Rank–Polyserial Correlation: A Quest for a “Missing” Coefficient of Correlation

Abstract

In the typology of coefficients of correlation, we seem to miss such estimators of correlation as rank–polyserial (RRPS) and rank–polychoric (RRPC) coefficients of correlation. This article discusses a set of options as RRP, including both RRPS and RRPC. A new coefficient JTgX based on Jonckheere–Terpstra test statistic is derived, and it is shown to carry the essence of RRP. Such traditional estimators of correlation as Goodman–Kruskal gamma (G) and Somers delta (D) and dimension-corrected gamma (G2) and delta (D2) are shown to have a strict connection to JTgX, and, hence, they also fulfil the criteria for being relevant options to be taken as RRP. These estimators with a directional nature suit ordinal-scaled variables as well as an ordinal- vs. interval-scaled variable. The behaviour of the estimators of RRP is studied within the measurement modelling settings by using the point-polyserial, coefficient eta, polyserial correlation, and polychoric correlation coefficients as benchmarks. The statistical properties, differences, and limitations of the coefficients are discussed.