Calculator
Example Data Table
| Model | Log Likelihood | k | n | Expected AIC | Expected BIC |
|---|---|---|---|---|---|
| Linear model | -120 | 3 | 80 | 246.000 | 253.146 |
| Quadratic model | -115 | 4 | 80 | 238.000 | 247.528 |
| Cubic model | -113 | 5 | 80 | 236.000 | 247.910 |
Formula Used
AIC formula: AIC = 2k - 2LL
AICc formula: AICc = AIC + [2k(k + 1) / (n - k - 1)]
BIC formula: BIC = ln(n)k - 2LL
RSS log likelihood estimate: LL = -(n / 2)[ln(2π) + 1 + ln(RSS / n)]
Delta formula: Delta = model score - best score
Weight formula: Weight = exp(-0.5 × Delta) / sum of all exp(-0.5 × Delta)
How To Use This Calculator
- Enter one or more candidate model names.
- Select whether each row uses log likelihood or RSS conversion.
- Enter the parameter count, including all estimated parameters.
- Enter the same sample size basis for every compared model.
- Press calculate to show results below the header.
- Use CSV or PDF buttons to save your results.
Understanding AIC and BIC
AIC and BIC help compare statistical models. They do not prove that a model is true. They rank choices by balancing fit and simplicity. A model with better likelihood usually fits data better. Yet extra parameters can overfit. These scores add a penalty for complexity.
Why the Scores Matter
Manual calculation is useful when checking software output. It also helps students see how each part works. AIC uses a lighter penalty. BIC uses a stronger penalty when sample size grows. Because of that, BIC often favors simpler models. AIC can favor prediction quality when the candidate set is reasonable.
Required Inputs
You need the log likelihood, parameter count, and sample size. The log likelihood should come from the same data set for every model. The parameter count should include all estimated terms. In many regression cases, that includes the variance term. If you use RSS instead, assume normal errors. The calculator converts RSS into a log likelihood estimate.
Reading Results
Lower AIC and BIC values are usually better. The absolute value is less important than the difference. Delta AIC shows how far each model is from the best AIC score. Akaike weights turn those gaps into relative support values. BIC weights offer a similar comparison under the BIC scale.
Good Calculation Habits
Always compare models fitted on identical observations. Do not compare scores from different response variables. Do not mix likelihood formulas from different assumptions. Check whether constants were included in the log likelihood. Some packages report comparable values, but some omit constants. Consistency matters more than the printed scale.
Practical Example
Suppose Model One has k equal to three. Its log likelihood is negative one hundred twenty. Its AIC is two times three minus two times negative one hundred twenty. The result is two hundred forty six. With n equal to eighty, BIC adds three times natural log of eighty. The result is about two hundred fifty three point fifteen.
Final Advice
Use these criteria as guides, not as final proof. Review residuals, assumptions, theory, and validation results. A small score advantage may not matter practically. A simple model can be best when accuracy is similar. Record inputs before changing assumptions or removing observations later.
FAQs
What does AIC mean?
AIC means Akaike Information Criterion. It compares models by using log likelihood and a penalty for estimated parameters. Lower values usually suggest a better balance between fit and complexity.
What does BIC mean?
BIC means Bayesian Information Criterion. It uses log likelihood, parameter count, and sample size. Its penalty grows with sample size, so it often prefers simpler models.
Should AIC or BIC be lower?
Yes. Lower AIC or BIC values usually indicate a preferred model among the candidates. The score is comparative, so evaluate it against other models fitted to the same data.
What is k in the formula?
k is the number of estimated parameters. It may include slopes, intercepts, variance terms, or other fitted quantities. Use the same counting rule across every model.
What is LL in AIC and BIC?
LL is the log likelihood of the fitted model. It measures how well the model explains the observed data under its assumptions.
When should I use AICc?
Use AICc when the sample size is small compared with the parameter count. It adds a correction to reduce small sample bias.
Can I compare models with different data?
No. AIC and BIC comparisons should use models fitted to the same response, same observations, and compatible likelihood assumptions.
What are model weights?
Model weights convert delta scores into relative support values. They help show how strongly each model is supported within the candidate set.