Decomposition of the difference between two rates and its consistency when more than two populations are involved

In recent years, many social scientists, in dealing with data of various kinds, have attempted to decompose the difference between two rates into effects of changes in the underlying factors. AU these studies can be divided into two categories, viz., the rate as a function of two or more factors (their product being a special case), and the rate as computed from cross‐classified data involving one or more factors. In the absence of general methods of decomposition to be used under these two circumstances, the researchers have devised their own ad hoc methods which in general produce meaningful results, but are sometimes less than satisfactory in terms of mathematical rigor and elegance. The objective of the present paper is to put together the general formulas that a researcher is expected to need for the two categories of decomposition problems for any number of factors. Applications of the use of the formulas under these two situations are illustrated by four examples. The paper also suggests a solution to the problem of possible internal inconsistencies of the factor effects when more than two populations are involved, and provides an illustrative example.