Simple exact analysis of the standardised mortality ratio

Abstract

The standardised mortality ratio is the ratio of deaths observed, D, to those expected, E, on the basis of the mortality rates of some reference population. On the usual assumptions--that D was generated by a Poisson process and that E is based on such large numbers that it can be taken as without error--the long established, but apparently little known, link between the Poisson and chi 2 distributions provides both an exact test of significance and expressions for obtaining exact (1-alpha) confidence limits on the SMR. When a table of the chi 2 distribution gives values for 1-1/2 alpha and 1/2 alpha with the required degrees of freedom, the procedures are not only precise but very simple. When the required values of chi 2 are not tabulated, only slightly less simple procedures are shown to be highly reliable for D greater than 5; they are more reliable for all D and alpha than even the best of three approximate methods. For small D, all approximations can be seriously unreliable. The exact procedures are therefore recommended for use wherever the basic assumptions (Poisson D and fixed E) apply.