Testing the difference in AIC of two non-nested models The whole point of AIC or any other information criterion is that less is better. So if I have two models M1: y = a0 + XA + e and M2: y = b0 + ZB + u, and if the AIC of the first (A1) is less than that of the second (A2), then M1 has a better fit from the information theory standpoint. But is there any cutoff benchmark for the difference A1-A2? How much less is actually less? In other words, is there a test for (A1-A2) other than just eyeballing?
Edit: Peter/Dmitrij... Thanks for responding. Actually, this is a case where my substantive expertise is conflicting with my statistical expertise. Essentially, the problem is NOT choosing between two models, but in checking if two variables which I know to be largely equivalent  add equivalent amounts of information (Actually, one variable in the first model and a vector in the second. Think about the case of a bunch of variables as against an index of them.). As Dmitrij pointed out, the best bet seems to be the Cox Test. But is there a way of actually testing the difference between the information contents of the two models?
 A: I think this may be an attempt to get what you don't really want.
Model selection is not a science.  Except in rare circumstances, there is no one perfect model, or even one "true" model; there is rarely even one "best" model.  Discussions of AIC vs. AICc vs BIC vs. SBC vs. whatever leave me somewhat nonplussed.  I think the idea is to get some GOOD models.  You then choose among them based on a combination of substantive expertise and statistical ideas.  If you have no substantive expertise (rarely the case; much more rarely than most people suppose) then choose the lowest AIC (or AICc or whatever).  But you usually DO have some expertise - else why are you investigating these particular variables?
A: Is the question of curiosity, i.e. you are not satisfied by my answer  here ? If not...
The further investigation of this tricky question showed that there do exist a commonly used rule-of-thumb, that states two models are indistinguishable by $AIC$ criterion if the difference  $|AIC_1 - AIC_2| < 2$. The same you actually will read in wikipedia's article on $AIC$ (note the link is clickable!). Just for those who do not click the links:

$AIC$ estimates relative support for a model. To apply this in practice, we start with a set of candidate models, and then find the models' corresponding $AIC$ values. Next, identify the minimum $AIC$ value. The selection of a model can then be made as follows.
As a rough rule of thumb, models having their $AIC$ within $1–2$ of the minimum have substantial support and should receive consideration in making inferences. Models having their $AIC$ within about $4–7$ of the minimum have considerably less support, while models with their $AIC > 10$ above the minimum have either essentially no support and might be omitted from further consideration or at least fail to explain some substantial structural variation in the data.
A more general approach is as follows...
Denote the $AIC$ values of the candidate models by $AIC1$, $AIC2, AIC3, \ldots, AICR$. Let $AICmin$ denotes the minimum of those values. Then $e^{(AICmin−AICi)/2}$ can be interpreted as the relative probability that the $i$-th model minimizes the (expected estimated) information loss.
As an example, suppose that there were three models in the candidate set, with $AIC$ values $100$, $102$, and $110$. Then the second model is $e^{(100−102)/2} = 0.368$ times as probable as the first model to minimize the information loss, and the third model is $e^{(100−110)/2} = 0.007$ times as probable as the first model to minimize the information loss. In this case, we might omit the third model from further consideration and take a weighted average of the first two models, with weights $1$ and $0.368$, respectively. Statistical inference would then be based on the weighted multimodel.

Nice explanation and useful suggestions, in my opinion. Just don't be afraid of reading what is clickable!
In addition, note once more, $AIC$ is less preferable for large-scale data sets. In addition to $BIC$ you may find useful to apply bias-corrected version of $AIC$ criterion $AICc$ (you may use this R code or use the formula $AICc = AIC + \frac{2p(p+1)}{n-p-1}$, where $p$ is the number of estimated parameters). Rule-of-thumb will be the same though.
