Concept Architecture
Concept
Theoretically, a Fractional Polynomial is a flexible regression modelling approach that extends conventional polynomial regression by allowing predictor variables to be raised to non-integer and negative powers. It was developed to model continuous, non-linear relationships parsimoniously while avoiding the instability often associated with high-order conventional polynomials. In health economics, fractional polynomials are widely used in survival analysis, risk prediction and regression modelling to represent non-linear effects of continuous variables on costs, utilities and clinical outcomes.
Mathematically, a fractional polynomial represents the relationship between a predictor and an outcome as a linear combination of transformed predictor terms using powers selected from a predefined set. Model selection is performed by comparing alternative power combinations using likelihood-based criteria or deviance statistics. First- and second-degree fractional polynomial models provide flexible functional forms while requiring relatively few parameters.
In practice, fractional polynomials are fitted using maximum likelihood estimation within regression or survival models. Candidate powers are systematically evaluated, with the preferred model selected according to statistical fit, clinical plausibility and predictive performance. Fractional polynomials are frequently applied when modelling age, biomarker values, disease severity and treatment effects in health economic evaluations.
Purpose
Used to model non-linear relationships between continuous variables and outcomes while maintaining parsimonious, interpretable regression models for survival analysis, risk prediction and economic evaluation.
Mathematical Formulae
Primary Formula
? = ?? + ??x?? + ??x?�
where:
- ? = linear predictor
- x = continuous predictor
- p?, p? ? {?2, ?1, ?0.5, 0, 0.5, 1, 2, 3}
- x? = ln(x)
If p? = p?:
? = ?? + ??x? + ??x?ln(x)
Supporting Formulae
For a first-degree fractional polynomial:
? = ?? + ??x?
Parameters are estimated using maximum likelihood:
?? = arg max L(?)
Related Mathematical Methods
- Maximum likelihood estimation
- Generalised linear models
- Cox proportional hazards modelling
- Flexible parametric survival models
- Restricted cubic splines
- Model selection using AIC and BIC
Example
A health technology assessment examines the relationship between patient age and mortality following cardiovascular treatment. A linear model fails to capture the observed curvature. A second-degree fractional polynomial with powers (0, 1) is selected, producing the model ? = ?? + ??ln(age) + ??age, which provides a better fit and improves long-term survival predictions used in the economic model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POWER | =POWER(A2,2) | Calculate transformed predictor variables for selected polynomial powers. |
| LN | =LN(A2) | Calculate the logarithmic transformation when p = 0. |
| IF | =IF(Power=0,LN(A2),POWER(A2,Power)) | Apply the appropriate fractional polynomial transformation. |
| LINEST | =LINEST(Y_range,X_range,TRUE,TRUE) | Estimate regression coefficients for exploratory fractional polynomial models. |
VBA (Optional)
Automate evaluation of candidate fractional polynomial power combinations and identify the model with the best statistical fit.
Sources
- Royston P, Altman DG. Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling.
- Royston P, Sauerbrei W. Multivariable Model-Building: A Pragmatic Approach to Regression Analysis Based on Fractional Polynomials for Modelling Continuous Variables.
- Royston P, Parmar MKB. Flexible parametric proportional-hazards and proportional-odds models for censored survival data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
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 fractional polynomial?
A flexible modelling technique representing a non-linear relationship using a small number of power transformations of a continuous variable.
Source: Royston & Altman 1994
What does a fractional polynomial use to bend a relationship?
A fractional polynomial represents a curved relationship between a variable and an outcome using a small set of power transformations of that variable, chosen from a fixed menu of powers rather than only the usual squares and cubes. Combining one or two such terms lets the fitted relationship take many shapes while staying compact and estimated from few parameters. This suits modelling how risk varies smoothly with a continuous factor such as age or dose. It is an alternative to splines for the same purpose. Royston and Sauerbrei (2008) describe the method.
Source: Royston & Sauerbrei 2008
How do fractional polynomials work?
Fractional polynomials work by transforming a continuous variable using powers selected from a defined set, including fractional and negative powers and the logarithm, and combining one or two such transformations to model the relationship. The best-fitting powers are chosen by comparing the fit of candidate combinations. Using only one or two terms keeps the model parsimonious while capturing curvature, so the technique represents a non-linear effect with a small number of transformed terms rather than many polynomial or spline terms.
Source: Royston & Altman 1994
Why use fractional polynomials?
Fractional polynomials are used to model non-linear relationships between an outcome and continuous predictors flexibly yet parsimoniously, capturing curves that a straight line cannot while using few parameters. They offer a wider range of shapes than ordinary polynomials of the same order, avoiding the awkward behaviour of high-order polynomials, and they are simpler and more stable than some spline approaches. This makes them useful where a continuous variable has a non-linear effect that should be represented without excessive complexity.
Source: Royston & Altman 1994
How do fractional polynomials differ from splines?
Fractional polynomials and splines both model non-linear relationships flexibly but differ in approach. Fractional polynomials use a small number of power transformations of the variable, giving smooth global curves with few parameters, while splines use piecewise polynomials joined at knots, offering local flexibility that can capture more complex or localised shapes. Fractional polynomials are more parsimonious and stable but less able to fit highly complex patterns, whereas splines are more flexible but require choices about knots and can overfit, so the two suit different needs.
Source: Collett 2015
What are the limitations of fractional polynomials?
Fractional polynomials are limited in the complexity of shapes they can represent, since using only one or two power terms may not capture highly irregular or localised patterns that splines could. The chosen powers depend on the data, so selection uncertainty exists, and, as with any flexible fitting, extrapolation beyond the data can be unreliable. They are best for smooth, moderate non-linearity in a continuous predictor. These limitations mean fractional polynomials suit parsimonious modelling of non-linear effects rather than very complex relationships.
Source: Royston & Altman 1994
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 20 Oct 2025
Content version: 1.0.0
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