For \(J=2\) this model reduces to the logistic model, but the cumulative logit model is similarly able to transform the continuous predictor to the predicted probability scale, except that each outcome level gets a predicted probability function.
We have so far recalled that ROC curves are invariant to monotonic transformation of the continuous measurement, including transformation by a logistic regression model to the predicted probability scale. In addition, we have reviewed the cumulative logit model and its transformation of a single continuous predictor to a series of separate predicted probabilities for each level of the ordinal outcome. These probabilities are comprehensive and mutually exclusive with respect to the outcome, and are suitable for computing a series of "cumulative" ROC curves.
Stage 2: Cumulative ROC Curves
Calculation of the ROC curve on the predicted probability scale can be readily extended to count \(TP\), \(TN\), \(FP\), and \(FN\) for each cumulative logit, resulting in \(J-1\) cumulative ROC curves. For the cumulative logit associated with the \(j\text{th}\) outcome, let \(p_{jk}\) be the \(k\text{th}\) candidate cumulative probability cutpoint from among the \(\hat{\pi}_{ij}\), then one may count \(TP_{jk}\), \(TN_{jk}\), \(FP_{jk}\), and \(FN_{jk}\) with the indicator function \(I\left(\cdot\right)\) by comparing the outcome \(Y_i\) with \(\hat{\pi}_{ij}\) vs. \(p_{jk}\) for \(i=1,\dots,N \); \(j=1,\dots, J-1\); and \(k=1,\dots, N\):