What each criterion balances
Both criteria combine lack of fit with a penalty for the number of estimated parameters:
- AIC: −2 log-likelihood + 2k;
- BIC: −2 log-likelihood + k log(n).
Here k is the parameter count and n is the sample size under the criterion’s definition. Both reward better likelihood fit and penalize extra flexibility. Because log(n) exceeds 2 once n is larger than about 7, BIC usually penalizes added parameters more strongly in ordinary sample sizes.
Akaike developed AIC from an information-theoretic predictive perspective.1 Schwarz derived BIC from a large-sample Bayesian model-selection argument.2 That does not mean AIC is exclusively “for prediction” or that BIC invariably finds truth; the candidate set, data-generating assumptions, priors, and asymptotic conditions matter.
Read differences, not isolated values
For each criterion, subtract the minimum candidate value:
Δi = criterion i − minimum criterion
The best-supported candidate by that criterion has Δ = 0. Burnham and Anderson provide commonly used ΔAIC guidance: differences of 0–2 indicate substantial support, 4–7 considerably less support, and above 10 essentially no support within the candidate set.3
| ΔAIC | Attributed reading | Context | Category |
|---|---|---|---|
| 0–2 | Substantial empirical support | Burnham–Anderson information-theoretic candidate comparison3 | DISCIPLINE CONVENTION |
| 4–7 | Considerably less support | Same framework and candidate set3 | DISCIPLINE CONVENTION |
| >10 | Essentially no support | Same framework and candidate set3 | DISCIPLINE CONVENTION |
These are attributed interpretation conventions, not absolute AIC-quality thresholds. A ΔAIC of 1 does not make either model scientifically adequate.
Comparisons that are invalid
Do not compare criteria when models use different response data, incompatible likelihood definitions, or different handling of missing observations. REML-based information criteria for models with different fixed effects require particular care. Also check whether the software reports AIC, AICc, conditional AIC, or another variant.
The candidate set limits the conclusion. AIC can select the least bad model in a poor set. Inspect residuals, design, and out-of-sample behavior rather than treating the minimum as certification.
RMSE and MAE evaluate predictive error on an interpretable scale. R-squared answers another fit question and also lacks a universal good cutoff. The linear regression output guide shows where model criteria sit beside coefficients and diagnostics.
When AIC and BIC disagree
Disagreement is expected when the added fit is enough to offset AIC’s smaller penalty but not BIC’s larger one. Report both only if both serve a stated purpose. Do not average the ranks or choose whichever favors the preferred story.
A defensible reporting sentence
“Among models fitted to the same 240 observations and likelihood, model M2 had the lowest AIC (ΔAIC = 0), while BIC favored the simpler M1. We treated the disagreement as a penalty-and-goal difference and checked held-out error and residual diagnostics.”