A Note on the Relationship Between the Phi Coefficient and the Tetrachoric Correlation Under Nonnormal Underlying Distributions

The connection between the phi coefficient and the tetrachoric correlation is well-understood when the underlying distribution is bivariate normal. For many other bivariate distributions, the identity that links these two quantities together is not straightforward to formulate. Furthermore, even when this can be done, solving the equation in either direction may be far from trivial. We propose a simple technique that enables students and researchers to compute one of these correlations when the other is specified. Generalizing the normal-based results to a broad range of bivariate distributional setups is potentially useful in graduate-level teaching as well as in simulation studies that involve dichotomization and random number generation where the relationships between these correlation types need to be modeled