Comparison indices of CFA I am comparing different models in Confirmatory Factor Analysis (CFA) to decide what my optimal number of factors and factor structure should be. The main indices I have been using are chi square , Comparative Fit Index (CFI) , RMSEA and Incremental Fit index (IFI. I am aware that there is cut off values available for these indices. However, I am unsure if there is any way of saying statistically if one of the models is better than the other.
 A: AIC values can be used to compare the models, smaller values suggesting a better model. AIC values are a function of the log-likelihood and therefore can only be compared if the models are fitted on the same data set (which I guess you have done). There's no agreed cut point to what is a substantial AIC difference, I have seen at least 2, at least 4 and at least 10 used.
Often people just compare the fit and select the better fitting values although that can be problematic inasmuch as sometimes fit indices can contradict eg CFI being better in model A, RMSEA being better in model B.
Also, you may have very small differences which may suggest a more complicated and perhaps less practically useful structure is better, so you can be guided by theoretical and practical rationales.
more on AIC and model selection https://books.google.ca/books?id=fT1Iu-h6E-oC&pg=PA51&dq=model+selection+and+multi+dimensional+inference&hl=en&sa=X&ei=dUXLUp-3N4fK2wWBlYHQCg#v=onepage&q=model%20selection%20and%20multi%20dimensional%20inference&f=false
Anderson, D., & Burnham, K. (2004). Model selection and multi-model inference. Second. NY: Springer-Verlag, 63(2020), 10.
general AIC info https://en.wikipedia.org/wiki/Akaike_information_criterion
