No No No. the idea of correlating the error terms is to minimize the redundancy of items in measuring the same construct. The two items measuring two different constructs should not be redundant in the first place. So, correlating items from different constructs just does not make sense.

Hi Valentine,

I've correlated error terms in the past to achieve a good model fit, but some experts argue that there are never appropriate reasons to do that.

Check this website:

http://statwiki.kolobkreations.com/index.php?title=Confirmatory_Factor_Analysis

Regarding your question, I can say however, that you can't correlate error terms if they belong to different latent factors. They must be in the same factor. See Hermida (2015).

Hermida, R. 2015. "The Problem of Allowing Correlated Errors in Structural Equation Modeling: Concerns and Considerations," Computational Methods in Social Sciences (3:1), p. 5.

Hope this helps!

I recommend performing an SEM analysis

Hi Valentine,

I do agree with Dr. Pau Garcia Grau and thanks for his explanations.

The first problem with allowing measurement errors to correlate in structural equation models based on post hoc modifications is that it allows researchers to achieve good fit statistics in spite of omitting relevant variables from their models (Cortina, 2002). As explained by Landis et al. (2009), “to the degree that two residuals correlate, there is evidence that there exists a cause of both of the variables to which the residuals are attached but that is not specified in the model” (p. 17). When indicator variables are systematically influenced by the same extraneous variable in addition to the specified latent variables they represent, the influence of the extraneous variable may be estimated through measurement error correlations without a specification of what the influence is (Fornell, 1983). As a result of the estimation of such correlations, the fit of the model improves, but our understanding of the phenomenon in question does not. The second issue with allowing measurement errors to correlate in a post hoc fashion is that significant correlations are likely to be due to sampling error. Given that the number of off-diagonal elements of error covariance matrices can be very large, the probability of one or more such covariances being large simply because of sampling error is substantial. Several studies have shown that modified models often capitalize on the idiosyncratic features of sample data, and are likely to produce a departure from the true population model (Chou & Bentler, 1990; Green, Thompson, & Babyak, 1998; Green, Thompson, & Poirer, 1999; Lance, Conway, & Mulaik, 1988; MacCallum, 1986; MacCallum et al., 1992). To that end, Grant (1996) found that changing a hypothesized model to allow measurement errors to correlate based on specification search recommendations improved model fit in an initial sample, but failed to hold in cross-validation samples. A third problem with allowing measurement errors to correlate is that this practice may bias parameter estimates of both the measurement and structural model (Tomarken & Waller, 2003). For example, Gerbing and Anderson (1984) argue that even when correlated measurement errors do not significantly alter parameter estimates of a measurement or structural model, they can still mask the underlying structure of modeled relationships.

Reference:

Hermida, R. (2015). The problem of allowing correlated errors in structural equation modeling: concerns and considerations. Computational Methods in Social Sciences, 3(1), 5-17.

Thank you very much Dr. Pau Garcia Grau. Your anser is so helpful

Thank you Alaleh Dadvari. You are very kind

No No No. the idea of correlating the error terms is to minimize the redundancy of items in measuring the same construct. The two items measuring two different constructs should not be redundant in the first place. So, correlating items from different constructs just does not make sense.

Yes. I share the point of view of Zainudin. It is condition for that. See Hair et al. 2016.

Instead of correlating error terms betwen two latent factors maybe the best plausible explanation may be a cross-loading. For some cross loading is somewhat troublesome, because the assumption of local independence is violated. In the other hand, sometimes a low fit could be the result of no using the right estimator, this is a major flaw in AMOS, because it has too little options to conduct the estimation. Before making any data driven change to your model I strongly suggest to use another software, you can try LAVAAN package in R.

Thank you all for your wonderful comments

Hi Valentine,

The answer you the question " Can I correlate error terms of different latent factors as suggested by modification index in AMOS? " is Yes. However, maybe the question should have been " Should I correlate error terms of different latent factors as suggested by modification index in AMOS to improve model fit?" The answer to that question is less straightforward. There may the a situation where two similarly worded items from different latent variables are measuring something other than their respective latent variables, and a correlated error may be appropriate to indicate and represent this. So the judicious and transparent use of a correlated error may be appropriate. However, simply adding more and more correlated errors to a poorly fitting model in order to meet some pre-determined level of fit is not appropriate.

Mark

In most cases, we are tempted to correlate error terms of different latent factors, as suggested by the modification index in AMOS. However, experts have argued that there are never appropriate reasons to do that.

Besides the reference given by Pau Garcia Grau, you also find Laura Hildreth on her doctoral thesis on Residual analysis for structural equation modeling at https://lib.dr.iastate.edu/etd/13400/

Kind Regards

may be you should revise the whole model, by adding another latent variable that will contain the relevant variables... of course based on a solid theory