From these counts, the coordinates \(\left(FPR_{jk},TPR_{jk}\right)\) for the \(j\text{th}\) cumulative \(ROC_j\) curve can be computed, where \(FPR_{jk}=1-\left(TN_{jk}/\left[TN_{jk}+FP_{jk}\right]\right)\) and \(TPR_{jk}=\left(TP_{jk}/\left[TP_{jk}+FN_{jk}\right]\right)\). Continuing with the case of the ternary ordinal outcome, \(p_{1k}\) is the \(k \text{th}\) candidate cutpoint from the first cumulative logit, and \(p_{2k}\) from the second, so that the cumulative \(ROC_1\) curve discriminates between \(Y=1\ \text{vs.}\ Y=2\ \text{or}\ 3\), and the cumulative \(ROC_2\) curve discriminates between \(Y=1\ \text{or}\ 2\ \text{vs.}\ Y=3\). Analogous to the binary case, an optimal probability cutpoint for the \(j\text{th}\) outcome level \(\hat{\pi}_j^*\) may be selected from the cumulative \(ROC_j\) curve using an optimality criterion. The cutpoint on the scale of the continuous predictor \(X_j^*\) is recovered by cross-referencing \(\hat{\pi}_j^*\) with its corresponding observed measurement.
Alternatively, ROC curve analysis may be forgone altogether by computing the \(J-1\) cutpoints from the MLE cumulative logit regression parameters, where \(X_j^*=-\left( \hat{\alpha}_{j-1}/\hat{\beta}_{j-1} \right)\).