SEM: Collinearity between two latent variables that are used to predict a third latent variable

In some structural equation models that I use in my bachelor thesis, there is a substantial correlation between two latent variables that are used to predict a third latent variable. Now I know there are several ways of quantifying multicollinearity when it comes to observable variables, but what about exogenous latent variables? Can I simply use the same indices that I use for observable variables in regressions analysis (such as the VIF)? And are there any guidelines as to when collinearity between latent variables becomes a problem?

Rules of thumb may say that multicollinearity is a problem only if two variables correlate above, say, .9 or even more. If two of your latent variables correlated that much, or even in the range of .7 / .8, then you have a problem before it comes to predicting the third variable: Your measurement model seems to be not very well defined. Maybe, for example, the two latents would be better modeled as only one? I would care much more about the measurement model then about multicollinearity.

• I imagine we'd care more about your concerns with the measurement model(s) if you specified them. In Hendrik's case, he may have theoretical reasons for considering the two collinear latent factors meaningfully distinct. Regardless, if the rule of thumb you're describing is the more general one for observed variables, this answer may miss his point in asking about latent variables specifically. If you have reason to think rules for observed variables can be applied equally well to latent variables, those reasons would be useful for evaluating the applicability of other rules of thumb too. Commented Nov 13, 2013 at 8:59
• Sorry, @NickStauner, I do not have any resources at hand about transferring rules of thumb from observed to latent variables. I admit that people in specific situations might be interested in differentiating between two highly correlated latents. But---at least as I interpret Hendrik's Bachelor thesis problem of predicting an outcome rather than building up a new, theoretical sound factor model---I would still think that the measurement model is more problematic than multicollinearity...!?! Commented Nov 18, 2013 at 10:36

Latent variable models are simply used to attempt to estimate the underlying constructs more reliably than by simply aggregating the items. Thus, in the structural part of the model (i.e. the regression) the same issues apply as in a standard regression.

Apart from in extreme situations (e.g., a standardized regression coefficient greater than 1), there are no definitive cutoffs. But cutoffs can be interpreted as for standard regressions. Some suggest a VIF of >5 or >10 as problematic, for example, but it these numbers would be extremely lenient if you were interested in the effect on one variable on the DV after controlling for the others (https://doi.org/10.1081/QEN-120001878).

One thing to note is that although calculating a VIF is easy in a standard regression and many packages/programs will do this automatically, it is not easy in a latent variable model. The calculation of VIF for a variable requires regressing it on all other predictors in the regression, which in a latent variable model means this has to be done in a latent variable model. As a result of this complexity, it is not surprising that this cannot be easily automated (and as far as I am aware has not been done).

Note also that multicollinearity is actually more likely to be an issue in latent variable models than when you just add items together since you are greatly reducing the error included in the estimate and thereby increasing correlations between related constructs.

What to do if you detect problematic multicollinearity will vary on a case by case basis. In most cases, it would probably be advisable to alter the measurement model, but there may be cases where such a course would not make sense. In such cases, you would probably be advised to drop one or more variables instead.