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I have two questions about AICc and number of parameters.

I am using AICc to select the best of three models in a logistic regression - and each model has the same number of parameters. What are some compelling arguments for this method? I have G and C statistics, but these do not allow ranking. The formula for AICc in this case is as follows for for logistic regressions:

  • $K_1$ = sample size
  • $K_2$ = number of parameters (including the intercept and s)
  • $K_3$ = Log Likelihood of model
  • $K_4$ = AICc = Akaike’s Information Criterion logistic regression

Let $$K_4 = -2K_3 + 2K_2 +\frac{2K_2(K_2+1)}{(K_1-K_2-1)}$$

Additionally, for a different data set, in which the linear regression models being compared with AICc have different numbers of parameters, how might I interpret two models which have nearly equivalent AICc values: one with a single predictor variable (A), and the other model has an additional variable B (A+B)? B on its own has a significantly larger AICc value, so on its own it is one of the least likely models, but model A+B is nearly as likely as model A. Why isn't the addition of one variable, which is not a likely predictor on its own, causing a greater change in the AICc value when added to A? The formula for AICc is as follows and is used for regression models:

  • $K_1$ = sample size
  • $K_2$ = number of parameters (including the intercept and standard deviation)
  • $K_3$ = residual standard error
  • $K_4$ = AICc = corrected Akaike’s Information Criterion

Let $$K_4 = K_1\log \left( \frac{K_3^2(K_1-K_2+1)}{K1} \right) + 2K_2 + \frac{2K_2(K_2+1)}{K_1-K_2-1}$$

Scortchi - Reinstate Monica
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Brian
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    Take a close look at the residual standard error for models B, A and A+B. Is your question now a simpler one? (i.e. do you find that your surprise at A+B being about as good as A by AICc is now a question about why - if B is so poor on its own - the residual standard error for A+B improved so much over A alone?) – Glen_b Aug 20 '13 at 00:30
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    Note that when you're using AICc (or AIC) to compare models with the same no. parameters, you're just comparing likelihoods; which you might be happy to do even if you didn't have any compelling arguments for using AICc. – Scortchi - Reinstate Monica Aug 20 '13 at 08:41

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