8.3 Comparing SMRs
Proposition 8.3.1. The SMR is a weighted average of the stratum-specific ratios \(I_j/\lambda ^{\text {s}}_j\), with weights \(E_j/E\) proportional to the study population’s own expected deaths in each stratum.
Proof. \(O/E = \sum _j D_j/E = \sum _j (E_j/E)(D_j/E_j)\), and \(D_j/E_j = I_j T_j/(\lambda ^{\text {s}}_j T_j) = I_j/\lambda ^{\text {s}}_j\). □
So each SMR uses its own population’s weights. Two SMRs computed against the same standard are not standardised to a common structure, and their ratio can mislead when the stratum ratios vary. This is the price of indirect standardisation: an SMR compares one population with the standard, not two populations with each other.
Example 8.3.2. With standard rates \(1\) and \(10\) per \(1000\) in two age groups, population A has rates \(2\) and \(10\) and person-time \(9000\) and \(1000\) years; population B has rates \(2.2\) and \(11\) (\(10\%\) higher than A’s in both groups) and person-time \(1000\) and \(9000\). Then \begin {align*} \text {A}: \quad O &= 18 + 10 = 28, & E &= 9 + 10 = 19, & \text {SMR} &= 1.47 ; \\ \text {B}: \quad O &= 2.2 + 99 = 101.2, & E &= 1 + 90 = 91, & \text {SMR} &= 1.11 . \end {align*}
Population B has the higher rate in every age group, but the lower SMR. A’s SMR is dominated by the young stratum, where its rate is twice the standard; B’s by the old stratum, where its rate is only \(10\%\) above it. Direct standardisation with any standard gives a CMF of \(1.1\) for B against A.
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