Comparings BIC of lme models with and without a correlation function I have created an lme model using the same predictors both with and without a specified correlation structure based upon distance between the points (lat/long). 
I know that I am able to compare the AIC/BIC between different correlation structures (corSpher vs. corExp vs. corGaus), but am unsure if I can directly compare the AIC/BIC values of a model with no correlation (lme(y~x1+x2, random=~x1|A, data = K)), and one with a correlation  (lme(y~x1+x2, random=~x1|A), corr = corGaus(form=~long+lat), data = K)) structure?
If I am not able to compare the AIC/BIC, are there any recommend methods in comparing the fit/best model?
Any advice would be greatly appreciated.
Thanks
 A: On general cases like these are what AIC and BIC are for, comparing different models that cannot be compared with likelihood ratio tests alone (or there are to many comparisons or so). However, in case of mixed models some caveats are to add.
Copying some relevant parts from the R-SIG-mixed faq:

How can I test whether a random effect is significant?
  
  
*
  
*perhaps you shouldn't (if the random effect is part of the experimental design, this procedure may be considered 'sacrificial pseudoreplication' (Hurlburt 1984); using stepwise approaches to eliminate non-significant terms in order to squeeze more significance out of the remaining terms is dangerous in any case)
  [...]
  
  
  Can I use AIC for mixed models? How do I count the number of degrees of freedom for a random effect?
  
  
*
  
*Yes, with caution.
  [...]
  
*in cases when testing a variance parameter, AIC may be subject to the same kinds of boundary effects as likelihood ratio test p-values (i.e., AICs may be conservative/overfit slightly when the nested parameter value is on the boundary of the feasible space). [...]
  
*AIC also inherits the primary problem of likelihood ratio tests in the GLMM context — that is, that LRTs are asymptotic tests. [...]
  

More can be found there and I vaguely remember reading something on this issue in Pinheiro & Bates.
