A new correlation coefficient between categorical, ordinal and interval variables with Pearson characteristics

Abstract

A prescription is presented for a new and practical correlation coefficient, ϕK, based on several refinements to Pearson’s hypothesis test of independence of two variables. The combined features of ϕK form an advantage over existing coefficients. Primarily, it works consistently between categorical, ordinal and interval variables, in essence by treating each variable as categorical, and can therefore be used to calculate correlations between variables of mixed type. Second, it captures nonlinear dependency. The strength of ϕK is similar to Pearson’s correlation coefficient, and is equivalent in case of a bivariate normal input distribution. These are useful properties when studying the correlations between variables with mixed types, where some are categorical. Two more innovations are presented: to the proper evaluation of statistical significance of correlations, and to the interpretation of variable relationships in a contingency table, in particular in case of sparse or low statistics samples and significant dependencies. Two practical applications are discussed. The presented algorithms are easy to use and available through a public Python library.11https://github.com/KaveIO/PhiK.