AIC

Study guides: Exam SRM, Exam ASTAM, Exam MAS-I, Exam MAS-II, Exam PCPA

The Akaike Information Criterion (AIC) is a model selection measure that balances goodness of fit against model complexity by penalizing the log-likelihood for the number of parameters.

\text{AIC} = -2\ell(\hat{\boldsymbol{\beta}}) + 2p

\text{where } \ell(\hat{\boldsymbol{\beta}}) \text{ is the maximized log-likelihood and } p \text{ is the number of parameters}

• Lower AIC indicates a better model (better fit relative to complexity)
• AIC does not require models to be nested, unlike likelihood ratio tests — but the models must be fitted by maximum likelihood to the same data: the same observations of the same, untransformed response. A model for \ln Y cannot be compared with one for Y, and comparing a Poisson GLM with a Gamma GLM is meaningless, since one response is discrete and the other continuous
• Derived from an information-theoretic perspective (Kullback–Leibler divergence)
• For small samples, use the corrected AIC: \text{AIC}_c = \text{AIC} + \dfrac{2p(p+1)}{n-p-1}
• Compare with BIC, which uses a stronger penalty \ln(n) instead of 2, favoring simpler models more aggressively for large n

Selecting Between Two GLMs Using AIC A Poisson GLM with 3 predictors and an intercept has log-likelihood \ell_1 = -120. A simpler Poisson GLM, fitted to the same data, with 1 predictor and an intercept has \ell_2 = -124. Which model has lower AIC?

Answer The intercept is a parameter, so the models have p_1 = 4 and p_2 = 2. \text{AIC}_1 = -2(-120) + 2(4) = 240 + 8 = 248 \text{AIC}_2 = -2(-124) + 2(2) = 248 + 4 = 252 The 3-predictor model has lower AIC (248 < 252), so it is preferred — the improvement in fit outweighs the added complexity.