Concept Architecture
Concept
Theoretically, Root Mean Square Error is a measure of predictive accuracy that quantifies the typical magnitude of differences between observed and predicted values. It is derived from squared-error loss and assigns greater influence to large errors because deviations are squared before aggregation. Taking the square root returns the measure to the original units of the outcome, allowing direct interpretation alongside the observed variable.
Mathematically, Root Mean Square Error is the square root of the mean of the squared prediction errors. Each error is calculated as the difference between an observed value and its corresponding prediction. The measure is non-negative, equals zero only when all predictions are exact, and is more sensitive to large errors than Mean Absolute Error.
In practice, Root Mean Square Error is calculated from observed and predicted values in a validation sample, cross-validation fold, or external dataset. In health economics it is used to assess models predicting healthcare costs, quality-adjusted life years, resource utilisation, treatment outcomes, disease progression, and other continuous economic or clinical outcomes.
Purpose
Used to quantify predictive error in the original units of an outcome, compare competing prediction models, evaluate model calibration and validation, and identify models producing materially large prediction errors.
Mathematical Formulae
Primary Formula
RMSE = �[(1/n) ? ?(y? ? ??)�]
where:
- n = number of observations
- y? = observed value for observation i
- ?? = predicted value for observation i
Supporting Formulae
e? = y? ? ??
MSE = (1/n) ? ?(y? ? ??)�
RMSE = �MSE
Related Mathematical Methods
- Prediction Error
- Mean Squared Error
- Mean Absolute Error
- Residual Analysis
- Cross-Validation
- Model Validation
- Ordinary Least Squares
Example
A health economic model predicts annual healthcare costs for five patients.
| Patient | Observed Cost | Predicted Cost | Squared Error |
|---|---|---|---|
| 1 | �8,200 | �8,000 | 40,000 |
| 2 | �10,100 | �10,400 | 90,000 |
| 3 | �7,500 | �7,600 | 10,000 |
| 4 | �9,300 | �9,100 | 40,000 |
| 5 | �8,900 | �9,000 | 10,000 |
MSE = (40,000 + 90,000 + 10,000 + 40,000 + 10,000) � 5
MSE = 38,000
RMSE = �38,000
RMSE = �194.94
The model's predictions differ from observed annual healthcare costs by approximately �195 in root mean square terms.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SQRT | =SQRT(AVERAGE(D2:D101)) | Calculate RMSE from a column of squared cost or outcome errors. |
| SUMXMY2 | =SQRT(SUMXMY2(B2:B101,C2:C101)/COUNT(B2:B101)) | Calculate RMSE directly from observed and predicted values. |
| POWER | =POWER(B2-C2,2) | Calculate the squared prediction error for each observation. |
| AVERAGE | =AVERAGE(D2:D101) | Calculate Mean Squared Error before taking its square root. |
| COUNT | =COUNT(B2:B101) | Determine the number of paired observations included in the calculation. |
VBA (Optional)
Automate RMSE calculation across competing health economic models, outcomes, and validation samples.
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.
- Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning.
- James G, Witten D, Hastie T, Tibshirani R. An Introduction to Statistical Learning.
- Steyerberg EW. Clinical Prediction Models.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluations: The Manual.
Related Concepts (3)
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 root mean square error?
A measure of average prediction error, the square root of the average squared difference between predicted and observed values, in the outcome's units.
Source: Willmott & Matsuura 2005
What does the root mean square error express about prediction accuracy?
The root mean square error expresses the typical size of a model's prediction errors, calculated as the square root of the average squared difference between predicted and observed values. Taking the square root returns the measure to the outcome's own units, so it can be read as a representative error, while the squaring beforehand makes it weigh large misses more heavily than the mean absolute error does. This makes it sensitive to occasional big errors. The typical prediction miss, in the outcome's units, is what it captures. Steyerberg (2009) discusses such measures.
Source: Steyerberg 2009
How is the root mean square error calculated?
The root mean square error is calculated by taking the difference between each predicted and observed value, squaring it, averaging the squared differences, and then taking the square root of that average. So the root mean square error is calculated as the square root of the mean squared error, which expresses the error in the original units of the outcome and, through the squaring, gives disproportionate weight to larger errors, making it sensitive to outliers, and this is why it conveys the typical size of the prediction error on the data's scale while emphasising the larger deviations.
Source: Willmott & Matsuura 2005
How does the root mean square error differ from the mean absolute error?
The root mean square error is the square root of the average squared error, giving greater weight to large errors through the squaring, while the mean absolute error is the average of the absolute errors, weighting all errors by their size linearly. The root mean square error is more sensitive to outliers, and the mean absolute error more robust. So the two differ in how they weight errors, with the root mean square error emphasising large errors and the mean absolute error treating them proportionally, which is why the root mean square error is preferred when large errors are especially undesirable and the mean absolute error when robustness to outliers is wanted.
Source: Willmott & Matsuura 2005
When is the root mean square error used?
The root mean square error is used to assess and compare the accuracy of predictions on the original scale of the outcome, particularly when larger errors are especially important and should be penalised more heavily. So the root mean square error is used for evaluating predictive models where the emphasis on large errors is appropriate and interpretability in the data's units is wanted, which is why it is a common measure of prediction accuracy in regression and forecasting, though its sensitivity to outliers means the mean absolute error may be preferred when a more robust measure is desired, so the choice depends on the purpose.
Source: Willmott & Matsuura 2005
How is the root mean square error interpreted?
The root mean square error is interpreted as the typical magnitude of the prediction errors in the units of the outcome, with a smaller value indicating more accurate predictions and zero indicating perfect prediction, while its emphasis on large errors means it is raised more by occasional big errors than the mean absolute error. So the root mean square error is interpreted as an average error on the original scale, weighted toward larger deviations, which makes it informative about overall accuracy and about the presence of large errors, and comparing root mean square errors across models identifies those with better predictive performance, especially where large errors matter.
Source: Willmott & Matsuura 2005
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-187
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