Concept Architecture
Concept
Theoretically, Goodness of Fit describes the degree to which a statistical model adequately represents the observed data. It quantifies the agreement between model predictions and empirical observations and provides an indication of how well the selected model captures the underlying data-generating process. Goodness of fit is fundamental to statistical modelling because it supports model selection, model validation and assessment of predictive performance. In health economics, it is used when developing regression models, survival models, risk prediction equations, utility mapping functions and disease progression models.
Mathematically, goodness of fit is represented by measures that compare observed values with those predicted by the model. The specific mathematical representation depends on the modelling framework. For linear regression, the coefficient of determination (R�) measures the proportion of variation explained by the model. For likelihood-based models, goodness of fit is commonly assessed using likelihood statistics, deviance or information criteria. These measures quantify model adequacy while balancing explanatory performance and model complexity.
In practice, goodness of fit is evaluated using one or more recognised statistical measures appropriate to the modelling approach. Analysts examine residual patterns, explained variation, prediction error and likelihood-based criteria before selecting or validating a model. Within health economic evaluations, adequate goodness of fit is essential because poorly fitting statistical models may produce biased estimates of costs, health outcomes, transition probabilities or treatment effects that propagate throughout decision-analytic models.
Purpose
Used to evaluate how well a statistical model represents observed data, compare competing models and support model validation before model outputs are incorporated into health economic analyses.
Mathematical Formulae
Primary Formula
For linear regression, the canonical goodness-of-fit measure is the coefficient of determination:
R� = 1 ? [�???� (y? ? ??)�] / [�???� (y? ? ?)�]
where:
- y? = observed value
- ?? = predicted value
- ? = sample mean
- n = number of observations
Supporting Formulae
Residual Sum of Squares:
RSS = �???� (y? ? ??)�
Total Sum of Squares:
TSS = �???� (y? ? ?)�
Explained Sum of Squares:
ESS = TSS ? RSS
Related Mathematical Methods
- Coefficient of determination (R�)
- Adjusted R�
- Residual analysis
- Likelihood ratio methods
- Deviance analysis
- Akaike Information Criterion (AIC)
- Bayesian Information Criterion (BIC)
- Mean squared error
Example
A health economist develops a regression model predicting annual healthcare costs from age, disease severity and comorbidity for 800 patients.
The observed variation in annual costs is:
TSS = 5,000,000
The residual variation after fitting the model is:
RSS = 900,000
Therefore,
R� = 1 ? 900,000 / 5,000,000 = 0.82
The model explains 82% of the observed variation in healthcare costs, indicating a strong goodness of fit.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RSQ | =RSQ(B2:B801,C2:C801) | Calculates the coefficient of determination between observed and predicted outcomes. |
| SUMXMY2 | =SUMXMY2(B2:B801,C2:C801) | Calculates the residual sum of squares. |
| AVERAGE | =AVERAGE(B2:B801) | Calculates the mean observed outcome for total variation. |
| DEVSQ | =DEVSQ(B2:B801) | Calculates the total sum of squares. |
VBA (Optional)
Automate calculation of goodness-of-fit statistics for multiple candidate models and generate a comparative model performance report.
Sources
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Harrell FE. Regression Modeling Strategies. 2nd ed.
- Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning.
- Burnham KP, Anderson DR. Model Selection and Multimodel Inference. 2nd ed.
- ISPOR Good Practice Reports on statistical modelling and model validation.
Related Concepts (2)
Library
Publications
1
Modeling Good Research Practices — Overview: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-1 — Caro, Briggs, Siebert & Kuntz, Task Force Report 1 ed., 2012 (Value in Health / Medical Decision Making)
The overview paper of the seven-part ISPOR-SMDM modelling good-practice series, setting out best-practice recommendations across model design, technique selection, implementation, validation, parameterisation, uncertainty and use in decision making.
Journal ArticleView source →
Frequently Asked Questions (6)
What is goodness of fit?
A statistical measure of how well a model's predicted values correspond to observed data, used to assess whether it adequately fits the pattern.
Source: Pearson 1900
Why is a close fit to data not always a good sign?
A model that matches the observed data very closely may be capturing the random noise in that particular sample rather than the underlying pattern, a problem called overfitting. Such a model fits the data it was built on well but predicts new data poorly, so a high goodness of fit alone does not guarantee a good model. This is why fit is weighed against complexity and tested on independent data. Fit measures how well the model describes the sample, not how well it will generalise. Hastie and colleagues (2009) explain this caution.
Source: Hastie et al. 2009
How is goodness of fit measured?
Goodness of fit is measured by comparing a model's predictions with the observed data, using statistics that quantify the discrepancy, such as the chi-squared statistic introduced by Pearson, which sums the squared differences between observed and expected values relative to the expected. Other measures include the coefficient of determination and various fit indices. Smaller discrepancies indicate better fit. These measures give a numerical summary of how closely the model matches the data, allowing the adequacy of the fit to be judged.
Source: Pearson 1900
Why does goodness of fit matter?
Goodness of fit matters because a model is useful only if it adequately represents the data, so measuring how well its predictions match observations shows whether it captures the underlying pattern. Poor fit indicates the model is misspecified or missing important features, so its predictions may be unreliable. Assessing fit helps in choosing among candidate models and in judging whether a fitted model is adequate for its purpose, such as extrapolating outcomes, so goodness of fit informs both model selection and validation.
Source: Akaike 1974
How does goodness of fit inform model selection?
Goodness of fit informs model selection by indicating how well each candidate model matches the data, so that better-fitting models are favoured. However, fit alone can mislead, since more complex models fit better simply by having more parameters, risking overfitting. Model selection therefore balances fit against complexity, using criteria such as the Akaike information criterion that penalise the number of parameters. Goodness of fit thus contributes to selection but is weighed against parsimony to choose a model that generalises rather than merely fitting closely.
Source: Akaike 1974
What are the limitations of goodness-of-fit measures?
Goodness-of-fit measures can be misleading because a model can fit the observed data closely yet predict poorly on new data if it has overfitted, capturing noise, and because better fit can always be obtained by adding parameters. Fit to the data used for fitting does not test generalisation. Some measures depend on sample size or assumptions that may not hold. These limitations mean goodness of fit is interpreted alongside considerations of complexity, out-of-sample performance, and the model's purpose, rather than taken as sufficient alone.
Source: Pearson 1900
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 15 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/goodness-of-fit
- Term code
- HE-EM-MV-030
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