Dear Mikhail, In your diagram, there is 3 latent variables(as F1, F2 and F3). They has measure variables. F3 has no measure. At least latent variables was linked 1 measure. Because F3 is not real observation in your data. In general, latent variables was illustrated by common loading. So check the F3 which contains measure.

Path from F3 to F2 or  F3 to F1 has to be fixed "1"

Hi. Thank you for your answers. Are you sure? The variance of F3 is already constrained to 1.

Actually there is another potential problem. What do you think if it is overall statistically plausible to explain the trait F3 by only two sub-constructs - F1 and F2? Doesn't it need 3 or more variables?

Mikhail, in your model the higher-order model is not identified. As F3 (second-order factor) has only two indicators (first-order factors), you need to either constrain both factor loadings to be equal and fix F3’s variance to one or you need to fix both loadings to one and freely estimate F3’s variance.

The problem is comparable to a first-order model with one latent variable and only two indicators, which would also not be identified.

HTH! Regards, Karin

Thank you Karin! I think, this is exactly what has happened. I think, that I'd better to give up with this model and, instead, use just a covariance between F2 and F1 without second-order factor. There is indeed no logic in trying to determine a latent trait by just two variables. For my research, it is not crucial - using a higher-order factor or using just a covariance between first-order factors. I really appreciate your help, guys.

Karin Schermelleh-Engel

I ran the literally same model (with only 5 indicators per 2nd order factor). Both solutions that you mentioned worked to make the model identified.

However, I was wondering about possible theoretical reasons to privilege one over the other. May you have any ideas?

Also, technically, one may constrain the error terms of the two 1st order factors to be equal, and the model is again identified. However, here again, I was wondering about theoretical reasons that would justify it.

Any ideas on these points would be greatly appreciated.

A theoretical reason would be the aim to use the second-order factor as a predictor in a structural equation model. Then, a factor is needed because a covariance would not help.

You are right that fixing both specific variances of the first-order factors to one would also lead to an identified model. However, the interpretation would be difficult. For example, imagine that you have two scales, the first scale measures anxiety and the second scale depression. The higher-order factor would be negative affectivity, a stable tendency to experience negative emotions. If you fixed the error variances - which are actually the specific parts of the first-order factors independent of the common construct - to the same value, then you would expect that the amount of specific variance in the scales anxiety and depression would be equal. I think it would be hard to defend this assumption.

Kind regards,

Karin

Thanks much! Yes the "technical" solution of fixing error variances to equality be hard to interpret in most cases, including the ones that you mention.

I see the point you are making about using the 2nd order factor as a predictor in a SEM model.

However, would you think that the two solutions of (a) constraining both 2nd order factor loadings to be equal or (b) fixing both 2nd order loadings to one and freely estimating the 2nd order F's variance would also work in a CFA model that includes not only the 2nd order factor but also other constructs?

It may then be difficult to interpret the correlations/ covariances of other constructs with the 2nd order factor, as well as the loadings on the 2nd order factor. Right?

Of course, why not? If you standardize the parameters, both types of restrictions lead to the same standardized parameters. Therefore, these restrictions are interchangeable. Interpretation of correlations with other constructs should be no problem.

HTH, Karin