Concept Architecture
Concept
Theoretically, Simpson Method is a numerical integration technique that approximates the value of a definite integral by replacing the integrand over successive intervals with quadratic polynomials. The method is based on Newton?Cotes numerical integration and generally provides greater accuracy than the trapezoidal rule when the function is sufficiently smooth. In health economics, Simpson Method is used when analytical integration is impractical, for example when calculating cumulative costs, health outcomes or survival measures from continuous functions.
Mathematically, Simpson Method approximates the integral by fitting a second-degree polynomial through three equally spaced points and integrating the polynomial exactly over each pair of subintervals. Composite Simpson Method extends this approach across multiple intervals, requiring an even number of subintervals. The approximation error decreases rapidly as the interval width becomes smaller and is proportional to the fourth derivative of the underlying function.
In practice, Simpson Method is implemented whenever continuous functions must be integrated numerically. Health economists apply the method to estimate cumulative hazard functions, life expectancy, quality-adjusted survival, discounted costs and other quantities derived from continuous models when closed-form solutions are unavailable.
Purpose
Used to approximate definite integrals accurately when analytical integration is unavailable, particularly for estimating cumulative health outcomes, costs and survival measures in health economic analyses.
Mathematical Formulae
Primary Formula
For a single application of Simpson's Rule,
??? f(x) dx � (b ? a)/6 [f(a) + 4f((a + b)/2) + f(b)]
Supporting Formulae
Composite Simpson Method:
??? f(x) dx � (h/3) [f(x?) + 4�???,?,?,??? f(x?) + 2�???,?,?,??? f(x?) + f(x?)]
where
h = (b ? a)/n
and n is an even number of subintervals.
Related Mathematical Methods
- Newton?Cotes quadrature
- Numerical integration
- Trapezoidal rule
- Gaussian quadrature
- Adaptive quadrature
Example
A health economist estimates cumulative undiscounted costs over one year from quarterly observations:
| Time (years) | Cost rate (�/year) |
|---|---|
| 0.00 | 1,000 |
| 0.25 | 1,200 |
| 0.50 | 1,500 |
Applying Simpson Method over the interval:
???�? f(t) dt � (0.5/6)(1000 + 4 ? 1200 + 1500)
= (0.5/6)(7300) = 608.3
The estimated cumulative cost over the first six months is approximately �608.30.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(B2:B10) | Aggregates weighted function values. |
| PRODUCT | =(H1/3)*H2 | Calculates the composite Simpson estimate, where H1 contains the interval width and H2 contains the weighted sum. |
| INDEX | =INDEX(B2:B10,ROW(A1)*2) | Retrieves alternating function values for Simpson weighting. |
| IF | =IF(MOD(A2,2)=0,2,4) | Applies Simpson weights to interior observations. |
VBA (Optional)
Automate composite Simpson integration for continuous cost, utility or survival functions using user-defined interval widths.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Burden RL, Faires JD. Numerical Analysis.
- Atkinson KE. An Introduction to Numerical Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
Related Concepts (4)
Library
Publications
1
Modeling Good Research Practices — Overview: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-1 — Caro, Briggs, Siebert & Kuntz, Task Force Report 1 ed., 2012 (Value in Health / Medical Decision Making)
The overview paper of the seven-part ISPOR-SMDM modelling good-practice series, setting out best-practice recommendations across model design, technique selection, implementation, validation, parameterisation, uncertainty and use in decision making.
Journal ArticleView source →
Frequently Asked Questions (6)
What is Simpson's method?
A numerical integration technique approximating a definite integral by fitting a series of parabolic segments to the function, more accurate than linear approximation.
Source: Press et al. 2007
When does Simpson's method lose accuracy?
Simpson's method fits smooth parabolic arcs to a function, which works well when the function itself is smooth, but its accuracy falls where the function bends sharply, jumps, or is not well behaved between the sampled points. Over such regions a parabola cannot follow the true shape, and the estimate drifts. Using more, closer points restores accuracy by shrinking the span each arc must cover. The method therefore suits smooth integrands and struggles with irregular ones. Press and colleagues (2007) note this limitation.
Source: Press et al. 2007
How does Simpson's method work?
Simpson's method works by dividing the integration interval into an even number of equal subintervals, fitting a parabola through the function's values at the ends and midpoint of each pair of subintervals, and computing the area under that parabola, then summing these areas across all pairs. Because parabolas fit smooth curves better than straight lines, this gives greater accuracy than the trapezoidal rule for the same number of points, provided the function is reasonably smooth, so the function need only be evaluated at the chosen points.
Source: Press et al. 2007
How does Simpson's method compare with the trapezoidal rule?
Simpson's method and the trapezoidal rule are both numerical integration methods, but they approximate the function differently: the trapezoidal rule uses straight-line segments, approximating the area by trapezoids, while Simpson's method uses parabolas over pairs of intervals. Because parabolas capture curvature that straight lines miss, Simpson's method is generally more accurate for smooth functions with the same number of points, at the cost of slightly more computation per interval, making it preferred where greater accuracy is wanted for well-behaved integrands.
Source: Press et al. 2007
When is Simpson's method accurate?
Simpson's method is accurate for smooth functions well approximated by parabolas over each pair of intervals, and its accuracy improves as the number of subintervals increases; it is exact for polynomials up to the third degree. For functions with sharp features, discontinuities, or rapid changes, the parabolic approximation may be poor unless many intervals are used. Its accuracy therefore depends on the smoothness of the function and the fineness of the subdivision, making it well suited to smooth integrands.
Source: Press et al. 2007
Where is Simpson's method used?
Simpson's method is used to approximate definite integrals in statistics, modelling, and other computations where an analytical solution is unavailable and the function is smooth, such as integrating over a distribution, computing an expected value, or finding an area under a curve. It offers a good balance of accuracy and simplicity for one-dimensional integrals of well-behaved functions. In health economic and statistical calculations, it is one of the standard numerical integration methods chosen for its accuracy on smooth integrands.
Source: Press et al. 2007
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 17 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/simpson-method
- Term code
- HE-EM-MV-071
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