Assessing the predictive performance of risk-based water quality criteria using decision error estimates from receiver operating characteristics (ROC) analysis

Errata

1. Assessing the predictive performance of risk-based water quality criteria using decision error estimates from receiver operating characteristics (ROC) analysis Volume 8, Issue 4, 674–684, Article first published online: 25 April 2012

Formulae for the FPE and FNE error rates were incorrect as originally published in Table 2. The correct formulae are: FPE = n(FP)/[n(TN) + n(FP)]; FNE = n(FN)/[n(FN) + n(TP)] as presented here in the corrected table.

Table 2. Performance metrics (after Linnet 1988) and example calculations for a 2 × 2 contingency table (i.e., error matrix) for ROC analysis with quadrant counts and ROC terms calculated for a hypothetical situation involving uncorrelated stressor and response variables, with total n = 1001 data pairs, and Ythr, and Xc set at the median values for each variablea
Indicator (stressor)
AttainingNonattainingTotal count
• aCounts are chosen to help illustrate the calculation of each term. FN = false negative; FNE = false negative error; FP = false positive; FPE = false positive error; NPE = negative predictive error; NPV = negative predictive value; PPE = positive predictive error; PPV = positive predictive value; ROC = receiver operating characteristics; Se = sensitivity; Sp = specificity; TN = true negative; TP = true positive.
Actual (response)
Nonattainingn(FN)n(TP)250+251 = 501
250251
Attainingn(TN)n(FP)252+248 = 500
252248
Total count250+252 = 502251+248 = 4991001
Prevalence = (n(TP)+n(FN))/(n(TP)+n(FP)+n(FN)+n(TN)) = 501/1001 ≈ 0.5
Nonerror rates:
Sp = n(TN)/[n(TN)+n(FP)] = 252/500 ≈ 0.5.
Se = n(TP)/[n(FN)+n(TP)] = 251/501 ≈ 0.5.
PPV = n(TP)/[n(TP)+n(FP)] = 251/499 ≈ 0.5.
NPV = n(TN)/[n(TN)+n(FN)] = 252/502 ≈ 0.5.
Accuracy = ½(Sp+Se) = (n(TP)+n(TN))/(n(TP)+n(FP)+n(FN)+n(TN)).
=(251+252)/1001 ≈ 0.5.
Error rates:
FPE = n(FP)/[n(TN)+n(FP)] = 1−Sp = 248/500 ≈ 0.5.
FNE = n(FN)/[n(FN)+n(TP)] = 1−Se = 250/501 ≈ 0.5.
PPE = n(FP)/[n(FP)+n(TP)] = 1−PPV = 248/499 ≈ 0.5.
NPE = n(FN)/[n(FN)+n(TN)] = 1−NPV = 250/502 ≈ 0.5.