A necessary and sufficient condition of comparability for using standardized mortality ratio (SMR)

A necessary and sufficient condition of comparability for using SMR was studied mathematically by considering the equivalence between SMR and CMF, as CMF was a perfectly comparable index. This condition was expressed by either proportionality of mortality vectors or proportionality of projected person-years to the plane spanned by mortality vectors of reference and index groups. We could obtain another expression of the condition, in which affect of distortions were easily understood, which consist of three factors: distortion of proportionality of mortality, distortion of person-years and similarity of distortions. Our results were applied to study the mortality of biliary tract cancer in Ibaraki Prefecture. Places where absolute difference between CMF and SMR exceeds some criterion (say, 0.15) were Satomi, Ushiboiri, Nihari in males and Gozenyama, Suifu and Asahi in females. All three distortion indices exceeded their upper 95% percentiles in Satomi in males.