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eduKateSG · Why Mathematics? · Akaike Information Criterion: Likelihood and Complexity
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 5–9
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 10–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–26
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a model should explain patterns without memorising every fluctuation? The Akaike Information Criterion, or AIC, combines maximised likelihood with a parameter penalty. It rewards fit through the likelihood and charges two units for each estimated parameter under its standard form.
AIC is useful for ranking candidate models fitted to the same response data and likelihood. Its absolute value has no standalone grade; differences among comparable models carry the information.
Did You Know? AIC can prefer a slightly more complex model when the gain in likelihood is large enough. Its aim is predictive information efficiency, not simply the fewest parameters.
Full chapter index
Chapters 1–4 · Understand the question
Chapters 5–9 · Reproduce the example
Chapters 10–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–26 · Study and verify
A highly flexible model can increase maximised likelihood by following sample-specific noise. AIC asks whether the improvement is large enough to pay a fixed complexity charge.
NIST gives AIC=2k-2 ln(L), where k is the number of fitted parameters and L is the maximised likelihood. NIST treats it as a screening criterion for candidate distributions and recommends fuller analysis after selection. Read the NIST reference.
Suppose k=4 and the maximised log-likelihood is -120. Then AIC=2×4-2(-120)=8+240=248.
| k | Log-likelihood |
|---|---|
| 4 | -120 |
If a second comparable model has AIC=252, the first has a four-unit advantage. Report Delta AIC values relative to the minimum, while remembering that cutoffs are conventions rather than proof.
| AIC | AICc when n=30 |
|---|---|
| 248 | 249.6 |
The -2 log-likelihood term falls when fit improves, while 2k rises with complexity. A lower total represents a better estimated trade-off under the criterion's assumptions.
Count every freely estimated parameter required by the likelihood, including variance or dispersion parameters when the model definition calls for them. Software conventions differ, so document them.
NIST gives AICc=AIC+2k(k+1)/(n-k-1). The extra term can be substantial when n is not large relative to k and approaches zero as n grows.
With AIC=248, k=4 and n=30, the correction is 2×4×5/(30-4-1)=40/25=1.6, so AICc=249.6.
AIC values are directly comparable only when candidates model the same response observations with likelihoods on the same scale. Different transformations or omitted rows can make a naive ranking invalid.
AIC=248 is neither good nor bad by itself. Its meaning comes from comparison with pre-specified, scientifically plausible alternatives.
NIST notes a close relationship between AIC and Mallows Cp for ordinary linear regression under relevant assumptions. They arise from different formulations but can lead to similar model comparisons.
AIC uses penalty 2k; BIC uses k ln(n). For n greater than about 7.4, BIC's per-parameter penalty is larger. Their theoretical objectives differ, so neither is universally superior.
Delta AIC values can be transformed into relative weights over the candidate set. These weights describe support within that set, not the probability that a model is literally true.
AIC can only choose among models offered. If every candidate omits a key mechanism or violates assumptions, the lowest AIC model may still be inadequate.
Trying thousands of transformations and reporting only the winner understates selection uncertainty. Pre-specify sensible candidates, preserve validation data and report near-ties.
For each candidate, record n, k, log-likelihood, -2 log-likelihood, penalty, AIC, AICc and Delta AIC. Verify one row manually.
Fit candidates with the same data pipeline and likelihood, extract converged log-likelihoods, confirm parameter counts and preserve warnings. Compare predictive diagnostics after ranking.
AIC supports ecological models, engineering reliability, economics, time series, epidemiology and machine learning. The transferable mathematics is disciplined comparison under uncertainty.
Fit two probability models to one dataset. Calculate AIC and AICc, then explain why a ranking can change in a small sample.
AIC is not an accuracy percentage, p-value or causal test. The smallest value does not guarantee future dominance, and models fitted to different responses should not be compared casually.
Suppose Model A has k=4 and log-likelihood -120, giving AIC 248. Model B adds one parameter. Its complexity penalty rises by 2, so its log-likelihood must improve by more than 1 to obtain a smaller AIC: the -2 log-likelihood term must fall by more than 2. If Model B reaches log-likelihood -118.8, its AIC is 2×5-2(-118.8)=247.6, a modest 0.4 advantage. That near-tie should be reported, not converted into a dramatic claim. Compare AICc, diagnostics and prediction before deciding. Record both candidate formulas so the numerical difference remains auditable.
The Mallows Cp guide develops a regression subset criterion. The PRESS guide supplies an empirical prediction check.
Students should show log-likelihood, k and the exact formula. Parents can ask whether AICc was needed, whether all models used identical observations and whether the candidate set was scientifically sensible.
- Recalculate for k=5 and log-likelihood -118.
- Compute the AICc correction when n=50.
- Explain why AIC values from different response transformations may not be comparable.
### Is lower AIC always better? Only among genuinely comparable candidates.
### Should small samples use AICc? Often, and the convention should be stated.
### Does AIC find the true model? That is not its promise.
Continue through the Mathematics Learning Hub. Build a candidate table with AIC, AICc, Delta AIC and one out-of-sample metric. Explain every model difference in plain language.
Extended practice laboratory
Practice case 1: Record log likelihood
Setup: Start from the article’s AIC worked example for practice case 1. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Multiply by minus two
Setup: Start from the article’s AIC worked example for practice case 2. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Count parameters
Setup: Start from the article’s AIC worked example for practice case 3. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Calculate 2k penalty
Setup: Start from the article’s AIC worked example for practice case 4. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Calculate AIC
Setup: Start from the article’s AIC worked example for practice case 5. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Compare second model
Setup: Start from the article’s AIC worked example for practice case 6. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Calculate Delta AIC
Setup: Start from the article’s AIC worked example for practice case 7. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Increase likelihood
Setup: Start from the article’s AIC worked example for practice case 8. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Add one parameter
Setup: Start from the article’s AIC worked example for practice case 9. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Find break-even gain
Setup: Start from the article’s AIC worked example for practice case 10. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Record n
Setup: Start from the article’s AIC worked example for practice case 11. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Calculate AICc denominator
Setup: Start from the article’s AIC worked example for practice case 12. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Calculate AICc correction
Setup: Start from the article’s AIC worked example for practice case 13. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Compare AIC with AICc
Setup: Start from the article’s AIC worked example for practice case 14. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Use a small sample
Setup: Start from the article’s AIC worked example for practice case 15. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Check identical rows
Setup: Start from the article’s AIC worked example for practice case 16. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Check response scale
Setup: Start from the article’s AIC worked example for practice case 17. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Check convergence
Setup: Start from the article’s AIC worked example for practice case 18. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Compare Mallows Cp
Setup: Start from the article’s AIC worked example for practice case 19. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Compare BIC
Setup: Start from the article’s AIC worked example for practice case 20. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Compare PRESS
Setup: Start from the article’s AIC worked example for practice case 21. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Inspect residuals
Setup: Start from the article’s AIC worked example for practice case 22. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Report near tie
Setup: Start from the article’s AIC worked example for practice case 23. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Reject absolute-grade claim
Setup: Start from the article’s AIC worked example for practice case 24. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write an AIC conclusion
Setup: Start from the article’s AIC worked example for practice case 25. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.