Concept Architecture
Concept
Theoretically, Harrell C-Index is a non-parametric measure of predictive discrimination for survival models that quantifies the ability of a model to correctly rank individuals according to their risk of experiencing an event. It extends the concept of the area under the receiver operating characteristic curve to time-to-event data by accommodating censored observations. In health economics, the Harrell C-index is widely used to evaluate prognostic models, survival prediction models and risk adjustment models.
Mathematically, the Harrell C-index is calculated as the proportion of all comparable pairs of individuals for which the predicted and observed event orderings are concordant. Only pairs for which the ordering of survival times can be determined contribute to the calculation. The index ranges from 0.5, representing random prediction, to 1.0, representing perfect discrimination.
In practice, the Harrell C-index is estimated after fitting a survival model such as a Cox proportional hazards model or a parametric survival model. Predicted risks are compared with observed survival outcomes across all comparable patient pairs. In health economics, the statistic is commonly reported when validating survival models used for extrapolation, economic evaluation and health technology assessment.
Purpose
Used to quantify the discriminative performance of survival prediction models by measuring their ability to correctly rank individuals according to event risk.
Mathematical Formulae
Primary Formula
C = (Nconcordant + 0.5 ? Ntied) � Ncomparable
where:
- C = Harrell concordance index
- Nconcordant = number of concordant pairs
- Ntied = number of tied prediction pairs
- Ncomparable = number of comparable subject pairs
Supporting Formulae
Concordance indicator:
I = 1, if predicted and observed order agree
I = 0, otherwise
Equivalent estimator:
C = (1 / Ncomparable) ? ? I?
where the summation is taken over all comparable subject pairs.
Related Mathematical Methods
- Cox Proportional Hazards Model
- Survival Analysis
- Time-to-Event Analysis
- Receiver Operating Characteristic Analysis
- Uno C-Index
- Model Validation
- Bootstrap Validation
Example
A health economist develops a survival model to predict mortality following heart failure treatment. Among 18,500 comparable patient pairs, 15,355 pairs are concordant and 290 pairs have tied predicted risks.
C = (15,355 + 0.5 ? 290) � 18,500
C = 15,500 � 18,500 = 0.838
A Harrell C-index of 0.838 indicates good discriminative ability, demonstrating that the model correctly ranks patient risk in approximately 84% of comparable cases.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| COUNTIFS | =COUNTIFS(B:B,">0") | Count comparable observations |
| SUMPRODUCT | =SUMPRODUCT(B2:B101,C2:C101) | Calculate concordant pair totals |
| IF | =IF(B2=C2,0.5,IF(B2>C2,1,0)) | Assign concordance scores |
| SUM | =SUM(D2:D1000)/E2 | Calculate overall C-index |
VBA (Optional)
Automate Harrell C-index estimation by evaluating all comparable patient pairs, calculating concordance statistics and generating validation reports for survival models.
Sources
- Harrell FE Jr, Lee KL, Mark DB. Multivariable Prognostic Models: Issues in Developing Models, Evaluating Assumptions and Measuring and Reducing Errors. Statistics in Medicine. 1996.
- Harrell FE Jr. Regression Modeling Strategies. 2nd ed.
- Uno H, et al. Evaluating Prediction Rules for t-Year Survivors with Censored Regression Models. Journal of the American Statistical Association. 2011.
- Steyerberg EW. Clinical Prediction Models. 2nd ed.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is the Harrell c-index?
A survival model discrimination measure generalising the ROC area under curve to time-to-event data, the probability the model ranks patients correctly.
Source: Harrell, Califf & Pryor 1982
How does the Harrell c-index measure discrimination for survival data?
The Harrell c-index measures how well a survival model discriminates by extending the idea behind the ROC area under curve to time-to-event data. It considers pairs of patients and asks whether the one who experienced the event sooner was given the higher predicted risk, counting the proportion of such comparable pairs the model ranks correctly. It handles censoring by using only pairs whose ordering can be determined despite incomplete follow-up. Judging correct risk ranking over time is what it captures. Collett (2015) discusses such measures.
Source: Collett 2015
How is the Harrell c-index interpreted?
The Harrell c-index is interpreted like the area under the curve: a value of 0.5 indicates discrimination no better than chance, and 1 indicates perfect discrimination, with higher values meaning the model better ranks patients by their risk of experiencing the event sooner. So the Harrell c-index is interpreted as the probability of correctly ordering comparable pairs of patients by predicted risk, with values further above 0.5 indicating better discrimination, providing an overall summary of a survival model's ability to distinguish those at higher and lower risk, though, like other discrimination measures, it does not assess calibration.
Source: Harrell, Califf & Pryor 1982
How does the Harrell c-index handle censoring?
The Harrell c-index handles censoring by considering only pairs of patients whose relative order of events can be determined despite censoring, that is pairs where it is known which patient experienced the event first, and computing concordance over these comparable, or usable, pairs. Pairs whose ordering cannot be determined because of censoring are excluded. So the Harrell c-index accommodates censored survival data by restricting the concordance calculation to comparable pairs, which allows discrimination to be assessed for time-to-event outcomes where not all event times are observed, extending the area under the curve to the survival setting.
Source: Harrell, Califf & Pryor 1982
How does the Harrell c-index relate to the area under the curve?
The Harrell c-index relates to the area under the curve as its generalisation to survival data: for a binary outcome the c-index equals the area under the ROC curve, and for time-to-event data it extends the same concordance concept to comparable pairs while handling censoring. So the Harrell c-index and the area under the curve share the interpretation of a concordance probability, with the c-index being the version suited to survival models, which is why it is used to summarise the discrimination of prognostic models for time-to-event outcomes just as the area under the curve summarises discrimination for binary outcomes.
Source: Hanley & McNeil 1982
What are the limitations of the Harrell c-index?
The limitations of the Harrell c-index include that it measures only discrimination, not calibration, so a model with a high c-index may still give inaccurate predicted risks; that it can be affected by the amount and pattern of censoring; and that, like other discrimination summaries, it may change little when a useful predictor is added to a strong model. So the Harrell c-index is used alongside calibration assessment and other measures, since a high value indicates good ranking of patients by risk but does not ensure accurate absolute predictions, and its dependence on the censoring in the data is considered when comparing values across studies or models.
Source: Harrell, Califf & Pryor 1982
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Dec 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/harrell-c-index
- Term code
- HE-ES-SA-075
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