A Necessary and Sufficient Condition of Comparability for Using Standardizes Mortality Ratio ( SMR )

A necessary and sutticient condition of comparability for using AMFS 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. J Epidemiol, 2001 ; 11 : 24-28


INTRODUCTION
One of the most frequent problems in descriptive epidemiological s tudies is the comparison with some measure (incidence, mortality or others) among community groups, to find high incident areas and to search for causation.If confounders like age, sex or race exist, then measures like crude rates often mislead due to the distortion of comparability.Various standardized indices have been proposed to adjust for these confounders.
Well-known indices are stratum-adjusted rates, which are weighted average with stratum-specific person-years by confounder(s).These rates are classified as a direct adjusted rate and an indirect one, or their ratio expressions, a comparative mortality figure (CMF) and a standardized mortality ratio (SMR)1).
In this paper, CMF and SMR are studied mathematically in the context of dealing with diseases like biliary tract cancer, where case facility rates are very high.In this case mortality is also meaningful in an epidemioglogical sense for causation when local cancer registration is not sufficiently applicable due to imprecise registry, instability and low registry proportion.
Minimum variance properties and interval estimation have been studied for these indices 2).We fmd in the literature that CMF is strictly comparable but unstable with small personyears, while SMR is not satisfied for its comparability 3.4) because of no common reference.However SMR is stable for using no stratum-adjusted estimators.
A sufficient condition of equivalence of CMF and SMR has been studied 1. 5, 6,7) as proportionality of every stratum-specific mortality between two groups.However, the necessary and sufficient condition has not been studied and therefore the effect of distortion of proportionality is not at all clear.
This study focused on (1) a necessary and sufficient condition of equivalence, and (2) the effect of distortion of proportionality.The comparability of CMF and SMR were further H.Takahashi,et al. 25 studied by a real life application.We analyzed the data on mortality due to biliary tract cancer in Ibaraki Prefecture in Japan.

DEFINITION
We denote a reference group So, and its relevant values characterized as *=0, and an index group Si and i=1.The group S; consists of J strata and we set j-th stratum-specific mortality as p;; and its person-years as nij (i=0,1).Then CMF and SMR are defined as,If we estimate every stratum-specific rate p;; as dij/nij by maxi-mum likelihood estimation with stratum-specific death number dij be where Do, D1 are total corresponding number of deaths.
These two indices are quite similar except that CMF is adjusted for reference person-years, while SMR is adjusted for index person-years.Additionally, they can be recognized weighted means of stratum-specific mortalities with weights being each person-years, which are also expressed by inner products in the J-dimensional Euclidean space: Now we can obtain the next necessary and sufficient condition of comparability based the equivalence of SMR and CMF using above expression.
[A necessary and sufficient condition of comparability for using SMR]A necessary and sufficient condition of CMF=SMR is n1(0)=kn0(0) or pi= lpo, (with k, l being constants).[Proof) We can also obt

EFFECT OF DISTORTION OF PROPORTIONALITY
In the previous section, we could obtain a necessary and sufficient condition of equivalence of SMR and CMF, as proportionality of projected person-years or proportionality of mortalities.In this section, we describe the effect of distortion of proportionality with no, n1 instead of n0(0),n1(0) From the expression (1), we can consider that * = (pl -(no, pi) / (no, po) po) would express distortion of proportionality of mortalities and r = (n1-(n1, po) / (no, po) noj) would express distortion of proportionality of person-years.Then effects of distortion of proportionality can be expressed by ( 2), where 11111 is a level of distortion of mortalities, and II r II is that of personyears and lcos*1 would be a level of similarity of both distortions.

A REAL LIFE APPLICATION
Using the data set available on average mortality of biliary tract cancer (ICD9= 156.0,156.1, 156.8, 156.9) in Ibaraki Prefecture (1992-1997), the comparability of SMR and CMF were analyzed in Table 1 in males and Table 2 in females.Cities and villages where absolute differences (i.e.SMR-CMF) exceeded 0.15 are Satomi (-0.878),Ushibori (0.196) and Niihari (-0.231) in males and Gozenyama (-0.178),Suifu (-0.164) and Asahi (-0.234) in females.If we set the criterion of difference as upper 5% percentiles of II 7 II, II r II and Icos* I, then they become 54.7, 0.233 and 0.116 for males and 30.5, 0.222 and 0.143 for females, respectively.
From this, we found in males that Satomi was caused by all three values II r/ II, II r I, lcos* I Ushibori was caused by mainly