This aticle is part of the mini-monograph “Health and Environment Information Systems for Exposure and Disease Mapping, and Risk Assessment.”

Appendix A

Generation of the observed cases for Simu 3 using the multinomial distribution.
For Simu 3, the number of cases for each of the 532 areas is generated using the multinomial distribution as follows:
where N is the total number of cases in the study region and is set equal (to the nearest integer) to the sum of the expected counts across all 532 areas. Hence N = 1,732 for the SF = 1 scenario and appropriate multiples of this for the other SFs.
The parameter θi represents the relative risk in area i relative to some nominal external reference rate. However, the constraint Σi Yi = N = Σi Ei imposed by the multinomial sampling effectively rescales the true relative risk in each area to be
The interpretation of θ* i is the relative risk in area i relative to the average risk in the study region.

Appendix B

Tradeoff between false-positive and false-negative rates for different decision rules.
Figure B1 shows three different loss functions representing weighted tradeoffs between the two types of errors: false positive and false negative, associated with the D(c, 1) decision rule for detecting raised-risk areas using the BYM model, plotted against cutoff c. Defining as in the text the false-negative rate to be the probability of failing to detect a true raised risk (i.e., 1 – sensitivity), and the false-positive rate to be the probability of false detection of a background area as corresponding to a raised risk (i.e., 1 – specificity), the three loss functions used are as follows:
Figure B1  Variation of the total error rate (loss function) as function of the cutoff probability c for different weighting of the two types of errors (false positive and false negative). Results shown are for the Simu 2 using the BYM model.   with each error being equally weighted.
where we weight the false negative error as twice as bad as the lack of specificity.
where we weight the lack of specificity as twice as bad as the false negative.
We wish to choose c to minimize the losses, and the graphs show that, on average, a value of around 0.7–0.8 is appropriate. Note that the plots in Figure B1 are based on Simu 2 with SF = 2 or 4. For a small number of other scenarios (mainly with SF = 1), a value of c < 0.7 was needed to minimize the loss. However, for consistency, we have used the same value of c (= 0.8) for all the BYM and L1-BYM results presented in this article.