Cstatistic

The cstatistic, commonly referred to as the concordance statistic or the area under the receiver operating characteristic curve (AUC), is a measure of discrimination for models that predict binary outcomes or time-to-event risks. It represents the probability that, for a randomly selected pair consisting of one positive and one negative case, the model assigns a higher predicted risk to the positive case. The statistic ranges from 0.5 to 1, where 0.5 indicates no discriminative ability beyond chance and 1 indicates perfect separation.

In binary outcome settings, the cstatistic equals the area under the ROC curve and is often interpreted as the model’s ability to rank-order cases by risk. In survival analysis, the c-statistic is generalized as the concordance index to accommodate censored data, assessing the agreement between predicted risk and observed event times. Variants include Harrell’s C, Uno’s C, and time-dependent AUCs.

Calculation typically involves computing the ROC curve from predicted probabilities to obtain the AUC, which can be done nonparametrically or via numerical integration. In time-to-event contexts, estimation methods account for censoring, and confidence intervals are commonly obtained by bootstrap or jackknife methods.

Interpretation and limitations: the c-statistic measures discrimination but not calibration, meaning a model can distinguish high from low risk yet misestimate absolute risk. It can be influenced by outcome prevalence and sampling schemes, and it does not reflect clinical usefulness without considering decision thresholds and calibration. It is widely used for model comparison and reporting predictive performance.

Software implementations include R packages such as pROC and survival, and Python tools like scikit-learn’s roc_auc_score and lifelines’ concordance_index.