The authors use empirical Bayes methods to obtain estimates of relative risks. A random-effects model is introduced, which assumes a parametric probability function for the distribution of relative risks. In particular, three models are defined in order to compare their results. First, a Gamma model is defined, where relative risks are assumed to follow a Gamma distribution while the observed cases follow a Poisson distribution. In this case, Maximum Likelihood can be used to estimate the posterior distribution for the relative risks, which also follows a gamma distribution.

A second model is defined by assuming a log-normal distribution for the relative risks. To obtain a closed form solution to this model the authors use a quadratic approximation for the Poisson likelihood of the log-relative risks. Maximum Likelihood estimation, using the EM algorithm, is applied to obtain the empirical Bayes estimates of the relative risks under this model. Under this second model, log relative risks are assumed to be correlated following a CAR model.

The third and final model is a nonparametric model where spatial correlation is ignored, and the distribution of the relative risks is estimated nonparametrically using the EM algorithm. The authors highlight the fact that the estimates provided are the empirical Bayes equivalent to the indirectly standardized rate estimates.

Data on cases of lip cancer in Scotland for the period between 1975 and 1980 are used to illustrate the models described in this article. The authors point out that the level of smoothing resulting from these models is determined exclusively by the data contrary to what occurs with other smoothed method in which the smoothing level is determined by a parameter defined a priori. Even though the authors choose the nonparametric mode as the preferred model for mapping data since it requires less assumptions compared to the other two methods, they point out that for very small areas with small cases it is important to account for the presence of spatial autocorrelation in the relative risks.

Last updated March 9, 2006