Concept Architecture
Concept
Theoretically, Mean Absolute Error (MAE) is a measure of predictive accuracy that quantifies the average magnitude of errors between observed and predicted values without considering their direction. It is based on the arithmetic mean of the absolute prediction errors and provides an intuitive measure of model performance expressed in the same units as the outcome variable. MAE exists to evaluate forecasting and prediction models while treating all prediction errors equally, regardless of whether they represent overestimation or underestimation.
Mathematically, Mean Absolute Error is calculated as the average of the absolute differences between observed and predicted values. Unlike measures based on squared errors, MAE does not disproportionately penalise large errors, making it less sensitive to outliers. It is widely used as an objective function in optimisation and as a performance metric for regression, forecasting and machine learning models.
In practice, MAE is calculated by comparing model predictions with observed outcomes in validation datasets. In health economics, it is used to evaluate prediction models for healthcare costs, quality-adjusted life years, hospital admissions, disease progression and resource utilisation. Lower MAE values indicate greater predictive accuracy and are commonly used when comparing alternative economic or statistical models.
Purpose
Used to quantify average prediction error, compare predictive models, assess forecasting accuracy and evaluate regression and machine learning models in health economic research.
Mathematical Formulae
Primary Formula
MAE = (1/n) ? ????� |y? ? ??|
where:
- n = number of observations
- y? = observed value
- ?? = predicted value
Supporting Formulae
Prediction error:
e? = y? ? ??
Absolute error:
|e?| = |y? ? ??|
Related Mathematical Methods
- Mean Squared Error
- Root Mean Squared Error
- Mean Absolute Percentage Error
- Median Absolute Error
- Cross-Validation
- Regression Diagnostics
- Model Validation
Example
A health economist develops a model to predict annual healthcare costs for five patients.
| Patient | Observed (�) | Predicted (�) | Absolute Error (�) |
|---|---|---|---|
| 1 | 4,800 | 4,600 | 200 |
| 2 | 6,300 | 6,500 | 200 |
| 3 | 5,700 | 5,400 | 300 |
| 4 | 7,100 | 6,900 | 200 |
| 5 | 5,100 | 5,400 | 300 |
Mean Absolute Error:
MAE = (200 + 200 + 300 + 200 + 300) � 5
MAE = �240
The model therefore predicts annual healthcare costs with an average absolute error of �240 per patient.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| ABS | =ABS(B2-C2) | Calculate the absolute prediction error for each observation. |
| AVERAGE | =AVERAGE(D2:D501) | Calculate the Mean Absolute Error across all observations. |
| SUM | =SUM(D2:D501) | Calculate the total absolute prediction error. |
| COUNT | =COUNT(D2:D501) | Determine the number of observations contributing to MAE. |
| IF | =IF(D2<500,"Acceptable","Review Model") | Flag predictions exceeding a predefined error threshold. |
VBA (Optional)
A VBA routine can automatically calculate Mean Absolute Error for multiple predictive models and generate comparative model performance reports.
Sources
- Willmott CJ, Matsuura K. Advantages of the Mean Absolute Error over the Root Mean Square Error in Assessing Average Model Performance. Climate Research. 2005;30:79?82.
- Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning. Springer.
- James G, Witten D, Hastie T, Tibshirani R. An Introduction to Statistical Learning. Springer.
- Kuhn M, Johnson K. Applied Predictive Modeling. Springer.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- ISPOR Good Practice Reports.
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 mean absolute error?
A measure of average prediction error magnitude between predicted and actual values, calculated without regard to whether errors are positive or negative.
Source: Willmott & Matsuura 2005
What does the mean absolute error measure about predictions?
The mean absolute error measures how far a model's predictions fall from the actual values on average, by taking the size of each error, ignoring its sign, and averaging across all cases. It expresses the typical magnitude of a prediction's miss in the original units, which makes it easy to interpret. Because it treats every error in proportion to its size, it is less swayed by a few large misses than the mean squared error, which squares them. The average size of a prediction's error is what it captures. Steyerberg (2009) discusses such measures.
Source: Steyerberg 2009
How is the mean absolute error calculated?
The mean absolute error is calculated by taking the absolute value of the difference between each predicted and actual value, summing these absolute errors, and dividing by the number of observations, giving the average absolute error. So the mean absolute error is calculated as the average of the absolute prediction errors, which measures the typical size of the errors on the original scale of the data, treating over- and under-predictions symmetrically and weighting each error by its magnitude alone, in contrast to the mean squared error, which squares the errors and so gives greater weight to larger ones.
Source: Willmott & Matsuura 2005
How does the mean absolute error differ from the mean squared error?
The mean absolute error averages the absolute differences between predictions and actual values, weighting all errors by their size, while the mean squared error averages the squared differences, giving disproportionately greater weight to larger errors. The mean absolute error is on the original scale and less sensitive to outliers, whereas the mean squared error penalises large errors more heavily. So the two differ in how they weight errors, with the mean absolute error treating them linearly and the mean squared error emphasising large ones through squaring, which makes the mean absolute error more robust to outliers and the mean squared error more sensitive to them.
Source: Casella & Berger 2002
When is the mean absolute error preferred?
The mean absolute error is preferred when a measure of average error on the original scale is wanted that is not overly influenced by a few large errors, since it weights all errors by their magnitude rather than squaring them, making it more robust to outliers than the mean squared error. So the mean absolute error is preferred when robustness to large errors and interpretability on the data's scale are valued, which is why it is used where outliers should not dominate the error measure, though the mean squared error may be preferred when large errors are especially undesirable and should be penalised more heavily, so the choice depends on the purpose.
Source: Willmott & Matsuura 2005
How is the mean absolute error interpreted?
The mean absolute error is interpreted as the average absolute size of the prediction errors, on the same scale as the data, so a smaller value indicates more accurate predictions and a value of zero perfect prediction. It directly conveys the typical magnitude of error. So the mean absolute error is interpreted straightforwardly as how far, on average, predictions deviate from actual values, with lower values meaning better accuracy, which makes it easy to understand and communicate, since it is expressed in the units of the outcome and represents the average error a prediction can be expected to have, ignoring the direction of the errors.
Source: Willmott & Matsuura 2005
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Verified by Dr Darrin Baines
British health economist
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Verification date: 18 Dec 2025
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