Concept Architecture
Concept
Theoretically, Taylor Series Expansion is a mathematical method that represents a sufficiently differentiable function as an infinite polynomial centred at a specified point. Developed from differential calculus, the expansion approximates complex non-linear functions using derivatives evaluated at a single location. Taylor series provide the theoretical foundation for numerical analysis, optimisation, asymptotic methods, statistical approximation, and computational modelling. In health economics, Taylor series are particularly important for deriving approximations used in uncertainty analysis, variance estimation, and economic modelling.
Mathematically, the Taylor Series Expansion expresses a function as the sum of successive derivative terms evaluated at an expansion point. The approximation becomes increasingly accurate as additional terms are included, provided the function is analytic in the neighbourhood of the expansion point. First-order expansions provide linear approximations, while higher-order expansions capture increasing levels of curvature. Numerous statistical methods, including the Delta Method, rely directly on first-order Taylor expansions.
In practice, Taylor Series Expansion is used to approximate otherwise intractable mathematical expressions encountered in statistical estimation and health economic modelling. Applications include variance approximation for transformed estimators, non-linear optimisation, probabilistic sensitivity analysis, Markov modelling, numerical solution algorithms, and approximation of complex likelihood functions.
Purpose
Used to approximate non-linear mathematical functions using polynomial expansions, simplify statistical estimation, derive variance approximations, support numerical computation, and improve analytical tractability 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)?/3! + ?
More generally:
f(x) = ????^� [f?�?(a) � n!] (x ? a)�
Supporting Formulae
First-order approximation:
f(x) � f(a) + f?(a)(x ? a)
Second-order approximation:
f(x) � f(a) + f?(a)(x ? a) + f?(a)(x ? a)�/2
Maclaurin series (a = 0):
f(x) = ????^� [f?�?(0) � n!] x�
Related Mathematical Methods
- Delta Method
- Numerical Analysis
- Optimisation
- Newton?Raphson Method
- Asymptotic Approximation
- Maximum Likelihood Estimation
- Numerical Integration
- Finite Difference Methods
Example
A health economist estimates the variance of a transformed cost-effectiveness estimator using the Delta Method.
The transformation is:
g(x) = ln(x)
Expanding around the estimated mean cost ?:
ln(x) � ln(?) + (1/?)(x ? ?)
If:
? = �12,000
A small change in cost of �300 is approximated by:
?ln(x) � (1/12,000) ? 300
= 0.025
Rather than evaluating the exact logarithmic transformation, the first-order Taylor expansion provides an accurate approximation suitable for variance estimation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POWER | =POWER(A2-ExpansionPoint,2) | Calculate higher-order Taylor terms. |
| FACT | =FACT(3) | Calculate factorial terms appearing in Taylor expansions. |
| LN | =LN(A2) | Compare exact function values with Taylor approximations. |
| EXP | =EXP(A2) | Evaluate exponential functions approximated by Taylor series. |
| SUM | =SUM(Term1:TermN) | Calculate polynomial approximations from Taylor expansion terms. |
VBA (Optional)
Automate construction of Taylor series approximations for user-specified functions and approximation orders within health economic models.
Sources
- Taylor B. Methodus Incrementorum Directa et Inversa. 1715.
- Burden RL, Faires JD. Numerical Analysis.
- Stoer J, Bulirsch R. Introduction to Numerical Analysis.
- Casella G, Berger RL. Statistical Inference.
- Oehlert GW. A Note on the Delta Method. The American Statistician. 1992.
- 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 (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a Taylor series expansion?
A mathematical technique approximating a complex function using a polynomial series based on its derivatives at a single point, underlying the delta method.
Source: Oehlert 1992
What does a Taylor series expansion approximate?
A Taylor series expansion approximates a complicated function near a chosen point by a polynomial built from the function's derivatives there, so that close to that point the simple polynomial behaves almost like the original. In statistics this is used to linearise awkward, nonlinear functions of estimates, replacing them locally with a straight-line approximation whose behaviour is easy to analyse. A first-order expansion keeps only the linear term, which underlies the delta method for approximating variances. Approximating a curve by a local polynomial is what it does. Kirkwood and Sterne (2003) discuss this.
Source: Kirkwood & Sterne 2003
How does a Taylor series expansion work?
A Taylor series expansion works by expressing a function near a chosen point as a sum of terms, each involving a successive derivative of the function at that point multiplied by a power of the distance from the point; keeping only the first-order term gives a linear approximation, and more terms improve accuracy near the point. So a Taylor series expansion works by building a polynomial approximation from the function's derivatives, with the approximation most accurate close to the expansion point, and in statistics the first-order expansion is commonly used to linearise a function so that its variance can be approximated from the variances of its inputs.
Source: Oehlert 1992
Why is a Taylor series expansion used in statistics?
A Taylor series expansion is used in statistics because linearising a complex function of random variables through a first-order expansion allows its approximate variance, and hence standard error, to be derived from the variances and covariances of the inputs, which is the basis of the delta method. So a Taylor series expansion is used to approximate the uncertainty of functions of estimates, such as ratios or transformations, by linearising them, which makes it possible to obtain standard errors for quantities that would otherwise be difficult, and this is why it underlies the delta method for propagating uncertainty through functions of random variables.
Source: Oehlert 1992
What is a first-order Taylor series expansion?
A first-order Taylor series expansion approximates a function near a point by a linear function, using only the value and the first derivative at that point, so the function is represented by its tangent line, or plane, there. So a first-order Taylor series expansion is the linear approximation of a function, which is the form most used in statistics, since linearising a function this way allows its variance to be approximated straightforwardly from the variances and covariances of its inputs, and this first-order approximation underlies the delta method, though it is accurate only near the expansion point and for functions that are not too strongly curved.
Source: Oehlert 1992
What are the limitations of a Taylor series expansion?
The limitations of a Taylor series expansion include that it is a local approximation, most accurate near the expansion point, so it can be inaccurate for strongly non-linear functions or far from the point, and when used with only the first-order term the approximation may be poor if the function curves markedly or the variability is large. So a Taylor series expansion is used with awareness that its accuracy depends on the function being reasonably approximated by the retained terms over the relevant range, which is why the delta method based on it can be unreliable for highly non-linear functions, in which case more terms or alternative methods such as the bootstrap may be needed.
Source: Oehlert 1992
Trust Record
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-212
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