package version 1.1.1. http://CRAN.R-project.org/package=pvrank
- Barton DE, Mallows CL (1965) Some aspects of the random sequence. Ann Math Stat 36:236–260MathSciNetCrossRefzbMATHGoogle Scholar
- Bhat DN, Nayar SK (1997) Ordinal measures for visual correspondence. In: Proceedings of the IEEE computer society conference on computer vision and pattern recognition, San Francisco (CA, USA), 18–20 June 1996, pp 351–357Google Scholar
- Blomqvist N (1950) On a measure of dependence between two random variables. Ann Math Stat 21:593–600MathSciNetCrossRefzbMATHGoogle Scholar
- Borroni CG (2013) A new rank correlation measure. Stat Pap 54:255–270MathSciNetCrossRefzbMATHGoogle Scholar
- Borroni CG, Zenga M (2007) A test of concordance based on Gini’s mean difference. Stat Methods Appl 16:289–308MathSciNetCrossRefzbMATHGoogle Scholar
- Cameron PJ, Wu T (2010) The complexity of the weight problem for permutation and matrix groups. Electron Notes Discret Math 28:109–116CrossRefzbMATHGoogle Scholar
- Campano F, Salvatore D (1988) Economic development, income inequality and Kuznets’ U-shaped hypothesis. J Policy Model 10:265–280CrossRefGoogle Scholar
- Conover WJ, Iman RL (1981) Rank transformations as a bridge between parametric and nonparametric statistics. Am Stat 35:124–129zbMATHGoogle Scholar
- Critchlow DE (1986) A unified approach to constructing nonparametric rank tests. Technical Report, Department of Statistics, Purdue University, Indiana, 86-15. Available at http://www.stat.purdue.edu/docs/research/tech-reports/1986/tr86-15.pdf
- Dallal GE, Hartigan JA (1980) Note on a test of monotone association insensitive to outliers. J Am Stat Assoc 75:722–725MathSciNetCrossRefGoogle Scholar
- Diaconis P, Graham RL (1977) Spearmans footrule as a measure of disarray. J R Stat Soci B 39:262–268MathSciNetzbMATHGoogle Scholar
- Fieller EC, Pearson ES (1961) Tests for rank correlation coefficients: II. Biometrika 48:29–40MathSciNetzbMATHGoogle Scholar
- Filliben JJ (1975) The probability plot correlation coefficient test for normality. Technometrics 17:111–117CrossRefzbMATHGoogle Scholar
- Fisher R, Yates F (1938) Statistical tables for biological, agricultural and medical research, 5th edn. Oliver and Boyd, EdinburghzbMATHGoogle Scholar
- Genest C, Plante J-F (2003) On Blest’s measure of rank correlation. Can J Stat 31:35–52MathSciNetCrossRefzbMATHGoogle Scholar
- Gideon RA, Hollister A (1987) A rank correlation coefficient resistant to outliers. J Am Stat Assoc 82:656–666MathSciNetCrossRefzbMATHGoogle Scholar
- Gini C (1914) Di una misura delle relazioni tra le graduatorie di due caratteri. Tipografia Cecchini, RomaGoogle Scholar
- Gordon AD (1979a) A measure of the agreement between rankings. Biometrika 66:7–15MathSciNetCrossRefzbMATHGoogle Scholar
- Gordon AD (1979b) Another measure of the agreement between rankings. Biometrika 66:327–332MathSciNetCrossRefGoogle Scholar
- Hájek J, Šidak Z, Sen PK (1999) Theory of rank tests, 2nd edn. Academic Press, San DiegozbMATHGoogle Scholar
- Hamming RW (1950) Error detecting and error correcting codes. Bell Syst Tech J 29:147–160MathSciNetCrossRefGoogle Scholar
- Hoeffding W (1948) A non-parametric test of independence. Ann Math Stat 19:546–557MathSciNetCrossRefzbMATHGoogle Scholar
- Hotelling H, Pabst MR (1936) Rank correlation and tests of significance involving no assumption of normality. Ann Math Stat 7:29–43CrossRefzbMATHGoogle Scholar
- Kemeny JG (1959) Mathematics without numbers. Daedalus 88:577–591Google Scholar
- Kendall MG (1938) A new measure of rank correlation. Biometrika 30:91–93CrossRefzbMATHGoogle Scholar
- King TS, Chinchilli VM (2001) Robust estimators of the concordance correlation coefficient. J Biopharm Stat 11:83–105CrossRefGoogle Scholar
