Probability of bivariate superiority: A non-parametric common-language statistic for detecting bivariate relationships

Researchers often focus on bivariate normal correlation (r) to evaluate bivariate relationships. However, these techniques assume linearity and depend on parametric assumptions. We propose a new nonparametric statistical model that can be more intuitively understood than the conventional r: probability of bivariate superiority (PBS). Our development of Bp, the estimator of a PBS relationship, extends Dunlap’s (1994) common-language transformation of r (CLr) by providing a method to directly estimate PBS—the probability that when x is above (or below) the mean of all X, its paired y score will also be above (or below) the mean of all Y. Probability of superiority is an important form of bivariate relationship that until now could only be accurately estimated when data met the parametric assumptions for r. We specify the copula that forms the theoretical basis for PBS, provide an algorithm for estimating PBS from a sample, and describe the results of a Monte Carlo experiment that evaluated our algorithm across 448 data conditions. The PBS estimate, Bp, is robust to violations of parametric assumptions and offers a useful method for evaluating the significance of probability-of-superiority relationships in bivariate data. It is critical to note that Bp estimates a different form of bivariate relationship than does r. Our working examples show that a PBS effect can be significant in the absence of a significant correlation, and vice versa. In addition to utilizing the PBS model in future research, we suggest that this new statistical procedure be used to find theoretically important but previously overlooked effects from past studies.