What is causing degrees of freedom and negative variance issues when estimating CFA with uncorrelated factors?

Question

Ultimately I want to estimate a second-order factor structure in AMOS.

As such I am trying to compare the fit of a first-order model in which the first-order factors are (a) constrained to be uncorrelated, and (b) free to correlate.

I am running into two problems when I try to run these models.

1. the chi-square and df's for the fully constrained and unconstrained models are coming out as the same (I am trying to constrain the model by naming the covariance parameters under the plugins tab which I think constrains them to be the same?) It is my understanding that they should be different and if the unconstrained model (factors free to correlate) has a smaller chi-square value then a second-order factor structure can be estimated.
2. If I constrain individual correlations (setting the covariance of the curved arrows between the latent variables in the measurement model to be 0) in several instances I get an error message that one of the variances is negative and the chi-square is not estimated.

Additional Information: I have 3 factors with 2 observed variables each. My sample size is 202. The second order factor is not included in these models. I am looking to compare a constrained model (constrain factors to be uncorrelated) to the same model with unconstrained correlations (allowing factors to correlate) to determine whether I can estimate a second-order factor (2 models), then I am also constraining correlations individually (there are 3 as there are 3 factors). Can I constrain the correlations to be uncorrelated by setting the covariances between each of the 3 factors to be 1?

Answer 1

Main Point

With only two observed variables per factor, the latent variable can not generally be estimated. Having correlations between factors presumably adds enough constraints to the model to allow estimation of the latent factors, but without those correlations, the latent factors are not estimatable.

What should you do?

1. Add more observed variables per factor. If you have three or more observed variables, your latent factors will generally be estimatable.
2. Constrain factor loadings to be equal. As a secondary option, you could constrain you factor loadings for the two items to be equal. Make sure they are on the same scale so that this makes sense.

I think the option of having at least three observed variables per factor is preferable.

Note also that setting the covariance between each factor to 1 will not create uncorrelated factors. If the factors have unit variance (i.e., like z-scores), then your factors will perfectly correlated (r=1). Thus, it would be equivalent to having a single factor. If the factors do not have unit variance, then it be constraining some other specific correlation structure that is related to the variance of each of the factors.

Comments below relate to the original post prior to the update.

Degrees of freedom

• If you constrain a parameter to be a constant, then you should get an additional degree of freedom in an SEM. Thus, when you specify $k$ latent factors to be uncorrelated, then that implies you are specifying $k(k-1)/2$ potential parameters to be zero and you should have that many additional degrees of freedom.
• If you are constraining $k$ latent factors to have a common covariance, then you should get $k(k-1)/2 - 1$ additional degrees of freedom, because you are still estimating one parameter.

So why might your degrees of freedom not be changing? Here are a few thoughts:

• Have you included the second order factor in your model testing the first-order model? If you have, remove the second-order factor.
• Have you specified constraints correctly in Amos?

Constraining factor correlations to be equal

• Note that in social sciences, it is often the correlation between factors that is conceptualised to be approximately the same from a theoretical perspective. Thus, you may want to ensure that your factors are all on the same scale before constraining covariances. For example, you could constrain the covariance of the factors to be one, rather than constraining one of the loadings, as is the default in Amos. By placing the factors on the same scale, constraining covariances to be equal is the same as constraining correlations to be equal.

Negative variances

• In terms of errors, it may be that the factors are so highly correlated that forcing the factors to be uncorrelated is causing estimation problems.
• You may want to just try removing the double headed arrows in Amos rather than specifying constraints to see whether that makes a difference to see whether that makes a difference
• You may have issues with number of items per factor.

There's also some discussion here