How is the ROC index (concordance) defined?

The ROC index, or concordance index, is defined as the percentage of pairs of observations that are correctly ordered by the model, taking into account both concordant and tied pairs. Concordant cases refer to instances where the model accurately predicts that one observation is more likely to be a positive case compared to another observation that is a negative case. Tied cases occur when the model provides equal predicted probabilities for two observations.

Thus, the correct answer captures both the concordant cases and tied cases, making it a comprehensive measure of the model’s ability to rank observations. The ROC index serves as an important metric in assessing the performance of classification models, particularly in scenarios where the focus is on the relative ordering of predictions rather than just absolute accuracy.

The other choices do not encompass the full definition of the ROC index. For instance, simply counting correct versus incorrect predictions fails to account for the nuanced evaluation of ordered pairs that the ROC approach requires. Similarly, considerations of model precision and the sum of model predictions do not directly relate to the definition of concordance as understood through the ROC index.