Concept Architecture
Concept
Theoretically, Loss of Significance is a numerical phenomenon in which substantial precision is lost during arithmetic operations involving nearly equal numbers. Also known as catastrophic cancellation, it arises because leading significant digits cancel during subtraction, leaving a result whose remaining digits are dominated by rounding errors introduced through finite-precision arithmetic. Loss of Significance is a fundamental concept in numerical analysis because it can greatly reduce computational accuracy despite the use of otherwise stable numerical methods.
Mathematically, Loss of Significance occurs when two approximate quantities with similar magnitudes are subtracted, causing relative error in the result to become much larger than the relative errors in the original values. Although the absolute rounding errors remain small, cancellation greatly amplifies their effect on the final computation. Numerical algorithms are therefore often reformulated to minimise cancellation and preserve significant digits.
In practice, Loss of Significance is important in health economics because optimisation routines, regression estimation, probabilistic sensitivity analysis, Markov modelling and simulation algorithms all involve repeated floating-point calculations. Understanding and avoiding catastrophic cancellation improves numerical stability, increases computational reliability and reduces the likelihood of inaccurate model outputs.
Purpose
Used to identify numerical instability caused by cancellation of significant digits, evaluate computational accuracy and support development of stable numerical algorithms for health economic modelling.
Mathematical Formulae
Primary Formula
Cancellation
y = a ? b
where:
a � b
Relative Error
Relative Error = |y ? ?| � |y|
where:
- y = exact result
- ? = computed result
Supporting Formulae
Absolute Error
AE = |y ? ?|
Machine Epsilon
� = smallest positive number such that
1 + � > 1
Related Mathematical Methods
- Floating-point arithmetic
- Finite precision
- Machine precision
- Rounding error
- Numerical stability
- Error propagation
Example
Suppose two computed values are
a = 1000.123456
b = 1000.123455
Their exact difference is
0.000001
Because both numbers are represented with finite precision, rounding errors in the leading digits largely cancel during subtraction, causing the remaining result to contain relatively little reliable information. This is an example of Loss of Significance, in which a numerically small difference is calculated from two much larger, nearly equal quantities.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Formula | =A2-B2 | Calculate the difference between nearly equal values. |
| ABS | =ABS((A2-B2)-C2) | Quantify the resulting numerical error. |
| ROUND | =ROUND(A2,10)-ROUND(B2,10) | Investigate the effect of finite precision on subtraction. |
| IF | =IF(ABS(A2-B2)<1E-8,"Potential Cancellation","Stable") | Identify possible loss of significance. |
| LN | =LN(A2) | Reformulate calculations to improve numerical stability where appropriate. |
VBA (Optional)
Automate detection of catastrophic cancellation, monitor numerical stability throughout health economic models and generate diagnostics identifying calculations susceptible to loss of significance.
Sources
- Higham NJ. Accuracy and Stability of Numerical Algorithms.
- Wilkinson JH. Rounding Errors in Algebraic Processes.
- Goldberg D. What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys.
- Burden RL, Faires JD. Numerical Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (7)
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 loss of significance?
The reduction in numerical accuracy caused by subtracting or combining nearly equal values, resulting in the loss of meaningful digits.
Source: Higham NJ. Accuracy and Stability of Numerical Algorithms. 2nd ed. SIAM; 2002. doi:10.1137/1.9780898718027.
What causes loss of significance?
Loss of significance is caused by subtracting or combining nearly equal values, which cancels their leading digits and leaves a result built mostly from the less reliable trailing digits. Because those leading digits carried most of the accuracy, their cancellation reduces the meaningful digits that remain. This cancellation of nearly equal values is what produces loss of significance Because subtracting close numbers cancels the digits that carried most of the accuracy, the danger is easy to overlook, which is why calculations are arranged to avoid combining nearly equal values.
Source: Higham NJ. Accuracy and Stability of Numerical Algorithms. 2nd ed. SIAM; 2002. doi:10.1137/1.9780898718027.
What happens to accuracy in loss of significance?
In loss of significance, numerical accuracy is reduced, since subtracting or combining nearly equal values strips away meaningful digits and leaves fewer reliable ones in the result. The figure that remains carries less precision than the inputs suggested. This reduction in accuracy through the loss of meaningful digits is the defining effect of loss of significance Because subtracting close numbers cancels the digits that carried most of the accuracy, the danger is easy to overlook, which is why calculations are arranged to avoid combining nearly equal values.
Source: Higham NJ. Accuracy and Stability of Numerical Algorithms. 2nd ed. SIAM; 2002. doi:10.1137/1.9780898718027.
Which operations tend to produce loss of significance?
Operations that subtract or combine nearly equal values tend to produce loss of significance, because the near cancellation removes the shared leading digits and exposes the less accurate remainder. Subtracting two close numbers is the classic case. This link to combining nearly equal values is why such operations are watched for loss of significance Because subtracting close numbers cancels the digits that carried most of the accuracy, the danger is easy to overlook, which is why calculations are arranged to avoid combining nearly equal values.
Source: Higham NJ. Accuracy and Stability of Numerical Algorithms. 2nd ed. SIAM; 2002. doi:10.1137/1.9780898718027.
Why is loss of significance a concern in computation?
Loss of significance is a concern because it reduces numerical accuracy by removing meaningful digits when nearly equal values are subtracted or combined, so a result can be far less precise than its inputs. In a longer calculation this degraded value may then spread. Recognising where nearly equal values are combined helps guard against loss of significance Because subtracting close numbers cancels the digits that carried most of the accuracy, the danger is easy to overlook, which is why calculations are arranged to avoid combining nearly equal values.
Source: Higham NJ. Accuracy and Stability of Numerical Algorithms. 2nd ed. SIAM; 2002. doi:10.1137/1.9780898718027.
How does loss of significance relate to rounding error?
Loss of significance is the reduction in accuracy from subtracting or combining nearly equal values and losing meaningful digits, while rounding error is the difference introduced when a value is rounded to fit a limited number of digits. Loss of significance exposes and magnifies the effect of rounding already present in the values. The two are connected, since finite-precision rounding is what makes the cancelled digits unreliable Because subtracting close numbers cancels the digits that carried most of the accuracy, the danger is easy to overlook, which is why calculations are arranged to avoid combining nearly equal values.
Source: Higham NJ. Accuracy and Stability of Numerical Algorithms. 2nd ed. SIAM; 2002. doi:10.1137/1.9780898718027.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 3 Apr 2026
Content version: 1.0.0
Canonical Identity
- Term code
- CS-NA-ST-001
Stable URI · Machine-readable · Resolvable · CC BY 4.0