title: Estimating confidence limits on a standardised mortality ratio when the expected number is not error free.
abstract: OBJECTIVE
  The aim was to demonstrate how the beta distribution may be used to find confidence limits on a standardised mortality ratio (SMR) when the expected number of events is subject to random variation and to compare these limits with those obtained with the standard exact approach used for SMRs and with a Fieller-based confidence interval.
  
  
  METHODS
  The relationship of the binomial and the beta distributions is explained. For cohort studies in which deaths are counted in exposed and unexposed groups exact confidence limits on the relative risk are found conditional on the total number of observed deaths. A similar method for the SMR is justified by analogy between the SMR and the relative risk found from such cohort studies, and the fact that the relevant (beta) distribution does not require integer parameters.
  
  
  METHODS
  Illustrative examples of hypothetical data were used, together with a MINITAB macro (see appendix) to perform the calculations.
  
  
  RESULTS
  Exact confidence intervals that include error in the expected number are much wider than those found with the standard exact method. Fieller intervals are comparable with the new exact method provided the observed and expected numbers (taken to be means of Poisson variates) are large enough to approximate normality. As the expected number is increased, the standard method gives results closer to the new method, but may still lead to different conclusions even with as many as 100 expected.
  
  
  CONCLUSIONS
  If there is reason to suppose the expected number of deaths in an SMR is subject to sampling error (because of imprecisely estimated rates in the standard population) then exact confidence limits should be found by the methods described here, or approximate Fieller-based limits provided enough events are observed and expected to approximate normality.
authors.affiliation: Department of Public Health Medicine, University of Sheffield Medical School.
authors.forename: P
authors.initials: P
authors.lastname: Silcocks
mesh_terms: D000293:Adolescent; D000328:Adult; D000368:Aged; D002648:Child; D002675:Child, Preschool; D015331:Cohort Studies; D016001:Confidence Intervals; D004812:Epidemiologic Methods; D006801:Humans; D007223:Infant; D007951:Leukemia, Myeloid / Q000401:mortality; D008297:Male; D008875:Middle Aged; D009026:Mortality; D012984:Software; D013223:Statistics as Topic / Q000379:methods