Concept Architecture
Concept
Theoretically, Simpson's Rule is a numerical integration method used to approximate the value of a definite integral by replacing the integrand with a sequence of quadratic polynomials. It is founded on numerical analysis and polynomial interpolation, providing greater accuracy than the trapezoidal rule for smooth functions. In health economics, Simpson's Rule is used to estimate cumulative quantities such as life expectancy, quality-adjusted life years, cumulative costs, and expected values when analytical integration is impractical.
Mathematically, Simpson's Rule approximates the area under a curve by fitting a second-order polynomial through successive groups of three equally spaced points. The definite integral is estimated as a weighted sum of function values, with alternating weights of four and two applied to interior points. The approximation error decreases rapidly for sufficiently smooth functions, making the method suitable for many continuous health economic models.
In practice, Simpson's Rule is implemented by partitioning the integration interval into an even number of equal subintervals, evaluating the function at each point, and applying the weighted summation formula. In health economics it is applied to integrate survival curves, estimate accumulated QALYs and costs, evaluate continuous-time disease models, and calculate expected outcomes in decision-analytic and epidemiological models.
Purpose
Used to accurately approximate definite integrals required for estimating cumulative health outcomes, costs, survival measures, and expected values in health economic analyses.
Mathematical Formulae
Primary Formula
??? f(x) dx � (h/3)[f(x?) + 4?????, odd?�?? f(x?) + 2?????, even?�?� f(x?) + f(x?)]
where:
- h = (b ? a)/n
- n is an even number of subintervals
- x? = a + ih
Supporting Formulae
Step size:
h = (b ? a)/n
Exact integral:
I = ??? f(x) dx
Related Mathematical Methods
- Numerical integration
- Trapezoidal rule
- Gaussian quadrature
- Adaptive quadrature
- Polynomial interpolation
Example
A survival model produces annual survival probabilities over four one-year intervals:
| Time (years) | Survival |
|---|---|
| 0 | 1.00 |
| 1 | 0.90 |
| 2 | 0.78 |
| 3 | 0.62 |
| 4 | 0.45 |
Using Simpson's Rule with h = 1:
(1/3)[1.00 + 4(0.90 + 0.62) + 2(0.78) + 0.45] = 3.03
The estimated expected survival over the four-year period is 3.03 life years.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B6,C2:C6) | Calculates the weighted Simpson's Rule summation. |
| SUM | =SUM(D2:D6) | Sums weighted function values before scaling. |
| ROW | =IF(MOD(ROW()-ROW($B$2),2)=1,4,2) | Assigns alternating Simpson weights to interior observations. |
| INDEX | =INDEX(B:B,ROW()) | Retrieves function values for weighted integration calculations. |
VBA (Optional)
A VBA procedure can automate Simpson's Rule integration for survival curves, cumulative cost calculations, or continuous-time model outputs across multiple scenarios.
Sources
- Burden RL, Faires JD. Numerical Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports on decision-analytic modelling.
Related Concepts (4)
Library
Publications
1
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation series.
BookView source →
Frequently Asked Questions (6)
What is Simpson's rule?
A numerical integration technique approximating a definite integral by fitting a series of parabolic curves to the function being integrated.
Source: Press et al. 2007
Who is Simpson's rule named after?
The rule is named after Thomas Simpson, an English mathematician of the eighteenth century, although the underlying idea predates him. It approximates the area under a curve by covering it with a series of small parabolic arcs rather than straight lines, and summing the areas beneath them. Because a parabola follows a curved function more closely than a straight segment does, the method is accurate for smoothly curving functions. It is a standard technique for numerical integration. Press and colleagues (2007) describe it.
Source: Press et al. 2007
How does Simpson's rule work?
Simpson's rule works by dividing the integration interval into an even number of equal subintervals, then, over each pair of adjacent subintervals, fitting a parabola through the function's values at the two ends and the midpoint, and computing the area under that parabola. Summing these areas across all pairs approximates the integral. Because parabolas fit smooth curves better than straight lines, Simpson's rule achieves greater accuracy than the trapezoidal rule for the same number of points, provided the function is reasonably smooth.
Source: Press et al. 2007
How does Simpson's rule compare with the trapezoidal rule?
Simpson's rule and the trapezoidal rule are both numerical integration methods, but they differ in how they approximate the function. The trapezoidal rule uses straight-line segments between points, approximating the area by trapezoids, while Simpson's rule uses parabolas fitted over pairs of intervals. Because parabolas capture curvature that straight lines miss, Simpson's rule is generally more accurate than the trapezoidal rule for smooth functions with the same number of points, at the cost of slightly more computation per interval.
Source: Press et al. 2007
When is Simpson's rule accurate?
Simpson's rule is accurate for smooth functions that are 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 very many intervals are used. Its accuracy therefore depends on the smoothness of the function and the fineness of the subdivision, being well suited to smooth integrands.
Source: Press et al. 2007
Where is Simpson's rule used?
Simpson's rule 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: 3 Oct 2025
Content version: 1.0.0
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