Estimating the degree of heterogeneity between event rates using likelihood.

The observed variability between mortality or morbidity rates in epidemiologic studies is partly due to random fluctuations. The same is true for rate ratios relative to reference rates. A method for estimating the distribution of true rate ratios is applied to a data set of perinatal mortality in 515 small areas in the North West Thames Health Region, England, in the period 1986-1990. Combining the random Poisson variability with the assumption that the true rate ratios are drawn from a gamma distribution (a family of positive unimodal distributions) produces a negative binomial log-likelihood for the dispersion parameter of the gamma. The maximum likelihood estimate of this parameter and its confidence interval are then found via direct numerical methods; alternatively, the hypothesis of no heterogeneity is tested by a likelihood ratio. The standardized mortality ratios (SMRs) for the data have an empirical distribution with 5th percentile at 0 and 95th percentile at 1.92, but their true variability, as described by the 5th to 95th percentiles of the fitted gamma distribution, is from 0.72 to 1.32. The likelihood ratio test confirmed this result, rejecting the hypothesis that the true rates are homogeneous (p = 0.015). The method requires only modest computing resources and is useful when assessing the need for more detailed study.