Convergence

Concept

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Theoretically, Convergence is the property whereby a sequence, iterative procedure, estimator or numerical approximation approaches a stable limiting value as the number of iterations, observations or computational steps increases. The concept is central to numerical analysis, optimisation, simulation and statistical estimation because it determines whether a computational or inferential procedure produces a meaningful and reproducible result. In health economics, convergence is essential when estimating model parameters, calibrating decision models, solving optimisation problems and analysing stochastic simulations.

Mathematically, convergence is represented by a limit. A sequence {x?} converges to L when the distance between x? and L becomes arbitrarily small as n increases. In iterative algorithms, convergence is assessed by examining successive parameter estimates, objective-function values or residuals against a predefined tolerance. Different forms include pointwise convergence, convergence in probability, almost-sure convergence and convergence in distribution.

In practice, convergence is evaluated using stopping criteria such as changes in parameter estimates, log-likelihood values, objective functions or simulated outcomes. Health economic analysts routinely assess convergence in probabilistic sensitivity analysis, Bayesian estimation, Markov chain Monte Carlo simulation, model calibration and numerical optimisation to ensure that reported costs, effects and decision statistics are stable.

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Purpose

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Used to determine whether an iterative, statistical or simulation-based procedure has reached a stable solution that is sufficiently accurate for interpretation and decision-making.

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Mathematical Formulae

Primary Formula

lim??� x? = L

Supporting Formulae

�-convergence criterion:

|x? ? L| < �

Successive-iteration criterion:

|x??? ? x?| < �

Relative convergence criterion:

|x??? ? x?| / max(1, |x?|) < �

Objective-function criterion:

|f(x???) ? f(x?)| < �

Monte Carlo standard error:

MCSE = s / �n

Related Mathematical Methods

• Fixed-Point Iteration
• Numerical Optimisation
• Maximum Likelihood Estimation
• Markov Chain Monte Carlo
• Monte Carlo Simulation
• Model Calibration
• Root-Finding Methods

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Example

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A health economic model is calibrated to observed five-year survival by iteratively updating a transition probability.

Iteration 24:

p?? = 0.08314

Iteration 25:

p?? = 0.08311

Convergence tolerance:

� = 0.0001

Absolute change:

|0.08311 ? 0.08314| = 0.00003

Because 0.00003 < 0.0001, the calibration algorithm satisfies the convergence criterion and terminates.

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Excel Implementation

Function	Example Formula	Health Economics Application
ABS	=ABS(B3-B2)	Calculate the change between successive iterations
IF	=IF(ABS(B3-B2)<$F$1,"Converged","Continue")	Apply a predefined convergence tolerance
MAX	=MAX(1,ABS(B2))	Scale a relative convergence criterion
SQRT	=STDEV.S(C2:C1001)/SQRT(COUNT(C2:C1001))	Calculate Monte Carlo standard error
Solver	Minimise calibration error subject to parameter constraints	Estimate model parameters until the optimisation procedure converges

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VBA (Optional)

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A VBA procedure can repeat model calculations until parameter changes or objective-function differences fall below a specified convergence tolerance.

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Sources

• Burden RL, Faires JD. Numerical Analysis.
• Robert CP, Casella G. Monte Carlo Statistical Methods.
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• ISPOR-SMDM Modeling Good Research Practices Task Force Reports.