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.