AUTHOR=Metsämuuronen Jari 

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

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 8 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2022.914932

DOI=10.3389/fams.2022.914932

ISSN=2297-4687

ABSTRACT=<p>In the typology of coefficients of correlation, we seem to miss such estimators of correlation as rank–polyserial (<italic>R</italic><sub><italic>RPS</italic></sub>) and rank–polychoric (<italic>R</italic><sub><italic>RPC</italic></sub>) coefficients of correlation. This article discusses a set of options as <italic>R</italic><sub><sub><italic>RP</italic></sub></sub>, including both <italic>R</italic><sub><italic>RPS</italic></sub> and <italic>R</italic><sub><italic>RPC</italic></sub>. A new coefficient <italic>JT</italic><sub><italic>gX</italic></sub> based on Jonckheere–Terpstra test statistic is derived, and it is shown to carry the essence of <italic>R</italic><sub><italic>RP</italic></sub>. Such traditional estimators of correlation as Goodman–Kruskal gamma (<italic>G</italic>) and Somers delta (<italic>D</italic>) and dimension-corrected gamma (<italic>G</italic><sub>2</sub>) and delta (<italic>D</italic><sub>2</sub>) are shown to have a strict connection to <italic>JT</italic><sub><sub><italic>gX</italic></sub></sub>, and, hence, they also fulfil the criteria for being relevant options to be taken as <italic>R</italic><sub><italic>RP</italic></sub>. 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 <italic>R</italic><sub><italic>RP</italic></sub> is studied within the measurement modelling settings by using the point-polyserial, coefficient <italic>eta</italic>, polyserial correlation, and polychoric correlation coefficients as benchmarks. The statistical properties, differences, and limitations of the coefficients are discussed.</p>