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I need a one-sentence explanation of the use of the AIC in model-building. So far I have "Simply put, the AIC is a relative measure of the amount of observed variation accounted for by different models and allows correction for the complexity of the model."

Any advice much appreciated.

R

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AIC is a number that is helpful for comparing models as it includes measures of both how well the model fits the data and how complex the model is.

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What would be the best explanation depends on what exactly is meant by "non-technical types". I like the statements that have been offered so far, but I have one quibble: They tend to use the term "complex", and what precisely that is understood to mean could vary. Let me offer this variation:

The AIC is a measure of how well a model fits a dataset, while adjusting for the ability of that model to fit any dataset whether or not it's related.

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  • $\begingroup$ +1. I think that this is the key: when is more flexible too flexible? Perhaps "adjusting for" might be rephrased "penalizing"? So it's something like The AIC is a measure of how well a model fits a dataset, penalizing models that are so flexible that they would also fit unrelated datasets just as well. $\endgroup$ – Wayne Mar 5 '12 at 18:03
  • $\begingroup$ I agree that it depends on what is meant by "non technical types". Most of the people I work with are PhDs in the social, medical or behavioral sciences; they are educated, but don't know statistics. But I'm not sure I like the last few words in your definition, as it's unclear what "related" means here. Maybe in my definition, I could substitute "number of independent variables" for "complex". I think a lot of people know what an IV is... Interesting question! $\endgroup$ – Peter Flom - Reinstate Monica Mar 5 '12 at 20:29
  • $\begingroup$ @PeterFlom, you're right--we're sort of trading off one possible confusion for another. In person, when you know who you're talking to, I doubt either of us would have any problems. The 1 sentence requirement is pretty tight, too; give me a couple of figures & 5 minutes & I can get anybody on-board. And just to be clear, it wasn't a criticism, I liked your answer & upvoted it. $\endgroup$ – gung - Reinstate Monica Mar 5 '12 at 22:14
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Here's a definition that locates AIC in the menagerie of techniques used for model selection. AIC is just one of several reasonable ways to capture the trade-off between goodness of fit (which is improved by adding model complexity in the form of extra explanatory variables, or adding caveats like "but only on Thursday, when raining") and parsimony (simpler==better) in comparing non-nested models. Here's the fine print:

  1. I believe the OP's definition only applies to linear models. For things like probits, the AIC are usually defined in terms of log-likelihood.
  2. Some other criteria are adjusted $R^{2}$ (which has the least adjustment for extra explanatory variables), Kullback-Leibler IC, BIC/SC, and even more exotic ones, like Amemiya's prediction criterion, rarely seen in the wilds of applied work. These criteria differ on how steeply they penalize model complexity. Some have argued that the AIC tends to select models that are overparameterized, because the model-size penalty is pretty low. The BIC/SC also increases the penalty as the sample size increases, which seems like a handy-dandy feature.
  3. A nice way to sidestep participating in America's Top Information Criterion, is to admit that these criteria are arbitrary and considerable approximations are involved in deriving them, especially in the non-linear case. In practice, the choice of a model from a set of models should probably depend on the intended use of that model. If the purpose is to explain the main features of a complex problem, parsimony should be worth its weight in gold. If prediction is the name of the game, parsimony should be less dear. Some would even add that theory/domain knowledge should also play a bigger role. In any case, what you plan to do with the model should determine what criterion you might use.
  4. For nested models, the standard hypothesis test restricting the parameters to zero should suffice.
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How about:

AIC helps you find the best-fitting model that uses the fewest variables.

If that is too far in the non-technical direction, let me know in comments and I'll come up with another.

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    $\begingroup$ AIC implements Occam's razor? $\endgroup$ – Dilip Sarwate Mar 5 '12 at 12:17
  • $\begingroup$ @Dilip - so far past the line, you can't even see the line anymore -- lol, thanks. I think I'll use something between Peter's and my own. $\endgroup$ – rosser Mar 5 '12 at 12:23
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AIC is a measure of how well the data is explained by the model corrected for how complex the model is.

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The flip side of @gung's excellent answer:

The AIC is a number that measures how well a model fits a dataset, on a sliding scale that requires more elaborate models to be significantly more accurate in order to rate more highly.

EDIT:

The AIC is a number that measures how well a model fits a dataset, on a sliding scale that requires models that are significantly more elaborate or flexible to also be significantly more accurate.

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  • $\begingroup$ +1, this is good, but to be honest, I like your "penalizing" suggestion better. I can imagine people who wouldn't understand what you mean by "elaborate". $\endgroup$ – gung - Reinstate Monica Mar 5 '12 at 18:30
  • $\begingroup$ @gung: I agree that this doesn't quite do it, but I was trying to bridge the gap from "models" to "explanations", where people can perhaps imagine that a more elaborate explanation would need to be more accurate in order to justify the extra words. $\endgroup$ – Wayne Mar 5 '12 at 19:31
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Let k be the number of parameters of a model and MaxL be the value of the likelihood function at its maximum. Then the Akaike Information Criterion is defined as $AIC=2k-2\ln\left(MaxL\right)$. The aim is to find a model which minimizes the AIC.

Given this definition, the AIC is a criterion used to choose the model which yields the best compromise between sparsity in the number of parameters and the maximum likelihood for the estimation of those parameters.

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