8.2 Indirect standardisation and the SMR
Direct standardisation needs stratum-specific rates for each population compared. When a population is small, as an occupational cohort usually is, those rates are unstable or, with few deaths, cannot be computed at all. Indirect standardisation reverses the roles: it applies standard rates to the study population’s own structure.
Definition 8.2.1. Let \(\lambda _j^{\text {s}}\) be standard stratum-specific rates. The expected number of deaths in the study population is \(E = \sum _j T_j \lambda ^{\text {s}}_j\), and with \(O = \sum _j D_j\) observed deaths the standardised mortality ratio is \[ \text {SMR} = \frac {O}{E} . \]
The SMR compares the deaths that occurred with those that would have occurred had the study population experienced the standard rates. Only the total number of deaths in the study population is needed, not their distribution across strata.
Theorem 8.2.2. Suppose \(D_j \sim \text {Poisson}(\theta \lambda ^{\text {s}}_j T_j)\) independently, so that the study rates are a constant multiple \(\theta \) of the standard rates, which are treated as known. Then \(O \sim \text {Poisson}(\theta E)\), the maximum likelihood estimate of \(\theta \) is the SMR, and exact confidence limits are \[ \frac {\chi ^2_{2O,\,0.025}}{2E} \qquad \text {and}\qquad \frac {\chi ^2_{2O + 2,\,0.975}}{2E} . \]
Proof. A sum of independent Poisson variables is Poisson with the sum of the means, \(\theta \sum _j\lambda ^{\text {s}}_j T_j = \theta E\). The likelihood is \(\prod _j(\theta E_j)^{D_j}e^{-\theta E_j}/D_j! \propto \theta ^O e^{-\theta E}\), with \(E_j = \lambda ^{\text {s}}_j T_j\), maximised at \(\theta = O/E\). The limits are those of (3.2) with person-time replaced by \(E\), by Lemma 3.3.6. □
Treating the standard rates as known is reasonable when they come from a national population, whose rates are estimated from vastly more deaths than the study’s. The test of \(\theta = 1\) uses the Poisson distribution with mean \(E\) directly; for large \(E\), \((O - E)/\sqrt {E}\) is approximately standard normal.
Example 8.2.3. Take as standard the national rates \(0.45\), \(2.8\) and \(18\) per \(1000\) person-years in the three age groups. For region A, \[ E_A = 60 \times 0.45 + 30 \times 2.8 + 10 \times 18 = 27 + 84 + 180 = 291 , \] against \(O_A = 320\), so \(\text {SMR}_A = 1.10\), with exact interval \((0.98,\ 1.23)\) and two-sided exact \(P = 0.098\). For region B, \(E_B = 13.5 + 84 + 720 = 817.5\) and \(O_B = 727\), so \(\text {SMR}_B = 0.89\), with interval \((0.83,\ 0.96)\). Region A’s mortality is about \(10\%\) above the national level, region B’s about \(11\%\) below. The ratio of SMRs, \(0.89/1.10 = 0.81\), happens to agree with the CMF here, because the regions’ rates are nearly proportional to the national ones; in general it need not (Section 8.3).
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