Solution. (a) \(w = (0.3,\ 0.3,\ 0.4)\): \(\text {DSR}_A = 0.15 + 0.9 + 8.0 = 9.05\) and \(\text {DSR}_B = 0.12 + 0.75 + 6.4 = 7.27\), region B’s crude rate, as it must be when B’s own structure is the standard. \(\text {CMF} = 0.803\). (b) \(\text {DSR}_A = 0.3 + 0.9 + 2.0 = 3.2\), \(\text {DSR}_B = 0.24 + 0.75 + 1.6 = 2.59\), \(\text {CMF} = 0.809\). The standardised rates themselves change greatly with the standard (from \(3.2\) to \(9.05\) for A), so they have no meaning except relative to one another; the CMF hardly changes, because the stratum ratios are nearly constant.

Solution. \(\Var (\log \text {CMF}) \approx (0.368/6.125)^2 + (0.188/4.93)^2 = 0.00362 + 0.00145 = 0.00507\), so \(\text {SE} = 0.0712\) and the interval is \(0.805\,e^{\pm 0.140} = (0.700,\ 0.925)\). Region B’s age-adjusted mortality is between \(7\%\) and \(30\%\) below A’s.

Solution. \(\text {SMR} = 18/9.6 = 1.88\). The limits are \(\chi ^2_{36,\,0.025}/19.2 = 1.11\) and \(\chi ^2_{38,\,0.975}/19.2 = 2.96\). Under \(\theta = 1\), \(O \sim \text {Poisson}(9.6)\), and \(P(O \geq 18) = 0.0098\), so the two-sided \(P\) is about \(0.020\). The comparison is with the general population, which includes people too ill to work; the healthy worker effect makes the expected count too high for an employed group and biases the SMR downwards. The effect is weaker for cancers than for all-cause mortality, but present.

Solution. \(O \sim \text {Poisson}(\theta E)\) by Theorem 8.2.2, so \(E(\text {SMR}) = \theta E/E = \theta \) and \(\Var (\text {SMR}) = \theta E/E^2 = \theta /E\). The coefficient of variation is \(1/\sqrt {\theta E}\), which depends on the expected deaths only. Doubling the size of a cohort, or its follow-up, doubles \(E\) and reduces the coefficient of variation by a factor of \(\sqrt 2\).

Solution. The combined person-time is \(10{,}000\) in each stratum, so \(w = (0.5,\ 0.5)\). Then \(\text {DSR}_A = 0.5 \times 2 + 0.5 \times 10 = 6\) and \(\text {DSR}_B = 0.5 \times 2.2 + 0.5 \times 11 = 6.6\), giving \(\text {CMF} = 1.1\). Indeed, since \(I_{Bj} = 1.1\,I_{Aj}\) in both strata, Proposition 8.1.2 gives \(\text {CMF} = 1.1\) for any standard, while the ratio of SMRs is \(1.11/1.47 = 0.75\).

Solution. With \(\lambda ^{\text {s}}_j = I_j\), \(E = \sum _j T_j I_j = \sum _j D_j = O\). With \(w_j = T_j/\sum _l T_l\), \(\text {DSR} = \sum _j (T_j/\sum _l T_l)(D_j/T_j)\) is the crude rate. A standard is a reference, and comparing a population with itself gives the trivial answer. The standard should be external to the comparison or, for direct standardisation, a population whose structure is in the range of those compared.