I have calculated AIC and AICc to compare two general linear mixed models; The AICs are positive with model 1 having a lower AIC than model 2. However, the values for AICc are both negative (model 1 is still < model 2). Is it valid to use and compare negative AICc values?
$\begingroup$ when AIC became minimum? please answer me $\endgroup$– user34929Nov 17, 2013 at 20:31
$\begingroup$ what does it mean when the AIC of model 1 is smaller than model 2? Is model 1 closer to zero or more distant to zero? In other words, if AIC of model 1 is -390 and model 2 has -450, would I choose model 1 or model 2?? $\endgroup$– JensOct 27, 2016 at 14:30
All that matters is the difference between two AIC (or, better, AICc) values, representing the fit to two models. The actual value of the AIC (or AICc), and whether it is positive or negative, means nothing. If you simply changed the units the data are expressed in, the AIC (and AICc) would change dramatically. But the difference between the AIC of the two alternative models would not change at all.
Bottom line: Ignore the actual value of AIC (or AICc) and whether it is positive or negative. Ignore also the ratio of two AIC (or AICc) values. Pay attention only to the difference.
3$\begingroup$ I found all the answers to this question helpful, but I think that this one is the most practical. $\endgroup$ Jul 27, 2010 at 15:46
1$\begingroup$ I am confused by the remark about changing units, because by definition AIC is unitless (it's an adjusted maximum log likelihood). A change in the data units would not change the maximized likelihood at all and therefore would not change the AIC either. (Regardless, your recommendation to pay attention only to the difference is not in question.) $\endgroup$– whuber ♦Sep 10, 2010 at 2:41
5$\begingroup$ @whuber: if the data are continuously distributed (which they might be, depending on whether the original poster really means "general" or "generalized" LMM) then the probability density has an implicit "delta-x" term in it, which is indeed affected by changing units. See also <emdbolker.wikidot.com/faq> $\endgroup$ Jan 31, 2011 at 0:03
4$\begingroup$ @Ben Thank you. When I wrote this I was confused between the AIC and the difference of AICs, thinking the latter was the former. It is correct that the choice of units introduces a multiplicative constant into the likelihood. Thence the log likelihood has an additive constant which contributes (after doubling) to the AIC. The difference of AICs unchanged. $\endgroup$– whuber ♦Jan 31, 2011 at 14:57
AIC = -2Ln(L)+ 2k
where L is the maximised value of Likelihood function for that model and k is the number of parameters in the model.
In your example -2Ln(L)+ 2k <0 means that the log-likelihood at the maximum was > 0 which means that the likelihood at the maximum was > 1.
There is no problem with a positive log-likelihood. It is a common misconception that the log-likelihood must be negative. If the likelihood is derived from a probability density it can quite reasonably exceed 1 which means that log-likelihood is positive, hence the deviance and the AIC are negative. This is what occurred in your model.
If you believe that comparing AICs is a good way to choose a model then it would still be the case that the (algebraically) lower AIC is preferred not the one with the lowest absolute AIC value. To reiterate you want the most negative number in your example.
Generally, it is assumed that AIC (and so AICc) is defined up to adding a constant, so the fact if it is negative or positive is not meaningful at all. So the answer is yes, it is valid.
$\begingroup$ Even if the constant is included, the AIC (AICc) can be negative. $\endgroup$ Jul 22, 2010 at 23:15
1$\begingroup$ That's what I've written. $\endgroup$– user88Jul 23, 2010 at 8:17
Yes it's valid to compare negative AICc values, in the same way as you would negative AIC values. The correction factor in the AICc can become large with small sample size and relatively large number of parameters, and penalize heavier than the AIC. So positive AIC values can correspond to negative AICc values.
Yes. It's valid to compare AIC values regardless they are positive or negative. That's because AIC is defined be a linear function (-2) of log-likelihood. If the likelihood is large, your AIC will be likely negative but it says nothing about the model itself.
AICc is similar, the fact that the values are now adjusted change nothing.