# Modernising Receiver Operating Characteristic (ROC) Curves +
## 3 Materials and Methods
### Modernising ROC
#### 3.4.2 Linearised ROC Curves Adopting the Rasch Model  
In this work, as reported in Section 4, we have extended [PERSON]'s [12] approach by, for each axis of the linearised ROC curves separately, adopting the Rasch model. As a first step, we made logistic regressions, using WINSTEPS(r), of the Rasch formula (Equation (1)) to the cases which yielded estimates of the abilities, \(\theta\), of each instrument and the level of difficulty, \(\delta\), for each decision scenario (positives or negatives). How one scores the raw data (exemplified in Table 1) to produce an estimate of the probability, \(P_{success}\), of \"successful\" classification, is up for discussion. We tested two distinct scoring ways of applying the Rasch model and calculated:  
1. The classic \(P_{success,i,j}\) as given by the four expressions for \(P\) in Figure 11 for each decision scenario (positives or negatives) from the pair of values given horizontally for each target value, \(j\), for device \(i\), in the 3 rd and 4 th columns of Table 1;
2. and \(P_{success}\) calculated from the cumulative rates for TPR and FPR (Figure 11) summed vertically for these columns over the range of target value, \(j\), for each device \(i\), according to the ROC expressions. An assessment of these two alternative scorings in producing the most relevant results is one main aim of the current work. In a second step, we examined the relation between the Rasch parameters (Equation (1)) from the logistic regressions versus the assigned value, \(x\), separately for each decision scenario. As explained in Appendix A.3, there are theoretical reasons for expecting that in each case the task difficulty, \(\delta\), and device ability \(\theta\) will follow the relation:  
\[\delta-\theta=-z=\left(\frac{SL-x}{u}\right) \tag{2}\]  
That is, the GLM link function \(z\) will be proportional to the distance of the actual marker value, \(x=\mu_{1}\), from the specification limit, \(SL\), and inversely proportional to the classification uncertainty, \(u=\frac{\sqrt{2}r}{\sqrt{u}}\). This model of ability and level of difficulty was inspired by the work of [PERSON] and [PERSON] [19] on the 'difficulty of detecting a shift with a control chart', citing [PERSON] [20], as also explained further in Appendix A.3. Figure 2: A linearisation of the ROC curves (shown in Figure 1) by making linearised ROC plots of log-odds ratios according to the methodology proposed by [PERSON] [12]. (The dotted lines indicate linear regressions to each curve). We call the new kind of modernised ROC curves for \(z_{ROC}\) plots. In terms of the Rasch GLM, each axis of a \(z_{ROC}\) plot for each device, \(i\), would typically correspond to taking the given device ability, \(\theta_{i}\), while allowing the level of difficulty, \(\delta_{j}\), to linearly vary across the range of marker value (e.g., concentration) with the distance to the specification limit for each decision scenario (I: \(y\) axis: positives and II: \(x\) axis: negatives); separate determinations of \(\theta\) and \(\delta_{j}\) are made according to the principle of specific objectivity (Appendix A.2). Note that the classification uncertainty, \(u\), referred to in Equation (2), is not the measurement uncertainty in, for example, the marker value, \(x\), (e.g., hCG concentration level in the present case study) but expresses how steeply the operating characteristic curve (given by cumulative probabilities) rises from zero below the specification limit, \(SL\), to +1 above \(SL\). A \"crisp\" classification decision--with no uncertainty--would be represented by a discrete step-function rising infinitely rapidly from 0 to 1 at \(x=SL\). The level of difficulty in making the correct classification will increase with larger classification uncertainties and for test results closer to the specification limit.