Concept Architecture
Concept
Theoretically, Curve Fitting is the process of estimating a mathematical function that best represents the relationship between observed data points. It is founded on statistical modelling, optimisation and regression analysis and exists to summarise empirical data, estimate underlying functional relationships and support prediction. In health economics, curve fitting is widely used to estimate survival functions, disease progression, healthcare costs, utility trajectories and other model inputs from clinical and observational data.
Mathematically, curve fitting estimates the parameters of a chosen functional form by minimising the discrepancy between observed and predicted values according to a specified objective function. Depending on the modelling framework, parameter estimation commonly uses least squares, maximum likelihood estimation or Bayesian inference. The fitted curve is subsequently used for prediction, interpolation or extrapolation.
In practice, curve fitting is applied to clinical trial data, registry data and observational datasets to estimate parametric survival models, cost functions and utility curves. Model selection is based on statistical goodness-of-fit, visual inspection, clinical plausibility and information criteria, ensuring that fitted curves provide appropriate inputs for health economic decision models.
Purpose
Used to estimate mathematical relationships from empirical data, generate predictive models, support survival extrapolation and provide parameter estimates for health economic evaluation.
Mathematical Formulae
Primary Formula
?? = arg min ?(y? ? f(x?; ?))�
where:
- ?? = estimated model parameters
- y? = observed values
- f(x?; ?) = fitted function
- ?(y? ? f(x?; ?))� = residual sum of squares
Supporting Formulae
Residual:
e? = y? ? ??
Residual Sum of Squares:
RSS = ?(y? ? ??)�
Coefficient of determination:
R� = 1 ? (RSS � TSS)
Related Mathematical Methods
- Least squares estimation
- Maximum likelihood estimation
- Bayesian estimation
- Parametric survival modelling
- Non-linear regression
- Model selection using AIC and BIC
Example
Observed overall survival from a clinical trial is fitted using Weibull, Gompertz and log-normal survival models. Maximum likelihood estimation identifies the Weibull model as providing the lowest AIC while remaining clinically plausible. The fitted curve is subsequently extrapolated to estimate lifetime QALYs and costs within a cost-effectiveness model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B20,A2:A20,TRUE,TRUE) | Estimate parameters for linear curve fitting. |
| RSQ | =RSQ(B2:B20,C2:C20) | Assess goodness-of-fit between observed and predicted values. |
| SUMXMY2 | =SUMXMY2(B2:B20,C2:C20) | Calculate residual sum of squares for fitted models. |
VBA (Optional)
Automate fitting of alternative regression or survival curves and compare goodness-of-fit statistics across candidate models.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Collett D. Modelling Survival Data in Medical Research. CRC Press.
- Burnham KP, Anderson DR. Model Selection and Multimodel Inference. Springer.
- Seber GAF, Wild CJ. Nonlinear Regression. Wiley.
- 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 curve fitting?
The statistical process of identifying the mathematical function that best represents an observed pattern of data, such as observed survival times.
Source: Collett 2015
Why does curve fitting involve a choice among candidate functions?
Curve fitting does not reveal a single true function; it selects, from a family of candidate mathematical forms, the one whose parameters best match the observed pattern. Because several forms may fit the visible data almost equally well yet diverge where data are sparse, fitting involves choosing among them rather than reading off one answer. This choice matters most in survival analysis, where different fitted curves can agree over the follow-up period but predict very different long-term survival. The candidates must be compared, not assumed. Latimer (2013) describes this.
Source: Latimer 2013
How is curve fitting performed?
Curve fitting is performed by selecting a candidate function or distribution, estimating its parameters from the data, often by maximum likelihood or least squares, and assessing how well the fitted curve matches the observations using goodness-of-fit measures and visual comparison. Several candidates may be fitted and compared. The result is a fitted curve with estimated parameters that represents the data pattern, chosen to balance fit against simplicity and, where relevant, the plausibility of any extrapolation beyond the observed range.
Source: Collett 2015
How is curve fitting used in survival analysis?
In survival analysis, curve fitting is used to fit parametric survival distributions, such as exponential, Weibull, or log-normal, to observed survival data, producing a smooth survival curve. This fitted curve describes the data and, importantly, can be extended beyond the observed follow-up to extrapolate long-term survival, which is needed to estimate mean survival for economic evaluation. Because different distributions can fit the observed data similarly yet extrapolate differently, curve fitting in survival analysis considers both the fit and the plausibility of the projection.
Source: Latimer 2013
How is the best-fitting curve chosen?
The best-fitting curve is chosen by comparing candidate functions on how well they match the data, using goodness-of-fit statistics such as information criteria and visual inspection of the fitted curve against the observations, and, where the curve will be extrapolated, on the plausibility of the projection beyond the data. Because closeness of fit within the data does not guarantee a sensible extrapolation, the choice balances statistical fit against clinical plausibility, so the selected curve both represents the data and projects credibly where required.
Source: Latimer 2013
What are the limitations of curve fitting?
Curve fitting is limited by the risk of overfitting, where a flexible function matches the data closely but generalises poorly, and by the fact that a good fit within the observed range does not ensure a valid extrapolation beyond it, since different well-fitting curves can diverge markedly outside the data. The choice of function embeds assumptions about the shape of the relationship. These limitations mean fitted curves are assessed for both fit and plausibility, and, for extrapolation, treated with attention to the uncertainty in the projected portion.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 20 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/curve-fitting
- Term code
- HE-EM-SM-015
Stable URI · Machine-readable · Resolvable · CC BY 4.0