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Scottish Indices of Deprivation 2003

Appendix 3: Constructing the Comparative Mortality Factor

Indirect standardisation has frequently been used to study area differences in mortality rates. However there are problems with its use in this context. These are outlined below and a case is made for using direct standardisation instead, to produce a Comparative Mortality Factor.

If m _ij is the age/sex specific mortality rate in study area j, then the SMR for that area is:

d _j = deaths in study area j.

m _ij = the death rate in age/ sex group i in the study area j.

m _i = the death rate in age/ sex group i in the standard population.

p _ij= the population in age/ sex group i in the study area j.

From the third term in equation [1] it is clear that variation in the SMR results from both the difference in the demographic composition of the ward as well as in the age/sex specific death rates. The population structure weights the proportional difference between the expected age/sex specific death rate and the observed, so that the SMR can range theoretically from the highest proportional difference recorded in any one age/sex group, to the lowest. Unless the proportional difference between the standard population age/sex specific death rate and that of a study area is constant across age groups, or all wards have an identical demographic composition, then the SMR will not measure the relative health status of an area. As neither of these conditions are likely to be true of all wards in Scotland, the SMR cannot measure relative health status.

To visually illustrate the 'weighting' impact of the population structure on the SMR, a simple but slightly extreme situation is taken where the population is divided into two age groups (the young and old) and ward [A] has an age specific death rate of 10% amongst the young, and 25% amongst the old, compared to ward [B] which has rates of 9% and 24%, respectively. The first ward is clearly relatively less healthy than the second. If the SMRs are calculated for these two areas (the death rate in the standard population is 5% and 22%) and the proportion of the population that is 'young' is varied from 0 to 1 (i.e. 1 indicating all the population are young), two SMR lines can be plotted ( Figure A8.1). In the case where ward [A] has the same ratio of young to old as ward [B], ward [A] correctly has a higher SMR than [B]. However, this is not necessarily true if the population ratios are different. For example if ward [B] is made up of 80% young people and ward [A] has any proportion less than 70%, then ward [B] will have a higher SMR than [A]. Similarly the age specific death rate is constant for the two wards, and yet their SMRs vary considerably as the population structure is altered.

Figure A6.1

The Comparative Mortality Factor, equation [2], is not influenced by variation in the population structure between wards.

pi = the population in age/ sex group i in the standard population.

D= total deaths in the standard population.

Unlike the SMR, the CMF does not down-weight the proportional difference between the age/sex specific observed and expected rates if they relate to a small proportion of the study population. The CMF weights the proportional difference by a standard population structure ( p _i) that is constant across study areas. This does make the CMF in some senses more vulnerable to small number error in m _ij than the SMR. If for example an age/sex rate is based on a relatively small age/sex population (compared to other groups in the study population), then that rate will be down-weighted in the SMR. However the SMR will still be vulnerable to small number error if the overall ward population is small, so this alone does not seem sufficient justification for using the SMR over the CMF. Instead it would seem sensible to deal with the small number problem separately and then to use the CMF. This can be achieved by applying the 'shrinkage' method to the crude age/sex rates. The calculation of the 'shrunk' crude age/sex rates are shown below, and this results in mij being replaced by mij*, its 'shrunk' estimate. The CMF is therefore:

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