Concept Architecture
Concept
Theoretically, the C-Statistic is a discrimination measure that quantifies the ability of a predictive model to distinguish correctly between individuals who experience an event and those who do not. It is equivalent to the area under the receiver operating characteristic (ROC) curve for binary outcomes and represents the probability that a randomly selected individual with the event receives a higher predicted risk than a randomly selected individual without the event. In health economics, the C-statistic is widely used to evaluate risk prediction models, prognostic models and health economic decision models incorporating individual-level risk prediction.
Mathematically, the C-statistic is calculated as the probability of concordance between predicted risks and observed outcomes. Its value ranges from 0.5, indicating discrimination no better than random chance, to 1.0, indicating perfect discrimination. The statistic is estimated from all permissible pairs of observations and is mathematically equivalent to the area under the ROC curve for binary classification models.
In practice, the C-statistic is estimated after fitting logistic regression, survival models or other predictive models. It is routinely reported alongside calibration measures to assess predictive performance and is commonly used in health technology assessment, clinical prediction modelling and health services research to evaluate the usefulness of risk prediction tools.
Purpose
Used to quantify model discrimination, evaluate predictive accuracy, compare competing prediction models and assess the performance of clinical and health economic risk prediction models.
Mathematical Formulae
Primary Formula
C = P(?? > ??)
where:
?? = predicted risk for an individual with the event
?? = predicted risk for an individual without the event
Supporting Formulae
C = (Number of Concordant Pairs + 0.5 ? Number of Tied Pairs) / Number of Comparable Pairs
For binary classification:
C = AUC
0.5 � C � 1.0
Related Mathematical Methods
Receiver Operating Characteristic Curve
Area Under the Curve
Logistic Regression
Harrell's C-Index
Calibration Plot
Decision Curve Analysis
Risk Prediction Modelling
Example
A health economist develops a logistic regression model to predict hospital admission following treatment.
The fitted model is evaluated using 8,500 patient records.
Comparable patient pairs = 1,240,000
Concordant pairs = 1,060,000
Tied pairs = 18,000
C = (1,060,000 + 0.5 ? 18,000) / 1,240,000
C = 0.862
The model therefore demonstrates excellent discrimination between patients who were admitted and those who were not.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RANK.AVG | =RANK.AVG(B2,$B$2:$B$501) | Rank predicted probabilities |
| SORT | =SORT(B2:C501,1,-1) | Order observations by predicted risk |
| COUNTIFS | =COUNTIFS(C:C,1) | Count observed events |
| SUMPRODUCT | =SUMPRODUCT(...) | Calculate concordant and discordant pairs for C-statistic estimation |
| AUC Calculation | Custom implementation | Estimate discrimination from predicted probabilities and observed outcomes |
VBA (Optional)
Automate calculation of the C-statistic, concordance tables and model performance summaries for predictive health economic models.
Sources
Harrell FE. Regression Modeling Strategies.
Steyerberg EW. Clinical Prediction Models.
Hosmer DW, Lemeshow S, Sturdivant RX. Applied Logistic Regression.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is the c-statistic?
A measure of predictive discrimination equivalent to the area under the ROC curve, the probability of correctly ranking a random pair of individuals.
Source: Harrell, Califf & Pryor 1982
What probability does the c-statistic represent?
The c-statistic represents the probability that, given a random pair of individuals in which one experiences the outcome and one does not, a model assigns the higher predicted risk to the one who does. This makes it a measure of discrimination, how well the model separates future cases from non-cases, and for a binary outcome it equals the area under the ROC curve. A value of 0.5 means the model does no better than a coin toss, and 1 means it ranks every pair correctly. Ranking cases above non-cases is what it captures. Steyerberg (2009) describes this measure.
Source: Steyerberg 2009
How is the c-statistic interpreted?
The c-statistic is interpreted as the probability that, for a randomly selected pair of individuals in which one has the outcome and one does not, the model assigns a higher predicted risk to the one with the outcome. A value of 0.5 indicates no discrimination, better than chance, and 1 perfect discrimination, with higher values indicating better separation. So the c-statistic is interpreted as a measure of concordance between predicted risk and actual outcome, with values further above 0.5 indicating better discrimination, providing an overall summary of how well the model ranks individuals by risk, though it does not assess calibration or usefulness at a specific threshold.
Source: Harrell, Califf & Pryor 1982
How does the c-statistic relate to the area under the curve?
The c-statistic is equivalent to the area under the ROC curve for a binary outcome: both give the probability that the model ranks a randomly chosen case higher than a randomly chosen non-case, so the two are the same measure of discrimination. The term c-statistic, for concordance statistic, is often used, especially for models including survival data, where it generalises to concordance over comparable pairs. So the c-statistic and the area under the curve are essentially the same measure of discrimination for binary outcomes, with the c-statistic terminology extending naturally to time-to-event settings through the concept of concordance.
Source: Hanley & McNeil 1982
What does the c-statistic measure?
The c-statistic measures discrimination, the ability of a predictive model to distinguish individuals who experience an outcome from those who do not, by assessing how well the model's predicted risks rank cases above non-cases. It summarises this across all thresholds into a single value. So the c-statistic measures how well a model separates those with and without the outcome, capturing discrimination independently of any particular decision threshold, which makes it a standard summary of a prediction model's performance, though it addresses only discrimination and not calibration or the model's usefulness for a specific clinical decision.
Source: Harrell, Califf & Pryor 1982
What are the limitations of the c-statistic?
The limitations of the c-statistic include that it measures only discrimination, not calibration, so a model can have a high c-statistic yet give poorly calibrated probabilities; that it can be insensitive to improvements from adding a predictor to an already strong model; and that it summarises across thresholds, which may not reflect performance at the threshold used in practice. So the c-statistic is used alongside calibration assessment and other measures, since a high value indicates good separation of cases from non-cases but does not ensure accurate predicted risks or usefulness for a particular decision, which is why discrimination and calibration are evaluated together for prediction models.
Source: Harrell, Califf & Pryor 1982
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 12 Dec 2025
Content version: 1.0.0
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- Persistent URI
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- Term code
- HE-ES-SA-042
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