- Knuth DE (1973) The art of computer programming, 2nd edn. Vol 1: Fundamental algorithms. Addison-Wesley Publishing Company, ReadingGoogle Scholar
- Lieberson S (1964) Limitations in the application of non-parametric coefficients of correlation. Am Sociol Rev 29:744–746CrossRefGoogle Scholar
- Mango A (2006) A distance function for ranked variables: a proposal for a new rank correlation coefficient. Metodološki Zvezki 3:9–19Google Scholar
- Millard SP (2013) EnvStats: an R package for environmental statistics. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Moore GH, Wallis AW (1943) Time series significance test based on sign differences. J Am Stat Assoc 38:153–165MathSciNetCrossRefzbMATHGoogle Scholar
- Moran PAP (1950) A curvilinear ranking test. J R Stat Soc B 12:292–295MathSciNetzbMATHGoogle Scholar
- Niewiadomska-Bugaj M, Kowalczyk T (2006) A new test of association and other tests based on the Gini mean difference. Metron 64:399–409MathSciNetGoogle Scholar
- Peters CC (1946) A new descriptive statistic: the parabolic correlation coefficient. Psychometrika 11:57–69MathSciNetCrossRefzbMATHGoogle Scholar
- Rényi A (1959) On measures of dependence. Acta Math Hung 10:441–451MathSciNetCrossRefzbMATHGoogle Scholar
- Salama IA, Quade D (1997) The asymptotic normality of a rank correlation statistic based on rises. Stat Probab Lett 32:201–205MathSciNetCrossRefzbMATHGoogle Scholar
- Salama IA, Quade D (2001) The symmetric footrule. Commun Stat Part A Theory Methods 30:1099–1109MathSciNetCrossRefzbMATHGoogle Scholar
- Salvemini T (1951) Sui vari indici di concentrazione. Statistica 2:133–154Google Scholar
- Sarndäl CE (1974) A comparative study of association measures. Psychometrika 39:165–187CrossRefzbMATHGoogle Scholar
- Scarsini M (1984) On measures of dependence. Stochastica 8:201–218MathSciNetzbMATHGoogle Scholar
- Schweizer B, Wolff EF (1981) On nonparametric measures of dependence for random variables. Ann Stat 9:879–885MathSciNetCrossRefzbMATHGoogle Scholar
- Shaper AG, Wannamethee G, Walker M (1988) Alcohol and mortality in British men: explaining the U-shaped curve. Lancet 2:1267–1273CrossRefGoogle Scholar
- Spearman C (1904) The proof and measurement of association between two things. Am J Psichol 15:72–101CrossRefGoogle Scholar
- Spearman C (1906) “Footrule” for measuring correlation. Br J Psychol 2:89–108Google Scholar
- Sugano O, Watadani S (1993) An association analysis based on a statistic orthogonal to linear rank statistics. Behaviormetrica 29:17–33CrossRefGoogle Scholar
- Tarsitano A, Amerise LI (2015) On a new measure of rank-order association. J Stat Econom Methods 4:83–105Google Scholar
- Tarsitano A, Lombardo R (2013) A coefficient of correlation based on ratios of ranks and anti-ranks. Jahrnbucher fr Nationalkonomie und Statistik 233:206–224Google Scholar
- Theil H (1950) A rank-invariant method of linear and polynomial regression analysis. Part 3. Proceedings of Koninklijke Nederlandse Akademie van Wetenenschappen 53:1397–1412zbMATHGoogle Scholar
- Ulam SM (1961) Monte Carlo calculations in problems of mathematical physics. In: Beckenbach F (ed) Modern mathematics for the engineer: second series. McGraw-Hill, New YorkGoogle Scholar
- Wanniski J (1978) Taxes, revenues, and the ‘Laffer curve’. Public Interest 50:3–16Google Scholar
- Whitfield JW (1950) Uses of the ranking method in psychology. J R Stat Soc Ser B 12:163–170Google Scholar
- Yerkes RM, Dodson JD (1908) The relation of strength of stimulus to rapidity of habit-formation. J Comp Neurol Psychol 18:459–482CrossRefGoogle Scholar
- Zayed H, Quade D (1997) On the resistance of rank correlation. J Stat Comput Simul 48:59–81CrossRefzbMATHGoogle Scholar