Concept Architecture
Concept
Theoretically, Area Under the Curve (AUC) is a mathematical measure obtained by integrating a function over a specified interval. It represents the accumulated magnitude of a variable across a continuous domain and is founded on integral calculus. In health economics and biomedical research, AUC is widely used to quantify cumulative exposure, overall model performance or total health outcomes over time, depending on the context.
Mathematically, the Area Under the Curve is represented by the definite integral of a continuous function over a defined interval. When only discrete observations are available, the area is commonly approximated using numerical integration methods, most frequently the trapezoidal rule. The resulting value retains the units of the dependent variable multiplied by the units of the independent variable.
In practice, Area Under the Curve is estimated from observed measurements, survival curves, concentration-time profiles, Receiver Operating Characteristic curves and health economic models. It is widely applied in pharmacokinetics to estimate drug exposure, in diagnostic evaluation to quantify test discrimination and in health economics to calculate cumulative costs, health outcomes and quality-adjusted life-years.
Purpose
Used to quantify cumulative outcomes over a continuous interval, estimate total exposure or benefit, evaluate diagnostic discrimination and support pharmacokinetic, clinical and health economic analyses.
Mathematical Formulae
Primary Formula
AUC = ??? f(x) dx
where:
- AUC = area under the curve
- f(x) = function of interest
- a, b = integration limits
Supporting Formulae
Trapezoidal Rule:
AUC � ?[(y? + y???) / 2] ? (x??? ? x?)
Receiver Operating Characteristic:
AUC = ??? TPR(FPR) d(FPR)
where:
- TPR = true positive rate
- FPR = false positive rate
Related Mathematical Methods
- Definite Integral
- Trapezoidal Rule
- Numerical Integration
- Receiver Operating Characteristic Analysis
- Pharmacokinetic Analysis
- Survival Analysis
- Quality-Adjusted Life-Year Calculation
Example
Drug concentrations are measured at four time points.
| Time (hours) | Concentration (mg/L) |
|---|---|
| 0 | 0 |
| 2 | 10 |
| 4 | 6 |
| 6 | 2 |
Using the trapezoidal rule:
AUC = [(0 + 10)/2 ? 2] + [(10 + 6)/2 ? 2] + [(6 + 2)/2 ? 2]
AUC = 10 + 16 + 8 = 34 mg�hour/L
The cumulative drug exposure over six hours is 34 mg�hour/L.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(((B2:B4+B3:B5)/2),(A3:A5-A2:A4)) | Calculates AUC using the trapezoidal rule from observed measurements. |
| SUM | =SUM(C2:C10) | Sums trapezoidal areas to obtain total AUC. |
| IF | =IF(B3>B2,((B2+B3)/2)*(A3-A2),0) | Calculates individual trapezoidal segments when values are valid. |
| ROUND | =ROUND(D2,2) | Formats the final AUC estimate for reporting. |
VBA (Optional)
A VBA macro can automatically calculate Area Under the Curve using trapezoidal integration for pharmacokinetic profiles, survival curves or health economic outcome trajectories.
Sources
- Gibaldi M, Perrier D. Pharmacokinetics. 2nd ed.
- Hanley JA, McNeil BJ. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology. 1982;143(1):29?36.
- Pepe MS. The Statistical Evaluation of Medical Tests for Classification and Prediction.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
Related Concepts (3)
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 area under the curve?
A summary statistic representing a predictive model's discriminative ability, the area beneath its ROC curve, from 0.5 for chance to 1 for perfect.
Source: Hanley & McNeil 1982
What does the area under the curve say about a model's discrimination?
The area under the curve summarises how well a predictive model separates those who will have an outcome from those who will not, measured as the area beneath its ROC curve. It runs from 0.5, meaning the model does no better than chance, to 1, meaning it discriminates perfectly. Concretely it is the probability that the model assigns a higher risk to a randomly chosen case than to a randomly chosen non-case. How well a model tells cases from non-cases is what it captures. Steyerberg (2009) discusses such measures.
Source: Steyerberg 2009
How is the area under the curve interpreted?
The area under the curve is interpreted as the probability that the model assigns a higher predicted risk, or score, to a randomly chosen person with the condition than to a randomly chosen person without it. A value of 0.5 indicates no discrimination, better than chance, and 1 perfect discrimination, with intermediate values indicating moderate ability. So the area under the curve is interpreted as a measure of discrimination, with higher values meaning the model better separates cases from non-cases, and it provides an overall summary of discriminative performance across thresholds, though it does not assess calibration or the clinical usefulness at a specific threshold.
Source: Hanley & McNeil 1982
What does the area under the curve measure?
The area under the curve measures discrimination, the ability of a model or test to distinguish those who have the condition or outcome from those who do not, across all possible decision thresholds. It summarises the ROC curve, which shows the trade-off between sensitivity and the false positive rate as the threshold varies. So the area under the curve measures how well a model ranks individuals by their likelihood of the outcome, capturing discrimination independently of any single cut-off, which makes it a widely used summary of a predictive model's or diagnostic test's ability to separate cases from non-cases, though it addresses only discrimination and not other aspects of performance.
Source: Hanley & McNeil 1982
What are the limitations of the area under the curve?
The limitations of the area under the curve include that it measures only discrimination, not calibration, so a model can have a high area under the curve yet produce poorly calibrated probabilities; that it summarises performance across all thresholds, which may not reflect usefulness at the threshold relevant in practice; and that it can be insensitive to improvements from adding predictors. So the area under the curve is used alongside other measures, such as calibration assessment and clinical usefulness at relevant thresholds, since a high value indicates good separation of cases from non-cases but does not guarantee that the predicted probabilities are accurate or that the model is useful for a specific decision.
Source: Hanley & McNeil 1982
How does the area under the curve relate to the ROC curve?
The area under the curve is derived directly from the receiver operating characteristic, or ROC, curve, being the area beneath it. The ROC curve plots the true positive rate against the false positive rate as the decision threshold varies, showing the trade-off between sensitivity and specificity, and the area summarises this whole curve into a single number. So the area under the curve relates to the ROC curve as its summary measure, condensing the information across all thresholds into one value of discrimination, which allows models or tests to be compared without choosing a particular threshold, while the ROC curve itself displays the full range of trade-offs.
Source: Hanley & McNeil 1982
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 11 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/area-under-curve
- Term code
- HE-ES-SA-006
Stable URI · Machine-readable · Resolvable · CC BY 4.0