Example 13.3.1. For the British doctors’ data (Example 10.6.1), where the stratum rate ratios fall from \(5.74\) to \(0.90\) with age:

Here too most methods agree, because their weights happen to be similar: all put about \(85\%\) of the weight on the three oldest strata, where most of the deaths are. The inverse-variance estimate is lower because it averages on the log scale. A weighted geometric mean is never larger than the corresponding arithmetic mean, and the gap is large when the values are spread out: the ratio \(5.74\) in the youngest stratum pulls an arithmetic average up far more than a geometric one. The agreement of the other methods should not be mistaken for evidence that \(1.42\) is a meaningful summary. The deviance of \(12.1\) on \(4\) degrees of freedom shows that the common ratio model fails, and the proper report is the age-specific ratios or the trend model of Example 12.4.3.