Is RMSEA an absolute or relative fit statistics test? I've been studying Structural Equation Model and I have a doubt about RMSEA. 
In the book Principles and Practices of Structural Equation Modeling Kline treats RMSEA as an approximate fit index (p. 204). However, all other material I laid my hands on treats it as an absolute fit index (even in other questions here at stackexchange). 
What I am getting wrong?
 A: Slightly sarcastic analogy: Some people say that apples are food, but other people say that they are fruit. How can this be?  
Chi-square is a measure of exact fit - if chi-square is not significant, then then model is said to fit. However, chi-square almost never says that your model fits. The problem with chi-square is that it is a significance test, and significance tests don't answer the question we want to ask - we don't care if the fit is exact, we care if the model fit is close enough that it doesn't matter. (And if you have a model with a large enough N, chi-square will almost never fit.*)
So, other measures of fit were developed. The first of these was GFI, developed by Joreskog, but not really used any more (even by Joreskog). Then came CFI, RMSEA, SRMR, TLI/NNFI, etc. These are measures of approximate fit. They try to tell you if the model fit is close enough.
These measures can be divided into two kinds. Relative fit measures compare the fitted model to the null model - these are the incremental fit indices, CFI, TLI, etc. They look at the relative fit - on the grounds that if the null model is worse, it is harder to get good fit.
Absolute fit indices just look at the fit - e.g. SRMR, RMSEA. 
In the extreme case, where there were no relationships in your data - they are random, your RMSEA and SRMR will be very good, but your CFI and TLI would be had. 
A (very) small group of people believe that you should not use approximate fit indices and that they are not informative. Kline is one of those (Les Hayduk is another). This is unfortunate, because the rest of Kline's book is good.
*I once generated some data that I knew was a perfect fit to the model. I fixed all parameters to the third decimal place to their correct values. The sample size was large, and chi-square said the model didn't fit - because my correlation of [say] 0.154 was wrong, it should have been 0.154438. Nobody, ever, has cared about the 4th decimal point of a correlation. 
