Using a mixed effects model to estimate geographic variation in cancer rates

Commonly used methods for depicting geographic Variation in cancer rates ar e based on rankings. They identify where the rates are high and low but do not indicate the magnitude of the rates nor their variability. Yet such mea sures of variability may be useful in suggesting which types of cancer warr ant further analytic studies of localized risk factors. We consider a mixed effects model in which the logarithm of the mean Poisson rate is additive in fixed stratum effects (e.g., age effects) and in logarithms of random re lative risk effects associated with geographic areas. These random effects are assumed to follow a gamma distribution with unit mean and variance 1/al pha, similar to Clayton and Kaldor (1987, Biometrics 43, 671-681). We prese nt maximum likelihood and method-of-moments estimates with standard errors for inference on alpha(-1/2), the relative risk standard deviation (RRSD). The moment estimates rely an only the first two moments of the Poisson and gamma distributions but have larger standard errors than the maximum likeli hood estimates. We compare these estimates with other measures of variabili ty. Several examples suggest that the RRSD estimates have advantages compar ed to other measures of variability.