Concept Architecture
Concept
Theoretically, Newton-Raphson Method is an iterative numerical optimisation and root-finding method used to solve nonlinear equations by successively approximating the roots of a differentiable function. It is founded on first-order Taylor series expansion and exists to efficiently identify parameter values that satisfy nonlinear equations or optimise differentiable objective functions. In health economics it is commonly used in maximum likelihood estimation, nonlinear model calibration, and parameter estimation where analytical solutions are unavailable.
Mathematically, the Newton-Raphson method generates a sequence of parameter estimates by using the first derivative of the objective function to determine the direction and magnitude of each update. For optimisation problems, second derivatives or the Hessian matrix are incorporated to improve convergence towards a local optimum. Under appropriate regularity conditions and sufficiently accurate starting values, the method exhibits quadratic convergence.
In practice, the Newton-Raphson method is implemented by defining an objective or estimating equation, calculating the required derivatives, selecting initial parameter values, and iteratively updating estimates until convergence criteria are satisfied. In health economics it is applied in nonlinear regression, maximum likelihood estimation, survival analysis, calibration of decision models, and estimation of parameters within statistical and economic evaluation models.
Purpose
Used to solve nonlinear equations and estimate model parameters efficiently through iterative derivative-based optimisation in health economic modelling and statistical estimation.
Mathematical Formulae
Primary Formula
For root finding:
x??? = x? ? f(x?)/f?(x?)
where:
- x? = current estimate
- f(x) = nonlinear function
- f?(x) = first derivative
Supporting Formulae
Multivariable optimisation:
????? = ??? ? ????(???)?f(???)
where:
- ?f(??) = gradient vector
- ??(??) = Hessian matrix
Related Mathematical Methods
- Root finding
- Numerical optimisation
- Maximum likelihood estimation
- Gradient-based optimisation
- Hessian matrix methods
- Taylor series approximation
Example
A health economist estimates the parameter of a nonlinear survival model by solving the likelihood equation
f(?) = ?� ? 5 = 0.
Using an initial estimate of
?? = 2,
the Newton-Raphson update is
?? = 2 ? (2� ? 5)/(2(2)) = 2.25.
A second iteration gives
?? = 2.25 ? (2.25� ? 5)/(2(2.25)) = 2.2361,
which is close to the true solution
�5 = 2.2361.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POWER | =POWER(B2,2)-5 | Evaluates the nonlinear objective or estimating equation. |
| Solver Add-in | Minimise objective cell by changing parameter cells | Performs iterative optimisation of model parameters. |
| ABS | =ABS(B2-B3) | Monitors convergence between successive parameter estimates. |
| IF | =IF(ABS(B2-B3)<1E-06,""Converged"",""Continue"") | Applies convergence criteria during iterative estimation. |
VBA (Optional)
A VBA procedure can automate Newton-Raphson iterations by repeatedly updating parameter estimates until the specified convergence tolerance is achieved.
Sources
- Newton I. Method of Fluxions.
- Raphson J. Analysis Aequationum Universalis. 1690.
- 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 model calibration and parameter estimation.
Related Concepts (7)
Library
Publications
1
Conceptualizing a Model: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2 — Roberts, Russell, Paltiel, Chambers, McEwan & Krahn, Task Force Report 2 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on model conceptualisation — defining the decision problem, scoping, and choosing an appropriate model structure before implementation.
Journal ArticleView source →
Frequently Asked Questions (6)
What is the Newton-Raphson method?
A numerical technique finding a function's roots by iteratively improving an initial guess using the function's value and derivative at each step.
Source: Press et al. 2007
What does the Newton-Raphson method need at each step?
At each step the method uses both the value of the function and its slope at the current guess to work out where the function would cross zero if it continued along that slope, and takes that crossing as the next guess. Because it exploits the derivative, it homes in on a root rapidly when the starting guess is good. The reliance on the derivative is also its weakness, since it can fail where the slope is hard to compute, is zero, or the initial guess is poor. Press and colleagues (2007) describe the method.
Source: Press et al. 2007
How does the Newton-Raphson method work?
The Newton-Raphson method works by starting from an initial guess and, at each step, using the function's value and derivative there to find where the tangent line to the function crosses zero, which becomes the next, improved estimate. The update subtracts the function value divided by the derivative from the current guess. Repeating this moves the estimate toward a root, since the tangent approximates the function near the guess, and the process continues until successive estimates change negligibly.
Source: Press et al. 2007
Why does the Newton-Raphson method converge quickly?
The Newton-Raphson method converges quickly, with quadratic convergence, when the initial guess is close enough to a root and the function is well behaved, because near a root the tangent line approximates the function closely, so each step roughly squares the accuracy, doubling the number of correct digits. This rapid convergence makes it efficient for solving equations when a good starting point is available and the derivative can be computed, which is why it is a standard root-finding method.
Source: Press et al. 2007
What are the limitations of the Newton-Raphson method?
The Newton-Raphson method requires the function's derivative, which may be unavailable or costly, and it can fail: it may diverge or oscillate if the starting guess is poor, and it breaks down where the derivative is zero or near zero. It finds only one root at a time and may converge to an unexpected one. It also assumes the function is smooth. These limitations mean it needs a good starting point and a well-behaved function, and safeguards are often added.
Source: Press et al. 2007
Where is the Newton-Raphson method used in modelling?
In modelling and statistics, the Newton-Raphson method is used to solve equations that arise in estimation and calibration, such as finding parameter values that satisfy a condition or maximise a likelihood, since maximisation reduces to finding where a derivative is zero. It is used in fitting statistical models, calibrating health economic models to targets, and other computations requiring the solution of non-linear equations. Its speed makes it attractive where the derivative is available and a reasonable starting point can be found.
Source: Press et al. 2007
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-DM-070
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