Confidence intervals for weighted sums of Poisson parameters

Directly standardized mortality rates are examples of weighted sums of Poisson rate parameters. If the numbers of events are large then normal approximations can be used to calculate confidence intervals, but these are inadequate if the numbers are small. We present a method for obtaining approximate confidence limits for the weighted sum of Poisson parameters as linear functions of the confidence limits for a single Poisson parameter, the unweighted sum. The location and length of the proposed interval depend on the method used to obtain confidence limits for the single parameter. Therefore several methods for obtaining confidence intervals for a single Poisson parameter are compared. For single parameters and for weighted sums of parameters, simulation suggests that the coverage of the proposed intervals is close to the nominal confidence levels. The method is illustrated using data on rates of myocardial infarction obtained as part of the WHO MONICA Project in Augsburg, Federal Republic of Germany.