# empbaysmooth: Empirical Bayes Smoothing In DCluster: Functions for the Detection of Spatial Clusters of Diseases

## Description

Smooth relative risks from a set of expected and observed number of cases using a Poisson-Gamma 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).

## Usage

 `1` ```empbaysmooth(Observed, Expected, maxiter=20, tol=1e-5) ```

## 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 Poisson-Gamma model, as described by Clayton and Kaldor, is a two-layers Bayesian Hierarchical model:

O_i|theta_i ~ Po(theta_i E_i)

theta_i ~ 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):

nuhat/alphahat=(1/n)*sum_i(thetahat_i)

nuhat/alphahat^2 = (1/(n-1))*sum_i[(1+alphahat/E_i)*(thetahat_i-nuhat/alphahat)^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 nuhat

.

 `prob` It is a vector of probabilities of the Negative Binomial, calculated as alphahat/(alphahat+E_i .

## References

Clayton, David and Kaldor, John (1987). Empirical Bayes Estimates of Age-standardized Relative Risks for Use in Disease Mapping. Biometrics 43, 671-681.

## Examples

 ```1 2 3 4 5 6 7 8``` ```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) ```

### Example output

```Loading required package: boot