Concept Architecture
Concept
Theoretically, a Spline Model is a regression or survival modelling approach that represents relationships using piecewise polynomial functions joined smoothly at predefined knot locations. It provides a flexible alternative to conventional linear and standard parametric models by allowing complex, non-linear functional forms while maintaining mathematical continuity. In health economics, spline models are widely used to model hazards, survival functions, treatment effects and continuous covariates that exhibit non-linear behaviour.
Mathematically, a spline model expresses the response as a linear combination of polynomial basis functions defined over intervals separated by knots. Continuity constraints ensure that the fitted function and its derivatives remain smooth at each knot. Model parameters are estimated using least squares or maximum likelihood estimation depending on the regression framework, with restricted cubic splines representing one of the most commonly applied spline formulations in survival analysis.
In practice, spline models are implemented by selecting an appropriate spline type, determining the number and placement of knots, generating basis functions and estimating the corresponding coefficients within regression or survival models. They are extensively applied in flexible parametric survival analysis, prognostic modelling, dose-response analysis and health economic extrapolation to improve model fit without imposing restrictive assumptions regarding functional form.
Purpose
Used to model non-linear relationships, improve regression and survival model fit, represent complex hazard functions and support flexible extrapolation in health economic evaluation.
Mathematical Formulae
Primary Formula
f(x) = ?? + ????? ??B?(x)
Where:
f(x) = fitted spline function
??, ?? = estimated regression coefficients
B?(x) = spline basis functions
K = number of basis functions
Supporting Formulae
Regression model:
g(E(Y)) = ?? + ????? ??B?(x)
where g(�) is the appropriate link function for the statistical model.
There is no universally recognised canonical mathematical formula for spline basis functions because multiple equivalent parameterisations are used, including B-splines, natural splines and restricted cubic splines.
Related Mathematical Methods
- Restricted cubic splines
- B-splines
- Natural cubic splines
- Flexible parametric survival modelling
- Maximum likelihood estimation
- Generalised additive models
Example
A health economist models age as a predictor of mortality using a spline model with five knots. Rather than assuming mortality increases linearly with age, the fitted spline captures periods of relatively stable risk followed by progressively steeper increases at older ages, producing a more accurate survival model for long-term economic evaluation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(A2>Knot1,(A2-Knot1)^3,0) | Generates truncated polynomial basis functions. |
| MAX | =MAX(A2-Knot1,0)^3 | Creates spline basis components. |
| LINEST | =LINEST(Y2:Y100,BasisRange,TRUE,TRUE) | Estimates spline regression coefficients. |
| MMULT | =MMULT(BasisRange,CoefficientRange) | Calculates fitted spline values. |
VBA (Optional)
VBA can automate knot selection, spline basis construction and prediction of fitted values for regression and survival analyses.
Sources
Harrell FE. Regression Modeling Strategies. 2nd ed. Springer; 2015.
Royston P, Sauerbrei W. Multivariable Model-Building. Wiley; 2008.
Royston P, Parmar MKB. Flexible parametric proportional-hazards and proportional-odds models for censored survival data. Statistics in Medicine. 2002;21:2175?2197.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
NICE. Health Technology Evaluations: The Manual. National Institute for Health and Care Excellence; 2022.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 15: Cost-effectiveness modelling using patient-level simulation — Davis, Stevenson, Tappenden & Wailoo, TSD 15 ed., 2014 (NICE Decision Support Unit (University of Sheffield))
Guidance on individual patient-level (microsimulation) cost-effectiveness modelling — when to use it in preference to cohort models, how to structure it, and how to handle the associated computational and uncertainty challenges.
Frequently Asked Questions (6)
What is a spline model?
A model using piecewise polynomial functions joined smoothly at knots to represent a flexible, non-linear relationship such as time and hazard.
Source: Royston & Parmar 2002
How does a spline model achieve smoothness across its pieces?
A spline model is built from separate polynomial pieces, one between each pair of knots, but it would be of little use if the pieces met in sharp corners. Splines are constructed so that the pieces join smoothly, agreeing in value and slope where they meet and often in curvature too, giving a continuous, gently flowing curve. This smoothness lets the model bend to follow the data without the abrupt jumps a set of disconnected pieces would show. The joins are where the smoothness is enforced. Harrell (2015) describes spline construction.
Source: Harrell 2015
How does a spline model work?
A spline model works by dividing the range of a variable at points called knots and fitting a polynomial, often cubic, within each segment, with the segments joined at the knots under smoothness conditions so the overall function and its derivatives are continuous. The function can change shape at each knot, giving flexibility to follow the data. The number and placement of knots control how flexibly the spline fits, more knots allowing more complex shapes at the risk of overfitting.
Source: Royston & Parmar 2002
What are the types of spline?
Types of spline include cubic splines, using cubic polynomial pieces; restricted cubic splines, which add linear constraints beyond the outer knots for stability; B-splines, a computationally convenient basis; and penalised splines, which control smoothness through a penalty. They differ in the polynomial degree, constraints, and how flexibility is controlled. In survival analysis, restricted cubic splines are common, since their linear tails give stable extrapolation, but the general idea of piecewise polynomials joined at knots is shared across spline types.
Source: Collett 2015
Why use spline models?
Spline models are used to represent non-linear relationships flexibly, capturing complex shapes that fixed functional forms, such as a straight line or a single parametric distribution, cannot, while remaining smooth. They let the data determine the shape between knots, adapting to patterns like turning hazards or non-linear covariate effects. This flexibility, controlled by the number and placement of knots, makes spline models valuable where the true relationship is unknown or complex, including in flexible parametric survival modelling and regression.
Source: Royston & Parmar 2002
What are the limitations of spline models?
Spline models require choosing the number and placement of knots, which affect the fit and can be somewhat arbitrary, and too many knots risk overfitting, while extrapolation beyond the outer knots can be unstable unless constrained, as in restricted cubic splines. Their flexibility can make them harder to interpret than simple forms, and their behaviour in sparse regions may be unreliable. These limitations mean knot choices are made carefully and, for survival extrapolation, the spline's tail behaviour is checked for plausibility.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 23 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/spline-model
- Term code
- HE-EM-SM-080
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