We offer an approximation to central confidence intervals for directly standardized rates, where we assume that the rates are distributed as a weighted sum of independent Poisson random variables. Like a recent method proposed by Dobson, Kuulasmaa, Eberle and Scherer, our method gives exact intervals whenever the standard population is proportional to the study population. In cases where the two populations differ non-proportionally, we show through simulation that our method is conservative while other methods (the Dobson et al. method and the approximate bootstrap confidence method) can be liberal.
1997 by John Wiley & Sons, Ltd.
1. Introduction
In epidemiology it is common to compare incidence or mortality rates by directly standardized rates (DSRs) and to assume that one can model these DSRs as weighted sums of independent Poisson random variables where the weights are known. In this paper we offer an approximation to the confidence intervals for DSRs under this assumption. We refer to these new confidence intervals as gamma intervals, since the approximation is based on the gamma distribution. The gamma intervals perform at least as well as existing methods in all situations studied here, but perform especially better than existing methods when the number of counts in any specific cell is small and there is large variability in the weights. Large variability in the weights occurs when comparing cancer rates across disparate populations using a single standard. For example, Cancer Incidence in Five Continents, ยปol Iยป, compares data from 163 different populations using the world population standard. For many cancers there are less than 10 cases across all age groups. Since we can write the gamma intervals as a simple function of the inverse chi-squared distribution, they are practical to use in any situation.
Dobson, Kuulasmaa, Eberle and Scherer (hereafter DKES) introduced confidence limits for weighted sums of Poisson random variables that, unlike the traditional confidence limits based on the normal distribution (see Clayton and Hills), do not require large cell counts. The DKES limits are exact for the case when all the weights are equal, a case for which the DSR reduces to a scaled Poisson random variable. The gamma intervals are also exact in this case as well as when some of the weights are equal and the rest are zero.
Recently, Swift applied the approximate bootstrap confidence (ABC) method of DiCiccio and Efron to this problem. This method is not exact when the weights are equal but has fairly good