Description Usage Arguments Details Value References Examples
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.
1 | empbaysmooth(Observed, Expected, maxiter=20, tol=1e-5)
|
Observed |
Vector of observed cases. |
Expected |
Vector of expected cases. |
maxiter |
Maximum number of iterations allowed. |
tol |
Tolerance used to stop the iterative procedure. |
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]
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 . |
Clayton, David and Kaldor, John (1987). Empirical Bayes Estimates of Age-standardized Relative Risks for Use in Disease Mapping. Biometrics 43, 671-681.
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)
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Loading required package: boot
Loading required package: spdep
Loading required package: sp
Loading required package: Matrix
Loading required package: MASS
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