DCluster (version 0.2-7)

empbaysmooth: 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.

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 $$\frac{\widehat{\alpha}}{\widehat{\alpha}+E_i}$$ .

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):

$$\frac{\widehat{\nu}}{\widehat{\alpha}}=\frac{1}{n}\sum_{i=1}^n \widehat{\theta}_i$$

$$\frac{\widehat{\nu}}{\widehat{\alpha}^2}=\frac{1}{n-1}\sum_{i=1}^n(1+\frac{\widehat{\alpha}}{E_i})(\widehat{\theta}_i-\frac{\widehat{\nu}}{\widehat{\alpha}})^2$$

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

# NOT RUN {
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)
# }