Indirect standardization and multiplicative models for rates, with reference to the age adjustment of cancer incidence and relative frequency data.

THE OBJECT of standardization procedures is to summarize mortality or morbidity rates for the individual age groups or other strata of a population into a single index which indicates its position relative to other populations while accounting for differences in age structures. When both numerator and denominator data are available for all strata in each population, and when the aim is primarily descriptive, direct standardization [l] is the usual procedure. By combining age-specific rates with weights proportional to an external standard age distribution this technique produces a synoptic figure, the directly standardized rate, suitable for official publications. Comparisons of populations may be based on standardized rates compiled from several such sources, provided the same standard distributions have been used. Situations often arise, however, in which either the age-specific data required for direct standardization are not available or, if available, the numbers of cases in certain age groups are so small that the rates are unstable and statistical significance is at issue. For the comparison of a specific set of populations under such conditions, Part I of this paper proposes the fitting of a multiplicative model to the two dimensional table of age-specific rates. This approach leads to versatile tests for the equality of population rates. When there are differences among the populations, efficient maximum likelihood estimates of those differences are obtained, provided the model holds. (This qualification is discussed below.) However the approach does not produce the sort of synoptic figure required for comparisons with other populations not included in the analysis. The methodology proposed here may be considered as a refinement and extension of the method of ‘indirect’ standardization which produces the standardized mortality ratio (SMR) [l].