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The R documentation for either does not shed much light. All that I can get from this link is that using either one should be fine. What I do not get is why they are not equal.

Fact: The stepwise regression function in R, step() uses extractAIC().

Interestingly, running a lm() model and a glm() 'null' model (only the intercept) on the 'mtcars' data set of R gives different results for AIC and extractAIC().

> null.glm = glm(mtcars$mpg~1)
> null.lm = lm(mtcars$mpg~1)

> AIC(null.glm)
[1] 208.7555
> AIC(null.lm)
[1] 208.7555
> extractAIC(null.glm)
[1]   1.0000 208.7555
> extractAIC(null.lm)
[1]   1.0000 115.9434

It is weird, given that both the models above are the same, and AIC() gives the same results for both.

Can anyone throw some light on the issue?

Sudip Sinha
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2 Answers2

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According, to the help for these two function (use ?AIC and ?extractAIC) this is expected.

Note that the AIC is just defined up to an additive constant, because this is also the case for the log-likelihood. This means you should check whether 

extractAIC(full.modell) - extractAIC(null.modell)

and

AIC(full.modell) - AIC(null.modell)

give the same result. As long as they do, both functions are equivalent for all practical purposes.

Erik
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    I'm probably missing something, but I still don't understand why `extractAIC(null.lm) != AIC(null.lm)` while `extractAIC(null.glm) == AIC(null.glm)` even though `null.lm` is the same model as `null.glm`. Could you expand your answer a little? – smillig Nov 16 '12 at 18:26
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    @smillig `extractAIC` uses different methods for `lm` fits and `glm` fits, i.e., `extractAIC.lm` and `extractAIC.glm`. You can use `getAnywhere` to study their code. `AIC` uses the same method for both. – Roland Nov 17 '12 at 15:18
  • I have several pairs of models (with multiple predictors) for which both functions give different results. Model 1: y = x1 + x2, Model 2: y = z + x1 + x2*z. `extractAIC()` gives lower (negative) value for Model 1, while AIC gives lower (positive) value for Model 2. – Maxim.K Sep 10 '13 at 11:25
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    @Maxim.K You give little information on the type of variables and models used. If you did and there are some differences to this question it might be worthwhile to post this as a new question. Hard to say, without knowing the details. – Erik Sep 10 '13 at 12:47
  • @Erik I doubt it will be worth much if I say that z is continuous and x2 is categorical (dummified). One would need the data to reproduce and I cannot publish them I'm afraid. – Maxim.K Sep 11 '13 at 11:56
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There are two differences for a usual linear regression model (lm) between AIC and extractAIC:

  • AIC accounts for the estimation of the unknown variance of the error (i.e., scale) while extractAIC does not, hence $k$ is one less with extractAIC.
  • AIC uses the formula $n\log\frac{RSS}{n}+n+n\log\left(2\pi\right)$ for the -2 log likelihood, while extractAIC drops the additive constant, and uses only $n\log\frac{RSS}{n}$. (At least by default; you can set a custom scale.)

Of course, none of these matters, if the values are only used in comparison.

In addition to these, and as already noted above, extractAIC behaves differently for lm and glm (in contrast to AIC!), namely, it uses the above formula only for lm, for glm it switches to the aic function of the fitted model, which is different.

Tamas Ferenci
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  • The socond point does not seem to be correct, because `AIC` does not compute the log likelihood at all, but calls the `logLik` method on the model. Example: `> m ll -2*as.numeric(ll) + 2*attr(ll, "df") [1] 170.2094 > AIC(ll) [1] 170.2094 > extractAIC(m) [1] 2.00000 77.39732 ` – cdalitz Jul 16 '21 at 12:11
  • @cdalitz "AIC does not compute the log likelihood at all, but calls the logLik method on the model" That is correct. But for `lm` (to which my two points refer), the formula used is just the one I have given: compare `32 * log(stats:::deviance.lm(m)/32)+32+32*log(2*pi)` with `-2*ll` in your example. – Tamas Ferenci Jul 19 '21 at 20:19