Conventional methods of decomposing the difference between two rates, such as Kitagawa's classic component analysis, are confined to taking the average of compositional differences. I propose a more general modeling approach involving three steps: (1) A system of equations with the various additive components of the rate difference is set up; (2) unknowns (refined rate differences) are estimated with Clogg's purging method; (3) the components are calculated. I use an example of U.S. mortality data to compare the proposed method with the conventional ones. The method can be generalized to decompositions for multiple groups and for multiple confounding factors. Kitagawa's method is a special case of this general approach.