May Allah Almighty bless us with following the right direction. In Fact the issue of model fit has been handled by James Gaskin in his videos in the same way, that is, by correlating the errorterms of the independent variables. see please on you tube SEM Series Part 5a_ Confirmatory Factor Analysis Good luck
This is possible, as far as I know there is a limit of two modifications and these covariances should be discussed. Yet I would be glad to learn about references for that.
Typically all independent variables are allowed to correlate. Constraining the correlations to zero places a strict structure on the exogenous variables and is inconsistent with most regression models (e.g., traditional multiple regression).
You should always be cautious using post hoc correlations based on modification indices. SEM is a confirmatory tool so the best process is to anticipate what error terms might logically be correlated and model that first. For example, all negatively worded items might have correlated error. Or even all items measured at the same time might have correlated errors. Be careful or you can trick yourself into believing parameter estimates that are not real.
I would look more closely at the e8 - r2 correlation. With that correlation, I don't believe that part of your model is identified. Also, I just saw your path diagram - sorry if my previous post is confusing. All exogenous variables are often allowed to correlate. It looks like you are asking about correlations between indicators (endogenous) variables. I agree with statements above - you should be very cautious about adding parameters based on Mod indices. I recommend looking at some of MacCallum's work on this topic (e.g., http://psycnet.apa.org/psycinfo/1992-25917-001, http://psycnet.apa.org/journals/bul/100/1/107/)
You may consider the following:
Recognizing covariances between the error of observed variables: a tactic to fit SEM models with latent variables that must be taken very cautiously
•¡A tactic to fit SEM models that must be taken cautiously!
•Useful to improve absolute and baseline fit indexes.
•!Use it as your last resource after trying everything else!
•It may overestimate path coefficients generating an untruthful model.
•Aggregating a covariance relationship (two-point arrow) between two measurement errors that are correlated will help to increase the approximation of the model to the covariances of the data and to the saturated model. Also, it may increase the degrees of freedom. Therefore, it will improve the absolute fit indexes and the baseline fit indexes.
•Start aggregating covariance between errors of observed variables related to the same latent variable. If you find a considerable value of association between the two errors (let’s say 0.40 or above, leave it). If not, erase it (if not, you will start inadequately manipulating the model).
•Go one by one. As more error covariances are introduced you may see how your fit indexes improve.
•However, the moment you see that the path coefficients increase, you must stop. This is not logical. This is the moment in which you are starting to inadequately manipulate generating an untruthful model.
•Be aware that this comes mostly from my appreciation of playing with structural models. In literature, the references to this technique and its drawbacks are very vague and scarce.
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