empbaysmooth {DCluster}

Empirical Bayes Smoothing

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 \frac{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.

Usage

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 \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 \widehat{\theta}_i=\frac{O_i+\nu}{E_i+\alpha}. Estimators of \nu and \alpha (\widehat{\nu} and \widehat{\alpha},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

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 Age-standardized Relative Risks for Use in Disease Mapping. Biometrics 43, 671-681.

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)