Concept Architecture
Concept
Theoretically, Statistical Fit is the degree to which a statistical model reproduces the observed characteristics of empirical data. It reflects the agreement between observed outcomes and those predicted by the fitted model and provides an indication of whether the selected model adequately represents the underlying data-generating process. Statistical fit underpins statistical inference, prediction and model comparison, and is routinely evaluated when developing models for health economic analyses.
Mathematically, statistical fit is quantified using objective measures that compare observed data with model predictions or likelihoods. The appropriate measure depends on the modelling framework and may include the coefficient of determination, residual error statistics, likelihood functions, deviance or information criteria. These measures assess the extent to which the model explains the observed data while recognising that improved fit must be balanced against increased model complexity.
In practice, statistical fit is evaluated using multiple complementary statistics together with residual diagnostics and graphical assessment. Health economists assess statistical fit when selecting regression models, survival models, utility mapping equations and risk prediction models used to estimate costs, health outcomes or transition probabilities. Statistical fit is interpreted alongside calibration, discrimination and external validation to determine whether a model is suitable for decision-making.
Purpose
Used to quantify how well a statistical model represents observed data, compare competing models and support the selection of appropriate models for health economic evaluation.
Mathematical Formulae
Primary Formula
For linear regression, a recognised measure of statistical fit is the coefficient of determination:
R� = 1 ? [�???� (y? ? ??)�] / [�???� (y? ? ?)�]
where:
- y? = observed value
- ?? = predicted value
- ? = mean observed value
Supporting Formulae
Residual Sum of Squares:
RSS = �???� (y? ? ??)�
Log-likelihood:
?(?) = �???� ln f(y? | ?)
Akaike Information Criterion:
AIC = ?2?(?) + 2k
where k is the number of estimated parameters.
Related Mathematical Methods
- Goodness of fit
- Residual analysis
- Likelihood ratio test
- Akaike Information Criterion
- Bayesian Information Criterion
- Cross-validation
- Calibration assessment
Example
A health economist compares two regression models predicting annual healthcare costs.
Model A has:
- R� = 0.78
- AIC = 2,315
Model B has:
- R� = 0.83
- AIC = 2,274
Model B demonstrates better statistical fit because it explains a greater proportion of observed variation while also achieving a lower information criterion, indicating a superior balance between fit and complexity.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RSQ | =RSQ(B2:B501,C2:C501) | Calculates the coefficient of determination between observed and predicted outcomes. |
| SUMXMY2 | =SUMXMY2(B2:B501,C2:C501) | Calculates residual variation for assessing model fit. |
| DEVSQ | =DEVSQ(B2:B501) | Calculates total variation in observed outcomes. |
| MIN | =MIN(E2:E5) | Identifies the model with the lowest AIC or other fit statistic. |
VBA (Optional)
Automate calculation of statistical fit measures for multiple candidate models and produce 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.
- Burnham KP, Anderson DR. Model Selection and Multimodel Inference. 2nd ed.
- Harrell FE. Regression Modeling Strategies. 2nd ed.
- Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning.
- 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 statistical fit?
A measure of how well a specified statistical model's predictions align with observed data, judging whether the specification adequately fits the pattern.
Source: Pearson 1900
What does statistical fit measure and what does it overlook?
Statistical fit measures how closely a model's predicted values match a set of observed data, summarising their agreement in a number. What it overlooks is whether that agreement will hold for new data, since a model can fit its own sample tightly yet generalise poorly, and it says nothing about whether the model is clinically sensible. A good fit statistic is therefore necessary but not sufficient evidence of a good model. It reports agreement with the sample, not truth. Hastie and colleagues (2009) discuss these limits.
Source: Hastie et al. 2009
How is statistical fit measured?
Statistical fit is measured by comparing a model's predictions with observed data using statistics that quantify the discrepancy, such as the chi-squared statistic, the coefficient of determination, or measures based on the likelihood, and through goodness-of-fit tests that judge whether the data are consistent with the model. Smaller discrepancies indicate better fit. Visual comparison of predicted against observed values complements these measures. Together they show how closely the model matches the data, allowing the adequacy of the specification to be assessed.
Source: Pearson 1900
Why does statistical fit matter?
Statistical fit matters because a statistical model is useful only if it adequately represents the data, so measuring how well its predictions align with observations indicates whether the specification is sound. Poor fit signals misspecification or missing features, so the model's inferences and predictions may be unreliable. Assessing fit informs whether a model is adequate and helps select among candidates, though fit is weighed against complexity and generalisation, since fitting the observed data closely does not guarantee good performance on new data.
Source: Akaike 1974
How does statistical fit relate to model complexity?
Statistical fit relates to complexity because a more complex model, with more parameters, can fit observed data at least as well, so fit alone would favour complexity, risking overfitting where the model captures noise and predicts poorly on new data. Judging a model therefore balances fit against parsimony, using criteria such as the Akaike information criterion that penalise the number of parameters. Statistical fit is assessed alongside complexity, seeking a specification that fits the data well without unnecessary parameters, so it generalises rather than merely fitting closely.
Source: Akaike 1974
What are the limitations of statistical fit measures?
Statistical fit measures can mislead because a model may fit the observed data closely yet predict poorly on new data if it has overfitted, and better fit can always be obtained by adding parameters, so fit to the fitting data does not test generalisation. Some measures depend on sample size or on assumptions that may not hold, and with large samples a fit test may flag trivial discrepancies. These limitations mean statistical fit is interpreted alongside complexity, out-of-sample performance, and the model's purpose, not taken as sufficient alone.
Source: Pearson 1900
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 17 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/statistical-fit
- Term code
- HE-EM-MV-073
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