Relative improvement over chance (RIOC) and phi as measures of predictive efficiency and strength of association in 2×2 tables

In a 2×2 table, phi measures the strength of the relationship and relative improvement over chance (RIOC) measures the predictive efficiency. A simplified formula for RIOC is derived. From the linkage between RIOC and the adjusted standardized residual, the variance of RIOC is calculated, thereby allowing its statistical significance to be tested. A disadvantage of phi is that its maximum value can be less than 1. A corrected phi correlation, obtained by dividing phi by the maximum possible phi, is shown to be mathematically identical to RIOC. Hence, there is a linkage between the percentage of correct predictions and the percentage of variance explained. RIOC is unchanged even when the social costs and benefits of the different outcomes are differentially weighted. RIOC is also mathematically identical to a corrected kappa calculated in a similar fashion to the corrected phi. Therefore, RIQC has distinct advantages over phi and kappa.