C-Statistic: Definition, Examples, Weighting and Significance

ROC Curve > C-Statistic

You may want to read this article first: What is a Receiver Operating Characteristic (ROC) curve?.

What is a C-Statistic?

The C-statistic (sometimes called the “concordance” statistic or C-index) is a measure of goodness of fit for binary outcomes in a logistic regression model. In clinical studies, the C-statistic gives the probability a randomly selected patient who experienced an event (e.g. a disease or condition) had a higher risk score than a patient who had not experienced the event. It is equal to the area under the Receiver Operating Characteristic (ROC) curve and ranges from 0 to 1.

• A value below 0.5 indicates a very poor model.
• A value of 0.5 means that the model is no better than predicting an outcome than random chance.
• Values over 0.7 indicate a good model.
• Values over 0.8 indicate a strong model.
• A value of 1 means that the model perfectly predicts those group members who will experience a certain outcome and those who will not.

The C-statistic isn’t used very often as it only gives you a general idea about a model; A ROC curve contains much more information about accuracy, sensitivity and specificity.

Weighting

A weighted c-index is used when the cost of failing to predict a positive outcome (like a test for cancer) is higher than benefit of correctly predicting a negative outcome. Weighting penalizes models that result in small probability differences for positive and negative outcomes, but doesn’t change the value of the C-statistic. It can also be used to adjust for stratified random sampling.

Statistical Significance

Like most statistics, the C-statistic is sometimes paired with a confidence interval. For example, you might have a result of 0.63 with a confidence interval ranging from 0.53 to 0.73). In general, any result is not significant if it includes 0.5, even if it includes the relevant C-statistic. For example, a result of 0.63 with a CI ranging from 0.43 to 0.83 would not be significant because it includes 0.5 in that range.

Reference:
Hosmer DW, Lemeshow S. Applied Logistic Regression (2nd Edition). New York, NY: John Wiley & Sons; 2000.