Concept Architecture
Concept
Theoretically, Error Propagation is the mathematical process by which uncertainty or measurement error in input variables is transmitted through a mathematical model to produce uncertainty in the calculated output. It is founded on differential calculus, probability theory and statistical estimation and provides a framework for quantifying the effect of imperfect measurements on derived quantities. Error propagation is fundamental to uncertainty analysis, numerical methods and scientific computation.
Mathematically, Error Propagation is commonly approximated using a first-order Taylor series expansion, in which the variance of a derived quantity is expressed as a function of the variances and covariances of its inputs. For independent variables, the propagated variance is obtained by summing the squared partial derivatives of the function with respect to each input, weighted by the corresponding input variances. This approach provides an efficient approximation when measurement errors are relatively small.
In practice, Error Propagation is widely applied in health economics to quantify uncertainty in decision models, cost-effectiveness analyses, epidemiological estimates and simulation studies. It supports probabilistic sensitivity analysis, model validation and interpretation of uncertainty surrounding economic outcomes, enabling more informed healthcare decision-making.
Purpose
Used to quantify how uncertainty in input variables affects calculated outputs, evaluate model reliability, support uncertainty analysis and improve interpretation of health economic and statistical results.
Mathematical Formulae
Primary Formula
For independent variables,
Var(f) � �[(?f/?x?)� ? Var(x?)]
where:
- f = calculated function
- x? = input variables
- ?f/?x? = partial derivative of the function with respect to input i
- Var(x?) = variance of input i
Supporting Formulae
Standard Uncertainty
u(f) = �Var(f)
General Propagation Formula
Var(f) � ��[(?f/?x?)(?f/?x?)Cov(x?,x?)]
where Cov(x?,x?) is the covariance between inputs i and j.
Related Mathematical Methods
- Taylor series approximation
- Delta method
- Variance estimation
- Sensitivity analysis
- Monte Carlo simulation
- Probabilistic sensitivity analysis
Example
Suppose total treatment cost is calculated as:
Cost = Unit Cost ? Quantity
where:
- Unit Cost = �100 � �5
- Quantity = 20 � 1
Using first-order error propagation,
Var(Cost) � (20� ? 5�) + (100� ? 1�)
= 400 ? 25 + 10,000
= 20,000
Standard uncertainty
= �20,000
� �141.4
The calculated treatment cost therefore has an estimated uncertainty of approximately �141, reflecting uncertainty in both inputs.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SQRT | =SQRT(B2) | Calculate propagated standard uncertainty from variance. |
| POWER | =POWER(A2,2) | Calculate squared partial derivative terms. |
| SUM | =SUM(C2:C10) | Sum variance contributions from independent inputs. |
| MMULT | =MMULT(MMULT(Jacobian,Covariance),TRANSPOSE(Jacobian)) | Calculate propagated variance using the covariance matrix. |
| RAND | =RAND() | Support Monte Carlo simulation for uncertainty propagation. |
VBA (Optional)
Automate propagation of uncertainty through health economic models, calculate propagated variances and standard uncertainties, and generate uncertainty analysis and probabilistic sensitivity analysis reports.
Sources
- Bevington PR, Robinson DK. Data Reduction and Error Analysis for the Physical Sciences.
- Ku HH. Notes on the Use of Propagation of Error Formulas. Journal of Research of the National Bureau of Standards.
- Taylor JR. An Introduction to Error Analysis.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Morgan MG, Henrion M. Uncertainty: A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
Related Concepts (6)
Library
Publications
1
Accuracy and Stability of Numerical Algorithms — Nicholas J. Higham, 2nd Edition ed., 2002 (Society for Industrial and Applied Mathematics (SIAM))
The standard reference on floating-point error, conditioning, stability and the numerical reliability of matrix, polynomial and linear-algebra algorithms.
BookView source →
Frequently Asked Questions (6)
What is error propagation?
The process by which uncertainties or numerical errors accumulate and influence the results of subsequent calculations.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does error propagation describe?
Error propagation describes the process by which uncertainties or numerical errors accumulate and influence the results of subsequent calculations. An error present at one stage does not stay isolated but is carried forward, affecting everything computed from that value. Tracing this spread of error through a sequence of calculations is what error propagation captures Because an early error is carried into everything computed from it, a long calculation can end far less accurate than its inputs suggest unless the way errors spread is tracked.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How do errors spread in error propagation?
In error propagation, uncertainties or numerical errors accumulate and influence the results of subsequent calculations, so an error introduced early is carried into every value derived from it. Each calculation that uses an affected quantity passes the error along, and errors may combine as they go. This accumulation and onward influence is the mechanism error propagation describes Because an early error is carried into everything computed from it, a long calculation can end far less accurate than its inputs suggest unless the way errors spread is tracked.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Why does error propagation matter in a chain of calculations?
Error propagation matters in a chain of calculations because uncertainties or numerical errors accumulate and influence subsequent results, so a small initial error can grow or spread as it is carried forward. Later results may be far less accurate than the inputs suggest if propagation is ignored. Understanding error propagation therefore helps judge how trustworthy the final result of a long calculation is Because an early error is carried into everything computed from it, a long calculation can end far less accurate than its inputs suggest unless the way errors spread is tracked.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What can error propagation do to the accuracy of a result?
Error propagation can reduce the accuracy of a result, because the uncertainties or numerical errors it describes accumulate and influence the results of subsequent calculations. As errors are carried forward and combine, the final figure may carry more uncertainty than any single input. This potential degradation of accuracy is why error propagation is tracked through a computation Because an early error is carried into everything computed from it, a long calculation can end far less accurate than its inputs suggest unless the way errors spread is tracked.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does error propagation relate to rounding error?
Error propagation is the process by which uncertainties or numerical errors accumulate and influence subsequent calculations, while rounding error is the difference introduced when a numerical value is rounded to fit a limited number of digits. Rounding error is one source of the errors that then propagate. The two are connected, since rounding introduces errors that error propagation carries forward through later calculations Because an early error is carried into everything computed from it, a long calculation can end far less accurate than its inputs suggest unless the way errors spread is tracked.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Apr 2026
Content version: 1.0.0
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