Concept Architecture
Concept
Theoretically, Numerical Integration is a numerical analysis method used to approximate the value of a definite integral when an analytical solution is unavailable or impractical. It is founded on the principle of replacing a continuous function with a finite approximation whose integral can be calculated directly. The method exists to evaluate cumulative quantities represented by integrals, including expected costs, health outcomes, survival functions, probability distributions, and transition processes commonly encountered in health economic modelling.
Mathematically, numerical integration approximates the definite integral of a function over a specified interval by summing weighted function evaluations at selected points. Numerous recognised methods exist, including the trapezoidal rule, Simpson's rule, Gaussian quadrature, and adaptive quadrature. The choice of method determines the approximation accuracy and computational efficiency according to the characteristics of the integrand.
In practice, numerical integration is implemented by discretising the integration interval, evaluating the function at predetermined points, and combining these values using the selected numerical integration rule. In health economics it is applied to calculate life expectancy, quality-adjusted life years, cumulative costs, expected values, survival functions, continuous-time Markov models, pharmacokinetic models, and probability distributions within decision-analytic models.
Purpose
Used to approximate definite integrals required for estimating cumulative costs, health outcomes, survival measures, transition probabilities, and expected values in health economic models.
Mathematical Formulae
Primary Formula
The definite integral is
I = ??? f(x) dx
which is approximated numerically by
I � ????� w?f(x?)
where:
- w? = numerical integration weights
- x? = evaluation points
- f(x?) = function values
Supporting Formulae
Trapezoidal rule:
??? f(x) dx � (h/2)[f(x?) + 2????�??f(x?) + f(x?)]
Simpson's rule:
??? f(x) dx � (h/3)[f(x?) + 4??odd?f(x?) + 2??even?f(x?) + f(x?)]
Related Mathematical Methods
- Trapezoidal rule
- Simpson's rule
- Gaussian quadrature
- Adaptive quadrature
- Numerical approximation
- Monte Carlo integration
Example
A survival model estimates the probability of remaining alive over five years. The expected life years are calculated by integrating the survival curve.
Suppose annual survival probabilities are:
| Time (years) | Survival |
|---|---|
| 0 | 1.00 |
| 1 | 0.92 |
| 2 | 0.84 |
| 3 | 0.74 |
| 4 | 0.63 |
| 5 | 0.50 |
Using the trapezoidal rule,
(1/2)[1.00 + 2(0.92 + 0.84 + 0.74 + 0.63) + 0.50] = 3.88
The estimated expected survival is 3.88 life years over the five-year period.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(C3:C6) | Sums intermediate function values for numerical integration. |
| SUMPRODUCT | =SUMPRODUCT(B2:B7,C2:C7) | Calculates weighted numerical integration approximations. |
| OFFSET | =AVERAGE(B2:B3) | Calculates adjacent function averages for trapezoidal integration. |
| Solver Add-in | Optimises integration parameters where required | Supports numerical estimation within complex health economic models. |
VBA (Optional)
A VBA procedure can automate numerical integration across survival curves or continuous-time model outputs using trapezoidal, Simpson's, or Gaussian quadrature methods.
Sources
- 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.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes.
- Burden RL, Faires JD. Numerical Analysis.
- 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 numerical integration?
Techniques for approximating the value of a definite integral when an exact analytical solution is unavailable or impractical to derive.
Source: Press et al. 2007
What does numerical integration approximate?
Numerical integration approximates the area under a curve, the value of a definite integral, using arithmetic on function values rather than an exact formula. It is used when the integral cannot be solved algebraically, which is common for the expected values and accumulated quantities that arise in economic models. By sampling the function at chosen points and combining the results, it yields a numerical answer close enough for practical purposes. The accuracy improves with more or better-placed sample points. Press and colleagues (2007) describe these methods.
Source: Press et al. 2007
Why is numerical integration needed?
Numerical integration is needed because many integrals arising in practice have no closed-form solution or are too complex to derive analytically, yet their values are required for calculations such as computing expected values, probabilities, or areas. In statistics and modelling, integrals over probability distributions or model outputs often cannot be solved exactly, so they must be approximated numerically. Numerical integration provides accurate estimates of such integrals, making feasible computations that would otherwise be impossible analytically.
Source: Press et al. 2007
What are common methods of numerical integration?
Common methods include the trapezoidal rule, which approximates the area under a function by trapezoids between evaluation points; Simpson's rule, which fits parabolas for greater accuracy; and Gaussian quadrature, which chooses points and weights optimally for high accuracy with few evaluations. For high-dimensional integrals, simulation-based methods such as Monte Carlo integration are used. Each method trades accuracy against the number of function evaluations, and the choice depends on the function, the dimension, and the accuracy required.
Source: Press et al. 2007
How accurate is numerical integration?
The accuracy of numerical integration depends on the method, the number and placement of evaluation points, and the smoothness of the function. Smoother functions and more points generally give more accurate results, and methods such as Gaussian quadrature achieve high accuracy with few evaluations for well-behaved functions. Functions with sharp features or discontinuities are harder to integrate accurately. Adaptive methods adjust the points to control error, so accuracy can usually be improved at the cost of more evaluations.
Source: Press et al. 2007
Where is numerical integration used in health economics?
In health economics and statistics, numerical integration is used to compute integrals that lack analytical solutions, such as averaging outcomes over a distribution of parameters or patient characteristics, computing expected values, evaluating likelihoods in estimation, and calculating areas such as under survival curves. Health economic models often require integrating over uncertainty or heterogeneity, and numerical integration provides the estimates. It is one of several computational tools that make the analysis of such models feasible.
Source: Press et al. 2007
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Oct 2025
Content version: 1.0.0
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