13.2 A homogeneous example
Example 13.2.1. A cohort of workers is followed for cancer mortality. The exposed are older than the unexposed.
| Exposed | Unexposed | ||||
| Age | Deaths | Person-yrs | Deaths | Person-yrs | \(\theta _k\) |
| 20–34 | 6 | 12,000 | 8 | 30,000 | 1.88 |
| 35–49 | 27 | 15,000 | 18 | 20,000 | 2.00 |
| 50–64 | 60 | 10,000 | 26 | 8000 | 1.85 |
| 65–79 | 48 | 3000 | 15 | 2000 | 2.13 |
| Total | 141 | 40,000 | 67 | 60,000 | 3.16 |
The stratum rate ratios are all close to \(2\). The methods give:
| Method | Rate ratio | \(95\%\) interval |
| Crude | 3.16 | (2.36, 4.22) |
| Direct, combined standard | 1.96 | (1.46, 2.64) |
| Direct, exposed as standard | 1.97 | (1.45, 2.66) |
| Direct, unexposed as standard | 1.96 | (1.45, 2.64) |
| SMR, unexposed rates as standard | 1.97 | (1.66, 2.32) |
| Mantel–Haenszel | 1.96 | (1.46, 2.64) |
| Inverse-variance | 1.96 | (1.46, 2.63) |
| Poisson maximum likelihood | 1.96 | (1.46, 2.63) |
Every adjusted estimate is within \(1\%\) of \(1.96\), against a crude \(3.16\); the confounding by age is large, and every method removes it. The Poisson model’s residual deviance is \(0.16\) on \(3\) degrees of freedom, confirming homogeneity. The weights of Proposition 13.1.1 on the four strata are similar for all methods (for example \(0.07\), \(0.23\), \(0.43\), \(0.27\) for Mantel–Haenszel and \(0.05\), \(0.19\), \(0.45\), \(0.31\) for the SMR), and with nearly equal \(\theta _k\) the weights hardly matter.
The intervals are also nearly the same, with one exception. The SMR interval is narrower because it treats the unexposed rates as known constants, when in fact they are estimated from only \(67\) deaths. That is reasonable for national rates but not for an internal comparison group, and the narrow interval is misleading here.
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