Coefficients for Bivariate Relations

  • Gideon J. MellenberghEmail author


A bivariate relation is a relation between two (X and Y) variables. A large number of coefficients for bivariate relations was developed. A classification of coefficients is presented to facilitate the choice of an appropriate coefficient. The classification is based on two distinctions. First, three types of relations are distinguished: (1) symmetrical (the relation between X and Y is the same as between Y and X), (2) equality (symmetrical and equality of X- and Y-values), and (3) asymmetrical (one variable is the independent variable or predictor and the other variable is the dependent variable or criterion). Second, five types of variables are distinguished: (1) dichotomous (two unordered or ordered categories), (2) nominal-categorical (more than two unordered categories), (3) ordinal-categorical (more than two ordered categories), (4) ranked (rank numbers), and (5) continuous (values from a continuum). Crossing of these two distinctions yields 3 × 5 = 15 different combinations. An example of a coefficient is given for 13 of these combinations (coefficients for the two other combinations are not known to the author). The examples are restricted to coefficients for relations between two variables of similar type, for example, Cohen’s kappa for an equality relation between two nominal-categorical variables.


Cohen’s (weighted) kappa Cramer’s V-square Goodman and Kruskal’s gamma Goodman and Kruskal’s lambda Log odds ratio Pearson’s pmc R-square Somer’s d Spearman’s rank correlation Zegers and ten berge’s identity 


  1. Agresti, A. (1984). Analysis of ordinal categorical data. New York, NY: Wiley.Google Scholar
  2. Agresti, A. (2002). Categorical data analysis (2nd ed.). Hoboken, NJ: Wiley.Google Scholar
  3. Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20, 37–46.Google Scholar
  4. Cohen, J. (1968). Weighted kappa: Nominal scale agreement with provision for scaled disagreement or partial credit. Psychological Bulletin, 70, 213–220.PubMedGoogle Scholar
  5. Everitt, B. S. (1977). The analysis of contingency tables. London, UK: Chapman and Hall.Google Scholar
  6. Gibbons, J. D. (1971). Nonparametric statistical inference. New York, NY: McGraw-Hill.Google Scholar
  7. Hays, W. L. (1973). Statistics for the social sciences (2nd ed.). London, UK: Holt, Rinehart and Winston.Google Scholar
  8. Warrens, M. J. (2012). Some paradoxical results for the quadratically weighted kappa. Psychometrika, 77, 315–323.Google Scholar
  9. Wickens, T. D. (1989). Multiway contingency tables analysis for the social sciences. Hillsdale, NJ: Erlbaum.Google Scholar
  10. Zegers, F. E., & ten Berge, J. M. F. (1985). A family of association coefficients for metric scales. Psychometrika, 50, 14–17.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Emeritus Professor Psychological Methods, Department of PsychologyUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations