Empirical Bayes estimates

Description

Computes empirical Bayes estimates of observed event counts using the method of moments.

Usage

epi.empbayes(obs, pop)

Arguments

Details

The gamma distribution is parameterised in terms of shape (α) and scale (ν) parameters. The mean of a given gamma distribution equals ν / α. The variance equals ν / α^{2}. The empirical Bayes estimate of event risk in each unit of interest equals (obs + ν) / (pop + α).

This technique performs poorly when your data contains large numbers of zero event counts. In this situation a Bayesian approach for estimating α and ν would be advised.

Value

A data frame with four elements: gamma the mean event risk across all units, phi the variance of event risk across all units, alpha the estimated shape parameter of the gamma distribution, and nu the estimated scale parameter of the gamma distribution.

References

Bailey TC, Gatrell AC (1995). Interactive Spatial Data Analysis. Longman Scientific & Technical. London, pp. 303 - 308.

Langford IH (1994). Using empirical Bayes estimates in the geographical analysis of disease risk. Area 26: 142 - 149.

Meza J (2003). Empirical Bayes estimation smoothing of relative risks in disease mapping. Journal of Statistical Planning and Inference 112: 43 - 62.

Examples

data(epi.SClip)
obs <- epi.SClip$cases; pop <- epi.SClip$population

est <- epi.empbayes(obs, pop)
empbayes.prop <- (obs + est[4]) / (pop + est[3])
raw.prop <- (obs) / (pop)
rank <- rank(raw.prop)
dat <- data.frame(rank, raw.prop, empbayes.prop)

plot(dat$rank, dat$raw.prop, type = "n", xlab = "Rank", ylab = "Risk")
points(dat$rank, dat$raw.prop, pch = 16, col = "red")
points(dat$rank, dat$empbayes.prop, pch = 16, col = "blue")
legend(x = "topleft", legend = c("Raw estimate", "Bayes adjusted estimate"), 
   col = c("red","blue"), pch = c(16,16), bty = "n")

Results