10.6 Rate ratios
With \(D_{1k}\) cases in \(T_{1k}\) person-years among the exposed and \(D_{0k}\) in \(T_{0k}\) among the unexposed in stratum \(k\), \(D_k = D_{1k} + D_{0k}\) and \(T_k = T_{1k} + T_{0k}\), \[ \widehat {\text {IRR}}_{\text {MH}} = \frac {\sum _k D_{1k}T_{0k}/T_k}{\sum _k D_{0k}T_{1k}/T_k} , \] with \[ \widehat {\Var }(\log \widehat {\text {IRR}}_{\text {MH}}) = \frac {\sum _k D_k T_{1k}T_{0k}/T_k^2}{\bigl (\sum _k D_{1k}T_{0k}/T_k\bigr )\bigl (\sum _k D_{0k}T_{1k}/T_k\bigr )} . \] The test of no association uses Theorem 4.5.1: given \(D_k\), \(D_{1k}\) is binomial with probability \(T_{1k}/T_k\) under the null hypothesis, so \[ X^2 = \frac {\bigl (\sum _k D_{1k} - \sum _k D_k T_{1k}/T_k\bigr )^2}{\sum _k D_k T_{1k}T_{0k}/T_k^2} . \]
Example 10.6.1. In the British doctors’ study (Doll and Hill, 1966; data as in Breslow and Day, 1987) deaths from coronary heart disease among smokers and non-smokers were:
| Smokers | Non-smokers | ||||
| Age | Deaths | Person-yrs | Deaths | Person-yrs | IRR |
| 35–44 | 32 | 52,407 | 2 | 18,790 | 5.74 |
| 45–54 | 104 | 43,248 | 12 | 10,673 | 2.14 |
| 55–64 | 206 | 28,612 | 28 | 5710 | 1.47 |
| 65–74 | 186 | 12,663 | 28 | 2585 | 1.36 |
| 75–84 | 102 | 5317 | 31 | 1462 | 0.90 |
The crude rate ratio is \(1.72\) and the Mantel–Haenszel rate ratio \(1.42\), with \(\text {SE}(\log ) = 0.107\) and interval \((1.15,\ 1.76)\). The test gives \(630\) observed deaths among smokers against \(595.2\) expected, variance \(110.1\), \(X^2 = 11.0\). The crude and adjusted ratios differ, but the more striking feature is that the stratum ratios fall steeply with age, from \(5.7\) to \(0.9\). This is effect modification on the ratio scale, and here the summary \(1.42\) is a poor description: the relative effect of smoking on heart disease is large in middle age and vanishes in old age, while the rate difference (Problem 10.8.4) behaves quite differently.
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