# Modernising Receiver Operating Characteristic (ROC) Curves +
## 4 Results and Discussions
### Rasch Parameters Versus Assigned Value, \(x\)  
Using the property of specific objectivity offered by the [PERSON] model (Appendix A.2) allows the variation in task difficulty to be examined (Figure 4), irrespective of which of the five device types are being used.  
These observed trends can be compared with the predictions of our adopted model of the level of difficulty for correct classification, inspired by the work of [PERSON] and [PERSON] [19] regarding the 'difficulty of detecting a shift with a control chart', citing [PERSON] [20], as explained in Section 3.4.2 and Appendix A.3.  
As shown in Figure 4a,c, for TPR, the general trends in \(\delta\) versus \(x\) is, as expected, that task difficulty is highest at the lowest target concentrations and decreases linearly with \(x\), reaching a minimum difficulty at \(SL\) (where task difficulty \(\delta=0\), i.e., \(\delta=\overline{\theta}\) and \(P_{success}=50\%\)). The theory of [PERSON] and [PERSON] [19], Appendix A.3 makes no prediction regarding how task difficulty trends beyond the specification limit, \(SL\)). Apart from a rescaling, the same trend in \(\delta\) versus \(x\) is found for both scoring alternatives.  
The trend is the opposite for FPR (Figure 4b,d), where task difficulty is least at the lowest target concentrations and increases linearly with \(x\), reaching a maximum difficulty at \(SL\) (where task difficulty \(\delta=0\)). Otherwise, the same trend in \(\delta\) versus \(x\) is found for both scoring alternatives, as found also for TPR. An explanation of the opposite trends  
Figure 4: [PERSON] estimates of target value difficulty (\(\delta_{j}\)) based on \(P_{success}\) for (**a**) scoring alternative (i) for positives; (**b**) scoring alternative (i) for negatives; (**c**) scoring alternative (ii) for positives and (**d**) scoring alternative (ii) for negatives for a combination of all five device types used for pregnancy testing. Measurement uncertainty coverage factors \(k=2\).  
of TPR and FPR task difficulty versus concentration has to do with how, for each kind of decision risk, a \"successful\" classification is rated; arguably, a higher probability of false positive classification (\"fall-out\") should be valued negatively, in contrast to the positive rating of successful true positive classification (\"specificity\").  
What is more, we can make estimates regarding the specification limit from the intercept (\(\delta=0\)) on the \(x\)-axis of each plot shown in Figure 4 by making linear regressions of Equation (2) over the range up to where the line crosses the \(x\)-axis, with the following results:  
\[SL_{tp}=\frac{c_{tp}}{m_{tp}}=-\frac{0.8808}{-0.0357}=24.7\ IU/L\]  
\[SL_{tp\_roc}=\frac{c_{tp\_roc}}{m_{tp\_roc}}=-\frac{6.1137}{-0.2126}=28.8\ IU/L\]  
and  
\[SL_{fp}=\frac{c_{fp}}{m_{fp}}=-\frac{-0.8734}{0.0354}=24.7\ IU/L\]  
\[SL_{fp\_roc}=\frac{c_{fp\_roc}}{m_{fp\_roc}}=-\frac{-8.2758}{0.2558}=32.4\ IU/L\]  
where \(m\) and \(c\) are, respectively, the slope and intercept (on the \(y\)-axis) of each straight line fitted to the data of \(\delta\) versus \(x\) up to \(SL\) in accord with the theory (Appendix A.3). These estimates are reasonably close to the expected diagnostic value, such as \(SL_{ref}=25\) hCG (IU/L).