empbaysmooth
Empirical Bayes Smoothing
Smooth relative risks from a set of expected and observed number of cases using a PoissonGamma model as proposed by Clayton and Kaldor (1987) .
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.
 Keywords
 models
Usage
empbaysmooth(Observed, Expected, maxiter=20, tol=1e5)
Arguments
 Observed
 Vector of observed cases.
 Expected
 Vector of expected cases.
 maxiter
 Maximum number of iterations allowed.
 tol
 Tolerance used to stop the iterative procedure.
Details
The PoissonGamma model, as described by Clayton and Kaldor, is a twolayers Bayesian Hierarchical model:
$$O_i\theta_i \sim Po(\theta_i E_i)$$
$$\theta_i \sim Ga(\nu, \alpha)$$
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):
$$\frac{\widehat{\nu}}{\widehat{\alpha}}=\frac{1}{n}\sum_{i=1}^n \widehat{\theta}_i$$
$$\frac{\widehat{\nu}}{\widehat{\alpha}^2}=\frac{1}{n1}\sum_{i=1}^n(1+\frac{\widehat{\alpha}}{E_i})(\widehat{\theta}_i\frac{\widehat{\nu}}{\widehat{\alpha}})^2$$
Value

A list of four elements:
 n
 Number of regions.
 nu
 Estimation of parameter $nu$
 alpha
 Estimation of parameter $alpha$
 smthrr
 Vector of smoothed relative risks.
 size
 Size parameter of the Negative Binomial. It is equal to $$\widehat{\nu}$$ .
 prob
 It is a vector of probabilities of the Negative Binomial, calculated as $$\frac{\widehat{\alpha}}{\widehat{\alpha}+E_i}$$ .
References
Clayton, David and Kaldor, John (1987). Empirical Bayes Estimates of Agestandardized Relative Risks for Use in Disease Mapping. Biometrics 43, 671681.
Examples
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)