8.1 Direct standardisation
Definition 8.1.1. Given a standard population with weights \(w_j \geq 0\), \(\sum _j w_j = 1\) (its proportions of person-time in the strata), the directly standardised rate of population \(k\) is \[ \text {DSR}_k = \sum _j w_j I_{kj} , \] which is the crude rate the standard population would have if it experienced the stratum-specific rates of population \(k\). The ratio of two directly standardised rates is the comparative mortality figure (CMF), \(\text {DSR}_1/\text {DSR}_0\).
- 1.
- If the stratum rate ratio \(I_{1j}/I_{0j} = \theta \) is the same in every stratum, then \(\text {CMF} = \theta \) whatever the standard.
- 2.
- In general the CMF is a weighted average of the stratum rate ratios, with weights \(w_j I_{0j}/\sum _l w_l I_{0l}\).
- 3.
- If the counts \(D_{kj}\) are Poisson, \(\Var (\text {DSR}_k) = \sum _j w_j^2 I_{kj}/T_{kj}\), estimated by \(\sum _j w_j^2 D_{kj}/T_{kj}^2\).
Proof. (1) \(\text {DSR}_1 = \sum _j w_j\theta I_{0j} = \theta \,\text {DSR}_0\). (2) \(\text {DSR}_1/\text {DSR}_0 = \sum _j w_j I_{0j}(I_{1j}/I_{0j})/\sum _l w_l I_{0l}\). (3) \(I_{kj} = D_{kj}/T_{kj}\) with \(D_{kj}\) Poisson with mean \(\lambda _{kj}T_{kj}\) has variance \(\lambda _{kj}/T_{kj}\), and the strata are independent. □
The choice of standard matters little when the rate ratios are nearly constant across strata, and matters a great deal when they are not; then the CMF is an average of quite different numbers and the stratum-specific ratios should be reported instead of, or as well as, the summary. Common standards are the combined population of the groups compared, a national population, or a conventional “world” or “European” standard population that allows comparison across published studies.
An approximate interval for \(\text {DSR}_k\) is \(\text {DSR}_k \pm 1.96\,\text {SE}\); when the deaths are few, intervals based on the gamma distribution are better. For the CMF, the delta method gives \(\Var (\log \text {CMF}) \approx \Var (\text {DSR}_1)/\text {DSR}_1^2 + \Var (\text {DSR}_0)/\text {DSR}_0^2\).
Example 8.1.3. Take as standard the two regions combined: \(90{,}000\), \(60{,}000\) and \(50{,}000\) person-years, so \(w = (0.45,\ 0.30,\ 0.25)\). Per \(1000\) person-years, \begin {align*} \text {DSR}_A &= 0.45 \times 0.5 + 0.30 \times 3.0 + 0.25 \times 20 = 6.125 , \\ \text {DSR}_B &= 0.45 \times 0.4 + 0.30 \times 2.5 + 0.25 \times 16 = 4.93 . \end {align*}
The CMF of B relative to A is \(4.93/6.125 = 0.805\): after allowing for age, region B’s mortality is about \(20\%\) lower, in line with the stratum ratios \(0.80\), \(0.83\), \(0.80\). The standard errors are \[ \text {SE}(\text {DSR}_A) = \sqrt {0.45^2\frac {30}{60^2} + 0.30^2\frac {90}{30^2} + 0.25^2\frac {200}{10^2}} = 0.368 \] per \(1000\) (with person-time in thousands), and similarly \(\text {SE}(\text {DSR}_B) = 0.188\). The intervals are \((5.40,\ 6.85)\) and \((4.56,\ 5.30)\). Most of the variance of \(\text {DSR}_A\) comes from its oldest stratum, which carries a quarter of the weight but only a tenth of A’s person-time.
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