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I have recently begun using the Structural Equations Modeling (SEM) Method of confirmatory factor analysis for a research endeavor in educational science. My question is, suppose I have two latent variables each with many of its own manifest variables. If I observe a covariance between the two latent variables (I'm not sure if specifically what program I used to reach this point matters but if so I use the lavaan package in R), what does it mean?

I believe it means that it is a residual covariance indicating the presence of a common factor not shown by their predictors but I am unsure. If this is the case what would the level of residual covariance mean? i.e. is there a cut-off point where it could be considered statistically insignificant? Thank you!

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  • $\begingroup$ As the latent variables are standard normal the covariance is just correlation. I think it is just this – it seems reasonable for factors to correlate when they are identifying traits / behaviours - although you can set the correlation to zero if it suits your hypothesis. If you think their may be a second level of latent factors you can examine this. $\endgroup$
    – user20650
    Commented Mar 28, 2013 at 17:15
  • $\begingroup$ Thank you so much for your response! What you say makes sense. That the two latent variables would have some degree of correlation, in fact I had thought about the idea of a multilevel cfa, but is the covariance really just correlation? In my particular case, if that were so I would expect them to correlate much more but instead the value is around .03. Which I am assuming is rather low for a correlation but perhaps not for a residual covariance. (hence the part of my question about statistical significance). I know this is a little vague without a more concrete example. $\endgroup$
    – StatiStudent
    Commented Mar 28, 2013 at 17:35

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It's a partial correlation. It represents covariance (or correlation) between the factors that is not explained by the predictors. It means that there are common causes that you have not included, or that the two factors are causally related.

There's no cutoff for statistical non-significance, other than the cutoff that you usually use (i.e. p < 0.05). You should probably leave it in, because (a) you don't care, (b) you're getting a degree of freedom for free if you take it out only when it's non-sig, and (c) if you take it out you are hypothesizing that you have included in your model every common cause of those two factors - and that seems unlikely.

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  • $\begingroup$ Thank you for your response. I'm now wondering as far as interpreting the value of the covariance, in my model I use standardized estimates. When I estimate a value of .03 for the covariance would I be correct in interpreting that as a weak residual? At what point might I want to begin looking into the possibility of a multilevel cfa? $\endgroup$
    – StatiStudent
    Commented Mar 28, 2013 at 17:57
  • $\begingroup$ 0.03 is the standardized or unstandardized? If standardized, that's very small. By hierarchical, do you mean 2nd order? A 2nd order CFA with 2 (or 3, for that matter) first order factors is exactly equivalent to a regular CFA with a correlation. $\endgroup$
    – Jeremy Miles
    Commented Mar 28, 2013 at 20:39
  • $\begingroup$ The 0.03 was the standardized value and yes, by hierarchical I did mean second order. So, I'm taking from this that a regular cfa with the correlation kept in is the way to go as it is more concise then doing a hierarchical model in this particular case. Thanks! $\endgroup$
    – StatiStudent
    Commented Mar 29, 2013 at 12:54
  • $\begingroup$ Hi, if I find a positive residual covariance in the first lag of a cross-lagged SEM and a negative residual covariance in the second lag, how can I interpret that? Is there a way to see why it is happening descriptively? $\endgroup$
    – Blain Waan
    Commented Jun 28, 2018 at 20:41

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