Algebraic Relationships between Relative Risk, Phi and Measures of Necessity and Sufficiency in a 2×2 Table

Epidemiologists have long regarded relative risk (RR) as a key measure of association between two binary variables. Yet even when the sample is representative of the population, associations having a RR > 9 may have a relatively small value for Phi (φ < .3). This paper provide additional reasons for using RR instead of φ. (1) Formulas relating φ and RR are derived. A relative φ 2 is constructed and is shown to equal the attributable fraction in the population (AFP). RR is shown to increase monotonically with this relative φ 2 for a given exposure factor prevalence (H). (2) Constructs are created to measure degrees of necessity (N) and sufficiency (S). Related formulas are derived. RR is shown to increase monotonically with N for a given exposure factor prevalence (H). (3) Coordinates that will always generate admissible results are discussed. (4) RR is a good measure of association between two binary variables because it increases monotonically with AFP, with a relative φ 2 and with N for a given exposure factor prevalence (H). In Appendix B, auxiliary identities are presented including a Bayes’ rule comparison, the over-involvement rule and the non-response bias effect size.