Exact limits for the ratio of two SMR values.

SIR-Liddell has given some interesting extensions of the methodology for setting limits on an SMR based on the ratio DIE, where D is an observed number of deaths and assumed to follow a Poisson distribution, whileE is its expectation and is assumed to be based on such extensive data as to be considered essentially free of error. Like others before,23 Liddell considers that any exact procedure for setting limits on a Poisson parameter can be used for setting limits on quantities proportionate to that parameter. Liddell ties his procedure to the link between the Poisson and the x2 distributions. While he considers various approximate methods for setting limits on an SMR, he does conclude that for relatively small D (<15) the exact method is quicker and certainly more appropriate. A square-root transformation method is somewhat appropriate generally, while use of the Wilson-Hilferty approximation' to the x2 distribution is indicated as allowing almost unlimited extension, ie, close approximation for both small and large D, of the exact methodology. Also, Liddell warns us that even if an overall SMR is close to unity, it may differ importantly from unity in some specific stratum, eg, a particular age-race-sex combination. Liddell' faults Frentzel-Beyme5 for suggesting that a methodology reported by Ederer and Mantel' for setting limits on the ratio of two Poisson variates could be used for setting exact limits on an SMR, yet taking both numerator and denominator variation into account in circumstances where E cannot be considered as fixed. But Liddell fails to advise his readers of the important implications that the procedure described by Ederer and Mantel might have for their work. First off, it would be true that D, as a summation of Poisson variables over many strata, would itself follow a Poisson distribution. However, E is a more general linear combination of Poisson variables and so would not follow an exact Poisson distribution. Yet it might not be too inappropriate to act as though E were proportionate to the total number of deaths in the population which gave rise to the stratum-specific death rates on which E was based. It may be that this is what Frentzel-Beyme had in mind but on which he did not elaborate-his main interest having been in bringing out the methodology in the standard case. Actually, there are situations for which the Frentzel-Beyme suggestion would be fully appropriate. When the proportionate distribution by strata, eg, age x race X sex, is assumed the same for the study and reference populations, then the suggestion of Frentzel-Beyme applies. An example would be where the same population is studied in successive years or time periods over which no important changes are considered to occur. For that matter, the suggestion of Frentzel-Beyme should be applauded rather than faulted. Consider that D and E are both subject to chance variation. Then none of the methods premised on E being essentially error-free would be appropriate. But by applying the method described by Ederer and Mantel, we would be allowing for some degree of variation in E, which could be rather important where the variability inE is not trivial compared with the variability in D. And the closer the study population's relative distribution by strata is to that of the reference population, the more exact will be the procedure recommended by Frentzel-Beyme. But by title this letter is supposed to deal with ratios of SMRs, so here goes. Suppose our two SMRs are Du/E1 and D2/E2, with Di andD2 each subject to Poisson variation, while Eu and E2 are error-free. Then, as Ederer and Mantel6 bring out, conditional on the total Di+D2, D, will be distributed like the number of successes in D1+D2 independent binomial trials where the fixed probability of success is Xl/(Xl+X2). Here, Xi and X2 are the true Poisson expectations of Di and D2, respectively. The usual binomial tables can allow setting limits on Xl/(Xl+X2) and so, in turn, on X1/X2. Normal approximations could apply for results outside the range of the binomial tables. In his report, Liddell' did not take the opportunity to instruct his readers on how they might make an exact comparison of two SMRs. Certainly, if those readers are interested in evaluating a single SMR, they should be interested in comparing two different SMR values. And while Liddell has focused on the case of the denominator value, E, fixed, Frentzel-Beyme provided a clue for taking the variation inE into account. Certainly, the variation in E sometimes contributes importantly to the variation in DIE. We note that exact methodology does exist for setting limits on the ratio of two Poisson ratios or two Poisson products, eg, (Di/D2)f(D3/D4) = (DuD4)/(D2D3). Simply, it is the-methodology for setting exact limits on the odds ratio of a 2 x 2 contingency table. Finally, we note, as we have earlier,6 that the exact confidence limits on the ratio gest. P rocted by coright.