Concept Architecture
Concept
Theoretically, Taylor Series Method is a numerical and analytical approximation method that estimates the value of a function by truncating its Taylor series after a finite number of terms. The method replaces a complex function with a polynomial approximation constructed from derivatives evaluated at a chosen expansion point. It is widely used in numerical analysis, optimisation, statistical approximation, and scientific computing because polynomial functions are generally easier to evaluate than the original non-linear functions.
Mathematically, the Taylor Series Method approximates a function using successive derivative terms of increasing order. The approximation error decreases as additional terms are retained, provided the function is sufficiently smooth in the neighbourhood of the expansion point. First-order approximations produce linearisation, while higher-order approximations capture increasing curvature. The method forms the mathematical basis of numerous statistical procedures, including the Delta Method and many numerical solution algorithms.
In practice, the Taylor Series Method is applied to approximate non-linear functions arising in health economic models, statistical estimation, optimisation, and uncertainty analysis. It is commonly used when exact analytical solutions are unavailable or computationally expensive, including variance approximation, likelihood optimisation, Markov modelling, probabilistic sensitivity analysis, and numerical differential equation methods.
Purpose
Used to approximate complex mathematical functions using finite polynomial expansions, simplify statistical and numerical calculations, support optimisation and variance estimation, and improve computational efficiency in health economic modelling.
Mathematical Formulae
Primary Formula
For a function expanded about x = a:
f(x) � f(a) + f?(a)(x ? a) + f?(a)(x ? a)�/2! + ? + f?�?(a)(x ? a)�/n!
Supporting Formulae
First-order Taylor approximation:
f(x) � f(a) + f?(a)(x ? a)
Second-order Taylor approximation:
f(x) � f(a) + f?(a)(x ? a) + f?(a)(x ? a)�/2
Maclaurin approximation (a = 0):
f(x) � ????� [f???(0) � k!]x?
Related Mathematical Methods
- Taylor Series Expansion
- Delta Method
- Numerical Analysis
- Newton?Raphson Method
- Numerical Optimisation
- Asymptotic Approximation
- Finite Difference Method
- Maximum Likelihood Estimation
Example
A health economist requires an approximation to the logarithm of an incremental cost estimate during variance estimation.
Expanding around ? = �15,000 using the first-order Taylor method:
ln(x) � ln(15,000) + (1/15,000)(x ? 15,000)
If the observed cost is �15,450:
?ln(x) � (1/15,000) ? 450
= 0.03
The Taylor Series Method provides an accurate approximation without evaluating higher-order non-linear expressions directly.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POWER | =POWER(A2-ExpansionPoint,2) | Calculate higher-order polynomial terms. |
| FACT | =FACT(4) | Calculate factorial denominators for Taylor terms. |
| LN | =LN(A2) | Compare exact logarithmic values with Taylor approximations. |
| EXP | =EXP(A2) | Compare exponential functions with polynomial approximations. |
| SUM | =SUM(Term1:TermN) | Calculate the truncated Taylor approximation. |
VBA (Optional)
Automate generation of finite-order Taylor approximations for functions used in health economic models and numerical optimisation.
Sources
- Taylor B. Methodus Incrementorum Directa et Inversa. 1715.
- Burden RL, Faires JD. Numerical Analysis.
- Stoer J, Bulirsch R. Introduction to Numerical Analysis.
- Oehlert GW. A Note on the Delta Method. The American Statistician. 1992.
- Casella G, Berger RL. Statistical Inference.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (4)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is the Taylor series method?
The practical application of a Taylor series expansion to derive an approximate variance formula for a function of random variables.
Source: Oehlert 1992
What does the Taylor series method produce for a function of estimates?
The Taylor series method applies a Taylor series expansion in practice to derive an approximate formula for the variance of a quantity that is a function of random variables. By linearising the function around the estimated values, it turns a hard problem into a simple one, expressing the combined uncertainty in terms of the variances of the parts and how the function responds to each. This is exactly the machinery of the delta method, used for ratios such as the cost-effectiveness ratio. Approximating a combined variance by linearising is what it does. Drummond and colleagues (2015) discuss this.
Source: Drummond et al. 2015
How does the Taylor series method work?
The Taylor series method works by taking a function of random variables, approximating it with a first-order Taylor series expansion around the means, which linearises it, and then computing the variance of this linear approximation from the variances and covariances of the variables and the function's derivatives. So the Taylor series method works by linearising the function and propagating the uncertainty of its inputs through the derivatives, which yields an approximate variance for the function, and this is the mechanism of the delta method, allowing standard errors to be obtained for quantities built from estimated components without simulation.
Source: Oehlert 1992
What is the Taylor series method used for?
The Taylor series method is used to obtain approximate standard errors and confidence intervals for functions of estimates, such as ratios, products, or other transformations, when an exact variance is unavailable; in health economics it is applied to the variance of the incremental cost-effectiveness ratio. So the Taylor series method is used to quantify the uncertainty of derived quantities by linearising them, which provides a way to propagate uncertainty through functions of random variables, and it is the basis of the delta method, valuable where the standard error of a combined or transformed estimate is needed but not directly available.
Source: Oehlert 1992
How does the Taylor series method relate to the delta method?
The Taylor series method relates to the delta method as essentially the same technique: the delta method is the statistical application of a first-order Taylor series expansion to approximate the variance of a function of random variables, so the Taylor series method and the delta method describe the same procedure of linearising and propagating uncertainty. So the Taylor series method and the delta method are closely equivalent, with the Taylor series expansion providing the mathematical basis and the delta method being the named statistical method that applies it, which is why both derive approximate variances for functions of estimates through first-order linearisation.
Source: Oehlert 1992
What are the limitations of the Taylor series method?
The limitations of the Taylor series method include that, being based on a first-order linear approximation, it can be inaccurate for strongly non-linear functions or when the variability is large, and its approximate variances may then mislead, particularly for ratios with uncertain denominators. So the Taylor series method is used with awareness that its accuracy depends on the function being reasonably linear over the relevant range, which is why for problematic cases, such as cost-effectiveness ratios with uncertain denominators, resampling methods like the bootstrap are often preferred, since they avoid the linear approximation that the Taylor series method relies upon.
Source: Oehlert 1992
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 25 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-213
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