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I have fitted two models.

Model 1 has 264.514 less AIC than model 2, a drop from circa 7800 to circa 7500.

However, model 2 has adjusted R^2 equal to 40 percent, while model 1 has it at 36.8 percent.

What gives? I thought such a significant $\Delta AIC$ would most likely also increase the adjusted R squared?

If it helps: model 2 and model 1 are identical, except model 1 has an exponential correlation structure with a nugget.

DameVidao
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    Possible duplicate of [When AIC and Adjusted $R^2$ lead to different conclusions](https://stats.stackexchange.com/questions/140965/when-aic-and-adjusted-r2-lead-to-different-conclusions) and https://stats.stackexchange.com/questions/83133/comparing-aic-and-adjusted-r2 – Tim Jun 02 '17 at 13:05
  • @Tim- at least very similar :) – meh Jun 02 '17 at 13:28
  • @OP AIC and $R^2$ are measuring two very different things. – meh Jun 02 '17 at 13:28
  • Sorry-comment above got cut. AIC and $R^2$ are measuring two very different things. You can got a lot of information on AIC by searching 'what is AIC' on this sites search box ! Roughly, very roughly, AIC is a measure of how close your model is to the true underlying model. Among other things AIC penalizes model complexity. $R^2$measures how well the model fits the observed data. It tells one nothing about how well the model fits the data underlying the observations. – meh Jun 02 '17 at 13:37
  • So AIC is a measure of how well a model will predict new data and $R^2$ merely measures how well the model fits the given data. For example, if one adds noise variables to a model $R^2$ will improve (go up), but AIC will get worse (also go up). In particular adding noise to a model will add explanatory power to the model on the observed data, but predictions will go to hell. – meh Jun 02 '17 at 13:39
  • "It tells one nothing about how well the model fits the data underlying the observations." $R^2 = 1$ means what? – user158565 Jun 03 '17 at 00:17
  • I don't think these are exactly the same question and I think they might get different answers, so I am voting to leave both open. – Peter Flom Jun 03 '17 at 12:05
  • I vote to reopen this question. The question https://stats.stackexchange.com/questions/140965 has a good answer from @Carl explaining some differences but the other answers seem to catch on $R^2$ instead of $R^2_{adj}$. The question https://stats.stackexchange.com/questions/83133 seems to have changed direction and ended up being answered with an explanation of R functions to compute OLS-models and GLM-models. – Sextus Empiricus May 03 '18 at 07:38
  • I do not know how to retract my reopen vote but the following question https://stats.stackexchange.com/questions/197112 answered by Christoph Hank explains very well similarities and differences between $R^2_{adj}$ and AIC and you could write $R^2_{adj}$ as a function of AIC and $n$ and $k$. – Sextus Empiricus May 03 '18 at 07:49

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