# Modernising Receiver Operating Characteristic (ROC) Curves +
## 4 Results and Discussions
### Evaluating Device Parameters  
In order to evaluate the slope, \(M\), and intercept, \(C\), of the new kind of \(z_{ROC}\) plots for the different device types (Section 3.4.2), estimates of \(u\) and \(SL\) for each device were obtained by linear regression of the plots in Figure 5 for each coordinate \(z_{TPR}\) and \(z_{FPR}\), respectively, versus the target value, \(x\), according to the theoretical model (Equation (A2)). From the separate Rasch estimates of task difficulty and device ability, thanks to the principle of specific objectivity (Appendix A.2), it is straightforward to calculate \(z_{TPR}=\theta_{TPR}-\delta_{TPR}\) and \(z_{TNR}=\theta_{TNR}-\delta_{TNR}\) for curves versus assigned value, \(x\), Hcg (IU/L) for the five devices used for pregnancy testing, based on the two methods (Section 4.1), respectively.  
The plots (Figure 5) for each coordinate \(z_{TPR}\) and \(z_{FPR}\), respectively, versus the target value, \(x\), show sets of parallel lines where the common slope of each set of lines corresponds to the (inverse) classification uncertainty, \(u\), of the set of items from the principle of specific objectivity (Appendix A.2). The set of parallel lines for FPR are better resolved for the  
Figure 5: Experimental studies of Rasch parameters based on \(P_{success}\) for scoring alternative (ii) for (**a**) positives and (**b**) negatives versus assigned value, hCG (IU/L) for five device types used for pregnancy testing. Measurement uncertainty coverage factors \(k=2\).  
two methods, aligning with the Rasch histogram plots (Figure 3) which show more distinct device ability variation for the false positive rates. In fact, most of the differences between the curves in Figure 5 (particularly pronounced in Figure 5b) are due to the differences in Rasch abilities \(\theta_{TPR}\) and \(\theta_{FPR}\) among the different device types as estimated with a Rasch logistic regression as described in Section 4.1. (The closeness of the curves, particularly in Figure 5a, indeed makes it difficult to distinguish the symbols denoting them; this is a result, not a shortcoming in figure quality.)  
In contrast to the fitting procedure (Section 4.2) used over the range 6 < x < SL (IU/L) in Figure 4, for the \(z_{ROC}\) plots, an approximate linear relation is assumed across the whole range of target concentration values 6 < \(x\) < 60 (IU/L) in order to match the range adopted when plotting traditional ROC curves (Appendix A.1). Note that this procedure extrapolates the theory of [PERSON] [19] (Appendix A.3) beyond the specification limit, \(SL\).  
As an example, an analysis of the FPR plot (Figure 5b) yielded the following values for device 1:  
\[SL_{fp,1}=\frac{c_{fp,1}}{m_{fp,1}}=-\frac{5.3}{-0.29}=+18.43 IU/L\]  
and  
\[u_{fp,1}=-\frac{1}{m_{fp,1}}=-\frac{1}{-0.29}=-3.47 IU/L\]  
(The negative sign on the uncertainty has to do with the opposite slope of FPR rating, as discussed in Section 3.2)  
The parameter estimates obtained by these linear regressions across the whole range of target concentration values 6 < \(x\) < 60 (IU/L) (to match the range adopted when plotting traditional ROC curves) are given in Table 3. Uncertainties, quoted in parentheses, have coverage factors \(k\) = 2. It can be noted that the \(SL\) estimates with these linear regressions over the whole concentration range deviate more from the corresponding \(SL\) estimate (of 25 IU/L) than with the method described in Section 4.2. A principal reason for these discrepancies is that the theory of [PERSON] and [PERSON] ([19], Appendix A.3) does not strictly apply for concentrations around \(SL\) where the probability of successful classification asymptotically approaches 100%.