Concept Architecture
Concept
Theoretically, Bayesian Information Criterion (BIC) is a likelihood-based model selection criterion that compares competing statistical models by balancing model fit against model complexity. Derived by Gideon Schwarz as an approximation to the Bayesian model evidence, BIC imposes a stronger penalty for additional parameters than the Akaike Information Criterion, particularly as sample size increases. In health economics, BIC is used to compare regression models, survival models and other statistical models that provide inputs to decision-analytic evaluations.
Mathematically, BIC approximates the logarithm of the marginal likelihood of a model under regularity conditions. The criterion combines the maximised likelihood with a penalty that increases according to both the number of estimated parameters and the sample size. Models with lower BIC values are preferred because they provide a better trade-off between explanatory power and parsimony.
In practice, BIC is calculated after fitting competing models using maximum likelihood estimation. The criterion is frequently used to select parametric survival models, regression specifications, latent class models and other statistical models in health economics. BIC is interpreted comparatively, with only differences between models fitted to the same dataset being meaningful.
Purpose
Used to compare competing statistical models by balancing goodness of fit against model complexity, supporting parsimonious model selection for health economic analyses.
Mathematical Formulae
Primary Formula
BIC = k ln(n) ? 2ln(L?)
where:
- k = number of estimated model parameters
- n = sample size
- L? = maximised likelihood
Supporting Formulae
Difference in BIC:
?? = BIC? ? BIC???
Approximate Bayes factor between two models:
BF?? � exp(??BIC/2)
Related Mathematical Methods
- Maximum likelihood estimation
- Bayesian model selection
- Model selection
- Regression analysis
- Survival analysis
- Parametric survival modelling
- Akaike Information Criterion
Example
Three parametric survival models are fitted to survival data from 500 patients.
| Model | Parameters (k) | Log-likelihood |
|---|---|---|
| Exponential | 2 | -320 |
| Weibull | 3 | -312 |
| Gompertz | 3 | -315 |
For the Weibull model,
BIC = 3 ln(500) ? 2(?312)
= 3(6.2146) + 624 = 642.64
For the exponential model,
BIC = 2 ln(500) ? 2(?320) = 652.43
Because the Weibull model has the lower BIC, it is selected as the preferred survival model for extrapolation within the economic evaluation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LN | =B2*LN(C2)-2*LN(D2) | Calculate BIC from the number of parameters, sample size and maximised likelihood |
| MIN | =MIN(E2:E6) | Identify the lowest BIC among competing models |
| ABS | =E2-$E$7 | Calculate the difference in BIC relative to the preferred model |
| EXP | =EXP(-(F2/2)) | Approximate the Bayes factor from the BIC difference |
| SORT | =SORT(A2:E6,5,1) | Rank competing models according to BIC |
VBA (Optional)
Automate calculation and comparison of BIC values across competing statistical models and produce ranked model selection summaries.
Sources
- Schwarz G. Estimating the Dimension of a Model. Annals of Statistics. 1978;6(2):461-464.
- Burnham KP, Anderson DR. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. 2nd ed. Springer; 2002.
- Claeskens G, Hjort NL. Model Selection and Model Averaging. Cambridge University Press; 2008.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- NICE. Health Technology Evaluation Manual.
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 the Bayesian information criterion?
A statistic comparing candidate models fitted to the same data, similar to the Akaike criterion but penalising complexity more as sample size grows.
Source: Schwarz 1978
Why does the Bayesian information criterion favour simpler models more strongly as data grow?
The Bayesian information criterion, like the Akaike criterion, trades fit against complexity, but its penalty for each extra parameter increases with the size of the dataset. As the sample grows, the criterion demands a larger improvement in fit to justify an added parameter, so it tends to select simpler models than the Akaike criterion in large samples. The rationale is that with more data the true model can be identified more confidently, and needless parameters should be pruned. Schwarz (1978) derived it.
Source: Schwarz 1978
How is the Bayesian information criterion calculated?
The Bayesian information criterion is calculated from the model's maximised likelihood, its number of parameters, and the sample size: it equals the number of parameters times the logarithm of the sample size, minus twice the log-likelihood. Better fit lowers it, and more parameters raise it, with the penalty per parameter growing with the sample size. Lower values indicate a better balance of fit and parsimony, so candidate models fitted to the same data are compared, and the one with the lowest value is preferred.
Source: Schwarz 1978
How does the Bayesian information criterion differ from the Akaike criterion?
The Bayesian and Akaike information criteria both compare models by fit and complexity but differ in their penalty for parameters. The Akaike criterion applies a fixed penalty per parameter, while the Bayesian criterion's penalty increases with the logarithm of the sample size, so it penalises complexity more heavily in large samples. Consequently, the Bayesian criterion tends to select simpler models than the Akaike criterion, especially with much data, and the two can favour different models, so both are sometimes reported.
Source: Akaike 1974
Why does the Bayesian information criterion penalise complexity more with larger samples?
The Bayesian information criterion penalises complexity more as the sample grows because its penalty per parameter increases with the logarithm of the sample size, reflecting its derivation from approximating the models' posterior probabilities. With more data, the criterion demands a greater improvement in fit to justify additional parameters, favouring parsimony. This tends to select the simpler model when data are plentiful, in contrast to the Akaike criterion's fixed penalty, which is one reason the two criteria can disagree on the preferred model.
Source: Schwarz 1978
How is the Bayesian information criterion used?
The Bayesian information criterion is used to compare candidate models fitted to the same data and to choose among them, selecting the model with the lowest value as the best balance of fit and parsimony given the sample size. It is applied in model selection, including choosing among survival distributions for extrapolation, often alongside the Akaike criterion. Because it favours simpler models, particularly with large samples, it offers a stricter guard against overfitting, informing model choice together with judgement and other considerations.
Source: Schwarz 1978
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 13 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-MV-004
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