If $nu$ and $alpha$ are the two parameters of the prior Gamma distribution, smoothed relative risks are $(O_i+nu)/(E_i+alpha)$.
$nu$ and $alpha$ are estimated via Empirical Bayes, by using mean and variance, as described by Clayton and Kaldor(1987).
Size and probabilities for a Negative Binomial model are also calculated (see below).
See Details for more information.
empbaysmooth(Observed, Expected, maxiter=20, tol=1e-5)
$$O_i|\theta_i \sim Po(\theta_i E_i)$$
$$\theta_i \sim Ga(\nu, \alpha)$$
The posterior distribution of $O_i$,unconditioned to $theta_i$, is Negative Binomial with size $nu$ and probability $alpha/(alpha+E_i)$.
The estimators of relative risks are $thetahat_i=(O_i+nu)/(E_i+alpha)$. Estimators of $nu$ and $alpha$ ($nuhat$ and $alphahat$,respectively) are calculated by means of an iterative procedure using these two equations (based on mean and variance estimations):
$$\frac{\widehat{\nu}}{\widehat{\alpha}}=\frac{1}{n}\sum_{i=1}^n \widehat{\theta}_i$$
$$\frac{\widehat{\nu}}{\widehat{\alpha}^2}=\frac{1}{n-1}\sum_{i=1}^n(1+\frac{\widehat{\alpha}}{E_i})(\widehat{\theta}_i-\frac{\widehat{\nu}}{\widehat{\alpha}})^2$$
library(spdep) data(nc.sids) sids<-data.frame(Observed=nc.sids$SID74) sids<-cbind(sids, Expected=nc.sids$BIR74*sum(nc.sids$SID74)/sum(nc.sids$BIR74)) smth<-empbaysmooth(sids$Observed, sids$Expected)