A System for Risk Equalities for n Binary Factors

The interaction of two-level factors in the case of a two-level response, used in epidemiology, toxicology, evidence-based medicine and other medical disciplines, is studied from viewpoint of verifying the condition of equality of risk differences. The paper proposes an extension of the risk differences equality to the case of n binary factors. In the general case, this condition is presented in the form of a system of linear equations regarding risks. For two binary factors, this system is represented by the equality to zero of the Koopman’s ICDR coefficient. For three variables, the equations were obtained by applying the expectation operator to the zero-interaction equalities from ANOVA. It is proved that the risk differences equality system for n binary factors is stable under action of the experiment symmetry group, which is isomorphic to the group of all automorphisms of Boolean cube as a Hamming graph. For the risk differences equality system, an equivalent system of linear equations is obtained, from which the solution space of the risk differences equality system can be found. This equivalent system contains $O( 2 ^{n})$ equations, while the original risk differences equality system consists of $O( 3 ^{n})$ equations. In addition, the equivalent system also has a clear interpretation and is suitable for efficient statistical analysis of binary experiment data.