Freeing and interpretation of factor variance in multigroup CFA setting

[Fred]

Hello! I have a question regarding freeing the factor variance in a multigroup CFA setting and how to interpret this variance.

I have a one factor model with three indicators, for two groups, and I want to compare the latent mean difference between these two groups. I will do this with lavaan. I have established measurement invariance on loadings, intercepts and residuals (I know residual invariance isn't necessary for this estimate).

I restricted the factor mean for the first group to 0, and I restricted the factor variance for the first group to 1 to scale the factor. I would like to estimate all factor loadings so I'm using the fixed-factor method.

I understand that if I want to estimate the factor variance in group 2 I need to restrict the factor loadings to be equal across groups, and if I want to estimate the factor mean in group 2 I need to restrict the intercepts to be equal across groups. This I did. I will free the parameter of the factor mean for the second group, since that is what I want to estimate.

I know that if I freely estimate the factor variance in the second group, I will have 11 free parameters (3 loadings, 3 intercepts, 3 residual variances, 1 factor mean, 1 factor variance).

This is the same as when I fix one loading to 1 in both groups (11 free parameters as well: 2 loadings, 3 intercepts, 3 residual variances, 1 factor mean, 2 factor variances). As such, my intuition is to always freely estimate the factor variance in the second group.

My question is: could (and should) I restrict the variance of the factor to 1 in both groups? What does it mean if I restrict that second factor variance to 1 as well? Am I then still able to interpret the factor mean difference, or is the factor mean difference better estimated when I estimate the factor variance in the second group as well?

Thanks in advance!

[Terrence Jorgensen]

My question is: could (and should) I restrict the variance of the factor to 1 in both groups?

Not necessary, but you can also test that constraint if you are interested.

What does it mean if I restrict that second factor variance to 1 as well?

If you can constrain both factors to have variances == 1, then the estimated mean in group 2 would be Cohen's d.

Am I then still able to interpret the factor mean difference, or is the factor mean difference better estimated when I estimate the factor variance in the second group as well?

If constraining the factor variances to equality leads to significantly worse fit, then that can bias other model parameters, as well as the LRT when the less constrained model does not fit well.

Terrence D. Jorgensen

Postdoctoral Researcher, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen