Comparing standardized mortality ratios

It is well known that the ratio of two standardized mortality ratios (SMRs) is not in general an unbiased estimate of the underlying within-stratum ratio of rates of one subcohort relative to another. It is also established, although less well known, that a sufficient condition for unbiasedness is that the underlying stratum-specific rates in each of the two subcohorts be proportional to the reference population. Further, the ratio of SMRs is more precise than the wholly internal (Poisson regression) estimate of rate ratio. In data that are compatible with the proportionality assumption, use of the ratio of SMRs thus buys precision at the cost of increased vulnerability to bias. To further elucidate choice between methods, we derive expressions for the asymptotic precision of each. These show that improved precision of ratio of SMRs depends on the extent to which the distribution of expected deaths over strata is different in the two cohorts, or equivalently, on the variance over strata of the proportion of expected deaths falling in the first cohort. The results are illustrated by hypothetic examples.