empbaysmooth: Empirical Bayes Smoothing

Description Usage Arguments Details Value References Examples

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).

See Details for more information.

Usage

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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

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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
Loading required package: spdep
Loading required package: sp
Loading required package: Matrix
Loading required package: MASS

DCluster documentation built on May 2, 2019, 11:51 a.m.