In her work on the components of a difference between two rates, Kitagawa (1955) was successful in dividing the difference into the rate effect and the effect of the factor, for data classified by one factor. Her formulation for data classified by two factors, however, involves an interaction term which is difficult to interpret. Retherford and Cho (1973) devised a method that does not include any interaction terms. However, their method has other limitations, such as the dependence of the results on the order in which the effects of the factors are computed. In this paper, we provide a general method capable of handling any number of factors, which is developed along the lines suggested by Kitagawa and by Retherford and Cho but without the limitations of their methods.
Demographers are often interested in comparing rates (e.g., birth rates, mortality rates) in populations cross‐nationally and/or over time. Interpreting difference between rates requires an understanding of the various factors that comprise that rate. Crude birth rates, for example, depend not just on the fertility of women of childbearing age, but also the proportion of the population that consists of such women. In such cases, in which the overall rate of a phenomenon for a population depends on a number of factors, a detailed comparison of two such rates from two different populations can be made in two distinctly different but closely related ways. One way is to see how the overall rates would change if one of the factors varied as it did in the two populations, while the other factors were kept at the same levels. The rates obtained in this way are called the standardized rates with respect to the unchanged factors, and the process is called
standardization
. The other way of comparing the two overall rates is to break the difference between these two rates into additive components constituting the effects of the factors involved. The effects of the factors obtained in this way are called the decomposed effects and the process is called
decomposition
. These two processes of standardization and decomposition are closely linked because, if they are developed correctly, the difference between the two standardized rates from the two populations corresponding to the only factor that has changed should be equal to the effect of the same factor in the decomposition process. Authors who have contributed to this subject include Kitagawa (1955), Cho and Retherford (1973), Das Gupta (1978, 1991, 1993), and Kim and Strobino (1984)
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