Title The Phi - coefficient , the Tetrachoric Correlation Coefficient , and the Pearson - Yule Debate

Two measures of association for dichotomous variables, the phi-coefficient and the tetrachoric correlation coefficient, are reviewed and differences between the two are discussed in the context of the famous so-called Pearson-Yule debate, that took place in the early 20th century. The two measures of association are given mathematically rigorous definitions, their underlying assumptions are formalized, and some key properties are derived. Furthermore, existence of a continuous bijection between the phi-coefficient and the tetrachoric correlation coefficient under given marginal probabilities is shown. As a consequence, the tetrachoric correlation coefficient can be computed using the assumptions of the phi-coefficient construction, and the phi-coefficient can be computed using the assumptions of the tetrachoric correlation construction. The efforts lead to an attempt to reconcile the Pearson-Yule debate, showing that the two measures of association are in fact more similar than different and that between the two, the choice of measure of association does not carry a substantial impact on the conclusions of the association analysis.