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?
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.
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.
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.